Princeton - 102776 AI ENABLED SYNTHESIS OF ELECTROMAGNETIC STRUCTURES AND HIGH FREQUENCY CIRCUITS CROSS-REFERENCE TO RELATED APPLICATIONS This application claims priority to U.S. Provisional Patent Application No.63/660,874, filed June 17, 2024, entitled "Deep Learning Enabled Electromagnetic Structure Synthesis of Multi-Port Electromagnetic Structures and Circuits for Radio-Frequency Systems," which is incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under Grant Nos. N00014-22-1- 2302, N00014-23-1-2332, and N00014-23-1-2592 awarded by the Office of Naval Research (ONR), Grant Nos. FA9550-22-1-0455 and FA9550-23-1-0176 awarded by the Air Force Office of Scientific Research (AFOSR), and Grant No. W911NF-21-1-0314 awarded by the Army Research Office (ARO). The government has certain rights in the invention. FIELD OF INVENTION The present disclosure relates to electromagnetic structure design and synthesis for radio-frequency and millimeter-wave applications, and more particularly to methods, systems, and computer-readable storage media for using deep learning-enabled convolutional neural networks to synthesize arbitrary multi-port electromagnetic structures with designer-specified scattering parameters. BACKGROUND Radio-frequency (RF) and millimeter-wave integrated circuits serve as foundational components in modern wireless communication systems, radar applications, and high- resolution sensing technologies. The design of these circuits involves complex iterative processes that require co-design and optimization of active circuit elements alongside passive electromagnetic structures. These passive structures encompass a wide range of components including single-port elements such as antennas, two-port structures like matching networks and filters, three-port configurations such as diplexers and power dividers, and four-port networks including couplers for quadrature signal generation. Each type of structure
Princeton - 102776 traditionally requires distinct design approaches with different pre-selected templates and regular geometries. Conventional design methodologies for electromagnetic structures rely heavily on template-based approaches with finite parameter sets. Engineers typically begin with standard unit elements such as transmission lines, inductors, and capacitors to construct more complex structures. These template structures are then parameterized using geometric variables like widths and lengths, followed by time-consuming parameter sweeps and ad-hoc optimization methods. The design process often depends on domain expertise, intuition, and trial-and-error approaches, with no guarantee of achieving globally optimal solutions. Furthermore, each electromagnetic structure type requires separate optimization procedures, and changes in specifications necessitate restarting the entire design cycle. The template-based design paradigm presents several limitations that constrain achievable performance. By selecting a particular template at the outset, designers inherently narrow the available design space, potentially excluding configurations that could yield superior performance metrics in terms of efficiency, compactness, or spectral coverage. Many designed structures operate considerably below fundamental theoretical limits due to losses associated with energy storage elements. The iterative nature of conventional design processes makes them resource-intensive and time-consuming, sometimes requiring months of effort for complex circuit blocks and years for complete transceiver systems. The exponential growth in design space complexity poses additional challenges when moving beyond regular template-based geometries. For arbitrary-shaped structures discretized into pixel grids, the number of possible configurations scales exponentially with grid resolution. A modest 16×16 pixel grid yields approximately 1077 possible designs, making exhaustive exploration through traditional electromagnetic simulation-based optimization computationally intractable. This vast design space potentially contains solutions with performance characteristics that exceed those achievable through conventional approaches, but remains largely inaccessible using existing methodologies. Current machine learning approaches in electromagnetic design have focused primarily on surrogate modeling of specific template-based geometries or optimization of particular functional blocks with predefined topologies. These methods lack the generalization capability needed for universal electromagnetic structure synthesis and remain constrained to narrow design spaces defined by their training parameters. The absence of a unified approach for arbitrary multi-port electromagnetic structure synthesis limits the potential for discovering novel architectures and achieving performance breakthroughs.
Princeton - 102776 It has been appreciated that a method is needed that overcomes one or more of these problems. SUMMARY This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. In a first aspect, a method of creating an electromagnetic surrogate model is provided. The method comprises: generating training data by performing electromagnetic simulations on and/or measuring a plurality of pixelated electromagnetic structures, wherein each pixelated electromagnetic structure comprises a grid of pixels representing presence or absence of conductive material, and wherein the training data associates each pixelated electromagnetic structure with corresponding electromagnetic scattering and/or radiating parameters; and training a neural network (NN) using the training data to predict electromagnetic scattering parameters from pixelated electromagnetic structure representations. This method enables rapid and accurate prediction of electromagnetic behavior for complex structures without the need for time-consuming electromagnetic simulations, significantly accelerating the design process for radio-frequency and millimeter-wave components and circuits. The neural network may be a convolutional neural network (CNN). Utilizing a CNN allows the model to effectively capture spatial relationships and local interactions within the pixelated electromagnetic structures, leading to improved prediction accuracy for complex geometries. The CNN may include a first number of convolutional layers and a second number of fully connected layers, where output from a last convolutional layer is flattened and fed to the second number of fully connected layers. This architecture enables the network to extract relevant features from the pixelated input and process them to generate accurate predictions of electromagnetic scattering parameters. Each convolutional layer may include a plurality of learnable filters followed by batch normalization and leaky Rectified Linear Unit. These components help improve training stability, reduce overfitting, and allow for more effective learning of complex nonlinear relationships in the electromagnetic data.
Princeton - 102776 The neural network may include a residual neural network, U-net, a Vision Transformer, a You Only Look Once (YOLO) system, an EfficientNet, or a combination thereof. Incorporating these advanced neural network architectures can further enhance the model's ability to capture complex electromagnetic behaviors and improve prediction accuracy for a wide range of structures. The method may further comprise optimizing the NN using a cost function defining one or more desired electrical properties of a target structure. This optimization process allows the surrogate model to be tailored for specific design goals, improving its effectiveness in generating electromagnetic structures with desired characteristics. The method may further comprise verifying the functionality of output from an optimized NN using an electromagnetic solver. Verification ensures that the surrogate model's predictions align with accurate electromagnetic simulations, providing confidence in the model's reliability for design purposes. The method may further comprise performing circuit electromagnetic co-simulations to verify end-to-end performance of a circuit design, the circuit design based on the output. This step enables validation of the surrogate model's effectiveness in real-world circuit design scenarios, ensuring that the predicted electromagnetic behavior translates to desired circuit performance. The method may further comprise deploying the trained neural network as an electromagnetic surrogate model to predict electromagnetic behavior of electromagnetic structures without performing electromagnetic simulations. Deployment of the surrogate model enables rapid design iterations and exploration of large design spaces without the computational burden of full electromagnetic simulations for each candidate structure. The training may comprise transfer learning using a first set of electromagnetic simulations with relatively lower accuracy followed by retraining with a second set of electromagnetic simulations with relatively higher accuracy, where the first set and second set of electromagnetic simulations are subsets of the generated training data. This transfer learning approach allows for efficient use of computational resources by leveraging a large dataset of lower-accuracy simulations to establish initial model weights, followed by fine-tuning with a smaller set of high-accuracy simulations.
Princeton - 102776 The training may comprise transfer learning using a first set of electromagnetic simulations having first characteristics that result in a relatively lower training time, followed by retraining with a second set of electromagnetic simulations with second characteristics that result in a relatively higher training time. This approach enables efficient model training by initially using simpler, faster-to- compute electromagnetic structures to establish baseline model performance, followed by refinement using more complex, time-intensive simulations. The first characteristics may include structures having an air dielectric or a different dielectric(s) or metal structure(s) or stacks of these from a desired dielectric(s) or metal structure(s) or stacks of these to be used in a final structure. Using simplified structures for initial training allows for rapid model development, while subsequent fine-tuning with more realistic structures ensures accuracy for the intended application. The pixelated electromagnetic structures may comprise multi-port structures. Incorporating multi-port structures in the training data enables the surrogate model to predict behavior for a wide range of complex electromagnetic components, including power dividers, couplers, and other multi-port devices. The multi-port structure may include 2-6 ports. This range of port numbers covers many common electromagnetic structures used in radio- frequency and millimeter-wave circuit design. The multi-port structure may include more than 6 ports. Extending the model to handle structures with more than 6 ports enables prediction for highly complex electromagnetic components and systems. Ports of the multi-port structure may be positioned at edges of the grid. This configuration aligns with common design practices for planar electromagnetic structures and simplifies the integration of these structures into larger circuit designs. The method may further comprise augmenting the training data using geometric transformations of the pixelated electromagnetic structures. Data augmentation through geometric transformations increases the effective size of the training dataset, improving the model's ability to generalize and predict behavior for a wider range of structure configurations. The method may further comprise using the electromagnetic surrogate model within an optimization algorithm to synthesize electromagnetic structures having target scattering parameters. This application of the surrogate model enables rapid inverse design of electromagnetic structures, allowing designers to efficiently explore the design space and generate structures that meet specific performance criteria.
Princeton - 102776 The optimization algorithm may comprise a gradient-based method, a quasi-newton method, a second-order method, a trust-region method, a heuristic method, Bayesian optimization, or a combination thereof. These optimization methods provide a range of approaches for efficiently navigating the design space and converging on optimal structures that meet the desired electromagnetic performance criteria. The gradient-based method may include gradient-descent or conjugate gradient. These methods leverage gradient information to efficiently navigate the design space and converge on optimal solutions. The second-order method may include Newton's method. Newton's method can provide faster convergence for certain types of optimization problems, potentially leading to more efficient structure synthesis. The heuristic method may include a genetic algorithm, particle swarm optimization, Nelder- mean, or simulated annealing. These heuristic methods are well-suited for exploring complex, non-convex design spaces and can help avoid local optima in the structure synthesis process. The electromagnetic scattering parameters may comprise complex S-parameters across a frequency range. Predicting complex S-parameters across a frequency range enables comprehensive characterization of the electromagnetic behavior of structures, supporting broadband design and analysis. The pixelated electromagnetic structures, either by itself or in combination with traditional components or electromagnetic structures, may realize functions of one or more electromagnetic structures. This capability allows the surrogate model to be used for designing a wide range of functional electromagnetic components, either as standalone structures or integrated with conventional designs. The one or more electromagnetic structures may include an antenna, a filter, a power divider, a power combiner, a diplexer, a duplexer, a quadrature hybrid, a balun, a coupler, an attenuator, a phase shifter, or a combination thereof. This diverse set of electromagnetic structures covers many common components used in radio-frequency and millimeter-wave circuit design, demonstrating the broad applicability of the surrogate model. The electromagnetic surrogate model may enable co-design of the electromagnetic structures with active circuit elements having complex frequency-dependent impedances. This co-design capability allows for integrated optimization of passive electromagnetic structures and active circuit elements, leading to improved overall system performance and more efficient design processes. The grid of pixels may be an A x B grid, where A and B are identical.
Princeton - 102776 Using a square grid simplifies the representation of electromagnetic structures and can be well-suited for many planar designs. The grid of pixels may be an A x B grid, where A and B are different. A rectangular grid allows for more flexibility in representing structures with non-square aspect ratios, accommodating a wider range of design geometries. In a second aspect, a non-transitory computer-readable storage medium containing instructions that, when executed by one or more processing units, causes the one or more processing units to perform the method of creating an electromagnetic surrogate model is provided. This aspect enables the implementation of the electromagnetic surrogate model creation method in software, allowing for widespread deployment and use of the technology on various computing platforms. In a third aspect, a system is provided. The system comprises one or more processing units operably coupled to a non-transitory computer-readable storage medium containing instructions that, when executed by the one or more processing units, causes the one or more processing units to perform the method of creating an electromagnetic surrogate model. This system aspect provides a hardware implementation of the electromagnetic surrogate model creation method, enabling efficient execution of the model training and prediction processes. In a fourth aspect, a method of manufacturing a circuit is provided. The method comprises: receiving an electromagnetic structure, the electromagnetic structure having been designed utilizing inverse synthesis with a forward model created using the method of creating an electromagnetic surrogate model; and realizing the electromagnetic structure on a substrate. This manufacturing method leverages the electromagnetic surrogate model to enable rapid design and fabrication of optimized electromagnetic structures, streamlining the circuit manufacturing process. The substrate may be a printed circuit board (PCB), in a package, in an integrated chip, or in a combination thereof. This range of substrate options allows for the implementation of the designed electromagnetic structures in various circuit form factors and technologies. The printed circuit board may include flame retardant 4 (FR-4) epoxy laminate, a Rogers PCB, a Megtron PCB, a metal-core PCB, a low-temperature co-fired ceramic (LTCC) PCB, or polytetrafluoroethylene (PTFE).
Princeton - 102776 These PCB materials cover a wide range of performance characteristics and cost points, enabling the realization of electromagnetic structures for diverse applications and frequency ranges. The metal-core PCB may comprise aluminum. Aluminum-core PCBs provide enhanced thermal management capabilities, which can be beneficial for high-power or thermally sensitive designs. The integrated chip may include silicon, GaAs, GaN, InP, SiC, or diamond. This range of semiconductor materials enables the implementation of electromagnetic structures in various integrated circuit technologies, supporting diverse performance requirements and frequency ranges. In a fifth aspect, a method for designing circuits is provided. The method comprises: selecting, through a user interface, one or more forward models created using the method of creating an electromagnetic surrogate model; and designing or co-designing one or more circuits that include an active device using one or more inverse designed electromagnetic structures to achieve target performance. This design method enables efficient creation of complex circuits by leveraging the electromagnetic surrogate model for rapid optimization of passive structures in conjunction with active devices. The active device may include a transistor, a diode, or a combination thereof. These active devices are fundamental building blocks in many radio-frequency and millimeter-wave circuits, allowing for the design of a wide range of functional components and systems. The one or more circuits may define an amplifier with a desired response across frequency range. This application enables the design of broadband amplifiers with optimized passive matching networks for improved performance across wide frequency ranges. The one or more circuits may define a mixer with a desired response across frequency range. Optimizing mixer designs using the surrogate model can lead to improved conversion efficiency and bandwidth in frequency translation applications. The one or more circuits may define a power amplifier with a desired response across frequency range. This application allows for the design of efficient, broadband power amplifiers with optimized matching networks and harmonic terminations.
Princeton - 102776 The one or more circuits may define a combination of low noise amplifier and power amplifier connected with a switch or a duplexer. This configuration enables the design of integrated transceiver front-ends with optimized performance for both receive and transmit paths. The one or more circuits may define a phase shifter. Designing phase shifters using the surrogate model can lead to improved bandwidth and phase accuracy for phased array and beam-steering applications. The one or more circuits may define a variable gain amplifier. Optimizing variable gain amplifier designs can result in improved gain control range and linearity across operating conditions. The one or more circuits may define an oscillator with a phase locked loop. This application enables the design of frequency sources with optimized tank circuits and loop filters for improved phase noise and frequency stability. The one or more circuits may define an oscillator free of a phase locked loop. Designing free-running oscillators using the surrogate model can lead to improved frequency stability and phase noise performance without the complexity of a phase-locked loop. The one or more circuits may realize a radio frequency transmitter or receiver or both. This application demonstrates the broad utility of the surrogate model in designing complete radio frequency front-end systems with optimized performance. The circuit may be realized in an integrated chip or with multiple chips in a packaged system. This flexibility in implementation allows for optimized designs across various levels of integration and packaging technologies. The one or more circuits may be realized on a printed circuit board or in combination with one or more chips. This hybrid approach enables the design of systems that leverage the strengths of both printed circuit board and integrated circuit technologies. The one or more circuits may define a 1-port multi-band antenna, a 2-port band pass filter, a 3-port divider with 90° phase difference, a 4-port frequency diplexer, a 4-port hybrid coupler, an unequal power divider, a filtering divider with even split, a mm-Wave antenna, or a mm-Wave amplifier.
Princeton - 102776 This diverse set of circuit examples demonstrates the broad applicability of the surrogate model-based design method across various radio-frequency and millimeter-wave components and systems. BRIEF DESCRIPTION OF FIGURES Embodiments of the invention will be described, by way of example, with reference to the following drawings, in which: Figure 1 illustrates a flowchart for a method of electromagnetic structure synthesis using deep learning, according to aspects of the present disclosure. Figure 2A illustrates a top view of an electromagnetic structure, according to aspects of the present disclosure. Figure 2B illustrates a binary grid pattern of the structure of FIG.2A. Figure 2C depicts a graph showing scattering parameters plotted against frequency for the pixelated electromagnetic structure of FIG. 2A, according to aspects of the present disclosure. Figure 3 illustrates a block diagram of a system for implementing deep learning enabled electromagnetic structure synthesis, according to aspects of the present disclosure. Figure 4 illustrates an inverse generative network training and synthesis overview, according to aspects of the present disclosure. Figure 5 illustrates details of a CNN architecture where the input is the binary matrix representing the multi-port EM structure, and the output is the real and imaginary parts of the S-matrix over the frequency of interest (e.g., 30 - 100 GHz). Figure 6 illustrates efficient dataset generation by transforming one 4-port simulation to generate six unique 2-port structures through rotations and flips, thereby reducing the generation of training dataset by six times. Figure 7 illustrates transfer learning for efficient training dataset generation and generalizable learning model, including combining a large number of rapid in-air simulations with much smaller number of in-dielectric simulations to allow faster learning and easy portability across different chip dielectrics, as well as transfer learning across EM structure size. Figure 8 illustrates an optimal number of convolutional layers (12) and FC layers (five) that minimize the difference between test and training set rms error preventing overfitting.
Princeton - 102776 Figure 9 illustrates cost minimization performance of different pixel counts is compared with simulation requirements; while higher pixels allow better results as observed, it comes with additional computational burden. Figure 10 illustrates an output matching network (OMN) topology where multiple practical considerations are handled within the optimization. Figure 11 illustrates creating an offspring from parents by creating a random cut line and merging two parts from either of the parents. Furthermore, port and ac ground locations are taken randomly from either parents. In this figure, first, two EM structures illustrate the features transferred from parents. Figure 12 illustrates wall clock time allocated to different tasks of the optimization, where S-parameter prediction takes only 0.3 s, and the bulk of the time was spent on cost calculation (1.8 s) due to the optimization of the dc block capacitor. Figures 13-15 illustrate a schematic and small-signal measurements, and specifically a PA schematic and simulated performance of inter-stage and inverse synthesized OMNs (Fig. 13), a Chip micrograph (Fig.14) and measured S-parameters (Fig.15). Figure 16 illustrates a concurrent triple-band measurement setup. Figure 17 is a table showing a comparison with high-performance wideband silicon PAs. Figure 18 illustrates prediction of the scattering parameter of a single-port antenna structure (against a 3-D electromagnetic nite-element-method solver) embedded in a high- frequency dielectric between 20-40 GHz. Figure 19 illustrates a prediction of scattering parameters of two-port filter structures between 30-100 GHz realized with arbitrary transmission-line based and pixelated structures (16 × 16 pixel); The comparison against full 3-D EM simulation demonstrates the accuracy of the deep learning model. Figure 20 illustrates inverse synthesis of arbitrary multi-port electromagnetic structures with desired scattering and radiating properties, enabled through a deep learning-based forward electromagnetic emulator. The latter takes the image of the structure and predicts accurately its multi-port scattering and radiating properties across frequencies in the space of arbitrary- shaped planar structures. Figure 21 illustrates prediction of antenna radiation patterns between 20-40 GHz demonstrates ability to predict both scattering and radiating properties, allowing generalization as depicted Figure 18.
Princeton - 102776 Figures 22A-22E illustrate inverse synthesis of various multi-port electromagnetic structures, including: inverse synthesis of multi-band packaged mm-Wave antenna operating at 24 and 28 GHz (22A); Inverse synthesis of a 2-port on-chip bandpass filter with a pass band target of 5060 GHz (22B) (The 16 x 16 pixel, 300 x 300 µm filter is compact with each side being ^^/ 10 at 50 GHz); Inverse synthesis of a 3-port equal power divider with ^ 90∘ phase shift between 2 output ports at 60 GHz (22C); Inverse synthesis of a a 4-port 24 ൈ 24 pixel quadrature hybrid between 70 and 80 GHz, showing close to ideal amplitude balance and phase relationship (22D); and Inverse synthesis of on-chip frequency diplexer ( 20 ൈ 20 pixel 500 ൈ 500^^ m ) with channel bandwidths of 24 െ 40GHz and 60 െ 80GHz (22E). Figure 23 illustrates a comparison of inverse-designed 80 GHz band-pass lters (200 × 200 μm, 16 × 16 pixels) synthesized with GA and BPSO. Figure 24 illustrates an inverse-designed broadband mm-Wave amplifier architecture that relies on asymmetrical power division at the input, and phase compensating combiner to create broadband frequency response; the three-port input and output combiners are synthesized through inverse design, and they are co-designed with multi-stage integrated active circuit amplifiers (that have complex terminating impedances). Figure 25 illustrates a measured small signal response of the chip shows 3 dB bandwidth of 23.6-37.3 GHz with peak gain of ~17.5 dB, owing to the asymmetrical networks synthesized through the inverse design approach. Figures 26 illustrates a snapshot from the 50th iteration from the optimization of a 2- port EM structure, where parent selection was performed separately for each of the next generation members; due to the selection scheme, low-cost samples have a good chance of being selected as parents frequently. Moreover, a diverse set of parents were selected. This contributes to the exploratory aspect of the optimization algorithm. Further, it illustrates average cost value of parents versus cost values of offspring, where the best cost value for successive generation is improved in comparison to the current generation. Interestingly offspring cost minimization does not exactly correlate with the parents’. This is in fact due to non-convexity of the design space. Figure 27 illustrates a detailed schematic of a circuit. Common reference numerals are used throughout the figures to indicate similar features.
Princeton - 102776 DETAILED DESCRIPTION The present disclosure relates to deep learning enabled electromagnetic structure synthesis for radio-frequency and millimeter-wave applications. Traditional electromagnetic structure design relies on template-based approaches using predefined geometric elements such as transmission lines, inductors, and capacitors. These conventional methods limit the design space to parameterized structures and require time-consuming iterative optimization processes that may not achieve global optimality. The disclosed approach utilizes convolutional neural networks (CNNs) to create electromagnetic surrogate models that can predict electromagnetic scattering parameters of arbitrary pixelated structures without performing electromagnetic simulations. A pixelated electromagnetic structure comprises a grid of pixels where each pixel represents either the presence or absence of conductive material. This discretization approach enables exploration of a vast design space that encompasses non-intuitive geometric configurations beyond conventional template structures. As used herein, the term “pixel” refers to an element of an image, regardless of the dimensionality. Each image will be made of multiple pixels, and each pixel may be of any shape, and any size. In some implementations, the pixels may a geometric shape, such as a square, rectangular, round, or oval. In other examples, each pixel group may include a certain number of pixels from a center of the image. In some examples, the image may consist of pixels with identical shapes and sizes. In some examples, the image may include different pixel groups, each pixel group having different shaped pixels. In some examples, different pixels may be of different size. As will be understood, the grid of pixels may be of any dimensions. That is, the grid may have dimensions of A x B, where A and B are the same, or different. The convolutional neural network architecture includes multiple convolutional layers followed by fully connected layers. The convolutional layers extract spatial features from the pixelated structure representations, while the fully connected layers map these features to electromagnetic scattering parameters across frequency ranges. The network learns complex relationships between geometric patterns and electromagnetic behavior through training on datasets generated from electromagnetic simulations. Once trained, the electromagnetic surrogate model can be deployed to predict electromagnetic behavior of electromagnetic structures without performing electromagnetic simulations. This capability enables rapid synthesis of multi-port electromagnetic structures through optimization algorithms. Such optimization algorithm may include, e.g., Gradient-
Princeton - 102776 Based Methods such as gradient-descent, conjugate gradient, Quasi-Newton Methods or others, Second-Order Method such as Newton's Method, Trust-Region Methods or heuristic methods such as genetic algorithm, particle swarm optimization, Nelder–Mead Simulated Annealing or Bayesian optimization or a combination of one or more of these. The synthesis process can explore thousands of candidate structures within minutes, compared to conventional approaches that may require days or weeks of electromagnetic simulations. The electromagnetic surrogate model enables co-design of the electromagnetic structures with active circuit elements having complex frequency-dependent impedances. This co-design capability allows for simultaneous optimization of passive electromagnetic structures and active circuit components, facilitating the development of integrated radio- frequency and millimeter-wave systems with enhanced performance characteristics. The pixelated electromagnetic structures, either by itself or in combination with traditional components or electromagnetic structures, can realize functions of electromagnetic structures such as antennas, filters, power dividers, power combiners, diplexers, duplexers, quadrature hybrids, baluns, couplers, attenuators, phase shifters and others, across different frequency ranges and applications. FIG. 1 illustrates a method 100 of electromagnetic structure synthesis using deep learning. The method 100 includes a training phase 130 that establishes the foundation for electromagnetic surrogate modeling through data generation and neural network training. The training phase 130 begins with a generating training data step 102. The generating training data step 102 performs electromagnetic simulations on a plurality of pixelated electromagnetic structures, where each pixelated electromagnetic structure comprises a grid of pixels representing presence or absence of conductive material. The training data associates each pixelated electromagnetic structure with corresponding electromagnetic scattering parameters. The generating training data step 102 further comprises augmenting the training data using geometric transformations of the pixelated electromagnetic structures. The geometric transformations include rotations, flips, and translations to create multiple 2-port structures from a single 4-port electromagnetic simulation, thereby expanding the dataset diversity without requiring additional electromagnetic simulations. The training phase 130 continues with a training EM surrogate step 104. The training EM surrogate step 104 trains a neural network using the training data to predict electromagnetic scattering parameters from pixelated electromagnetic structure representations. In some implementations, the neural network may be a convolutional neural network (CNN). The convolutional neural network includes a first number of convolutional layers and
Princeton - 102776 a second number of fully connected layers, where the first number is larger than the second number. Output from a last convolutional layer may be flattened and fed to the fully connected layers. Each convolutional layer may include a plurality of learnable filters followed by batch normalization and leaky Rectified Linear Unit activation functions. In some implementations, the neural network may utilize a residual neural network, or U-net, or Vision Transformers, or YOLO (You Only Look Once) system, or EfficientNet, or a combination of one of or more of these, with or without a CNN. The training EM surrogate step 104 preferably includes a two-step process, a first step that incorporates assumptions that reduce complexity (resulting in, e.g., lower training time, lower costs, lower accuracy, etc.), followed by a second step that does not incorporate those assumptions (or incorporates different assumptions) that result in more complexity (resulting in, e..g, higher training times, higher cost, higher accuracy, etc.). For example, the training step may include transfer learning using a first set of electromagnetic simulations with relatively lower accuracy followed by retraining with a second set of electromagnetic simulations with relatively higher accuracy, where both sets are subsets of the generated training data. As a separate example, the training step may include transfer learning using a first set of electromagnetic simulations having first characteristics that result in a relatively lower training time, followed by retraining with a second set of electromagnetic simulations with second characteristics that result in a relatively higher training time. The transfer learning approach uses a two-step process where initial training uses a simplified stackup setup with air dielectric and 2 metal layers, followed by retraining with chip dielectric stack configurations. The training EM surrogate step 104 can use informed dataset generation to skew statistical properties towards well-matched cases using semi-optimization during dataset generation with search rules similar to genetic algorithm approaches. The training EM surrogate step 104 can include passivity enforcement using minimum perturbations to correct passivity violations in predicted S-parameters. The training phase 130 concludes with an optimizing EM surrogate step 106. The optimizing EM surrogate step 106 optimizes the convolutional neural network using a cost function defining one or more desired electrical properties of a target structure, where optimizing is free of an electromagnetic simulation step. The optimizing EM surrogate step 106 refines the electromagnetic surrogate model parameters to enhance prediction accuracy and generalization capabilities across different electromagnetic structure configurations and frequency ranges.
Princeton - 102776 FIGS. 2A and 2B illustrate a pixelated electromagnetic structure representation that demonstrates how complex electromagnetic geometries are discretized into binary grid patterns. The pixelated electromagnetic structure comprises a grid of pixels where each pixel represents either the presence or absence of conductive material. Black squares in the grid indicate regions with conductive material on a top metal layer, while white squares represent regions without conductive material or ground layer connections. The pixelated electromagnetic structure may include designated port locations positioned at edges of the grid, shown as diagonal striped areas in the figure. These port locations enable the structure to function as a multi-port electromagnetic device. The structure also includes short circuit connections, indicated by horizontal striped areas, which provide AC shorting locations that can be implemented with bypass capacitor arrays for VCC feed applications. In some implementations, the structure may be a single port structure. In some implementations, the structure may be a multi-port structure with at least 2 ports. There is theoretically no upper limit to the number of ports that can be incorporated, aside from limitations based on the arbitrary physical dimensions selected for the size of the structure. In some implementations, the multi-port structure may include 2-20, 2-10, or 2-6 ports. In some implementations, the multi-port structure may include more than 6, 10, or 20 ports. In some implementations, the multi-port structure may include more than 6 ports. In some implementations, the multi-port structure may include 6-1,000,000 ports. The binary representation allows for discretization of electromagnetic structures with specific pixel sizes that can be configured to ^^/100 at center frequency, where diagonal connections between pixels can have widths of 6.6 μm due to pixel overlaps. This discretization approach enables representation of complex electromagnetic geometries that would be difficult to parameterize using conventional template-based design methods. The pixelated electromagnetic structure representation supports various types of electromagnetic devices including antennas, filters, power dividers, power combiners, diplexers, and quadrature hybrids. The grid-based approach allows for synthesis of structures with arbitrary geometric configurations that can exhibit electromagnetic properties not achievable through conventional design approaches. For 4-port networks, the pixelated electromagnetic structures can incorporate quadrature symmetry to reduce required training data while maintaining design flexibility. The quadrature symmetry constraint allows the same underlying structure pattern to be used for
Princeton - 102776 multiple port configurations, thereby expanding the effective dataset size without requiring additional electromagnetic simulations. The pixelated representation enables exploration of a vast design space where each pixel can independently be set to conductive or non-conductive states. For a grid with N×N మ pixels, this approach yields 2ே possible structural configurations, providing access to electromagnetic designs that extend beyond human intuition and conventional design knowledge. FIG. 2C illustrates electromagnetic scattering parameters that serve as the output of a trained convolutional neural network. The electromagnetic scattering parameters comprise complex S-parameters across a frequency range, demonstrating the electromagnetic behavior prediction capabilities of the neural network surrogate model. The graph in FIG.2C shows three different scattering parameter curves plotted against frequency from 110 to 150 GHz. The s11 parameter represents the input reflection coefficient, remaining relatively constant near 0 dB across the frequency range. The s22 parameter shows the output reflection coefficient, exhibiting a characteristic dip reaching approximately -14 dB around 130 GHz before returning toward 0 dB. The s12 parameter demonstrates the forward transmission coefficient, showing a gradual decline from approximately -2 dB at 110 GHz to about -10 dB at 150 GHz. The neural network can be trained to predict specific frequency ranges including 24- 100 GHz for millimeter-wave applications and 20-40 GHz for antenna applications. The frequency range selection depends on the target application and the electromagnetic structures being synthesized. For millimeter-wave passive structures, the neural network predicts S- parameters across broad frequency ranges to enable wideband electromagnetic structure synthesis. For antenna applications, the frequency ranges can be tailored to specific communication bands or radar applications. The neural network can predict both scattering parameters and radiation patterns for antenna structures, with radiation patterns sampled at specific frequency points and angular cuts. For antenna radiation pattern prediction, the neural network outputs include φ=0° and φ=90° cuts sampled from θ=90° to θ=-90° over 72 angular points. The radiation patterns are sampled at 11 frequency points across the operating frequency range, providing 1584 total radiation pattern data points for each antenna structure prediction. The electromagnetic scattering parameters enable characterization of multi-port electromagnetic structures including power dividers, combiners, filters, and matching
Princeton - 102776 networks. The complex S-parameters include both magnitude and phase information, allowing the neural network to predict complete electromagnetic behavior including insertion loss, return loss, isolation, and phase relationships between ports. The frequency-dependent nature of the S-parameters captures resonant behavior, bandwidth characteristics, and frequency selectivity of the electromagnetic structures. The neural network prediction accuracy for electromagnetic scattering parameters enables rapid synthesis of electromagnetic structures without requiring electromagnetic simulations during the optimization process. The predicted S-parameters serve as inputs to cost functions that define desired electrical properties, allowing optimization algorithms to explore vast design spaces and synthesize structures with target electromagnetic characteristics across specified frequency ranges. The method 100 transitions from the training phase 130 to a synthesis phase 140, where the trained electromagnetic surrogate model is deployed for rapid structure generation and optimization. The synthesis phase 140 leverages the computational efficiency of the convolutional neural network to explore vast design spaces without requiring electromagnetic simulations during the optimization process. The synthesis phase 140 begins with a synthesizing structures step 108. The synthesizing structures step 108 utilizes the trained electromagnetic surrogate model within an optimization algorithm to synthesize electromagnetic structures having target scattering parameters. The optimization algorithm comprises a genetic algorithm or particle swarm optimization that can rapidly evaluate thousands of candidate structures using the neural network predictions. The synthesizing structures step 108 can use binary particle swarm optimization (BPSO) as an alternative to genetic algorithm approaches, where particles represent binary pixelated structures and velocity updates determine pixel state changes. The synthesizing structures step 108 includes co-optimization of discrete components like DC blocking capacitors through lookup tables and cascading with predicted S-parameters. The co-optimization process evaluates different capacitor values by cascading the capacitor S- parameters with the neural network predicted S-parameters of the pixelated structure, allowing simultaneous optimization of both passive electromagnetic structures and discrete circuit elements. The synthesizing structures step 108 can use a tandem neural network approach with an inverse generative network that directly outputs structures from S-parameter inputs, providing an alternative synthesis pathway that bypasses iterative optimization algorithms. The neural network architecture deployed in the synthesizing structures step 108 can include specific filter sizes varying from 12×12 to 3×3 across the convolutional layers, with
Princeton - 102776 each layer having 64 learnable filters. The varying filter sizes enable the network to capture electromagnetic features at different spatial scales, from broad geometric patterns to fine structural details that influence electromagnetic behavior. The optimizing EM surrogate step 106 can include DC path checking between shorting locations and input ports using path finding algorithms. The DC path checking ensures that synthesized structures maintain proper DC connectivity for biasing active circuit elements, where the path finding algorithms verify continuous conductive paths between designated shorting locations and input ports within the pixelated structure representation. The synthesis phase 140 continues with a verification step 110. The verification step 110 verifies the functionality of synthesized structures using electromagnetic solvers to confirm that the neural network predictions accurately represent the electromagnetic behavior of the optimized structures. The verification step 110 performs electromagnetic simulations on selected synthesized structures to validate the neural network predictions and ensure that the optimization process has produced structures with the desired electromagnetic characteristics. The synthesis phase 140 proceeds with a co-simulation step 112. The co-simulation step 112 performs circuit electromagnetic co-simulations to verify end-to-end performance of circuit designs that incorporate the synthesized electromagnetic structures. The co-simulation step 112 combines the electromagnetic behavior of the synthesized structures with active circuit elements to evaluate complete system performance, including gain, bandwidth, stability, and other circuit-level metrics. The synthesis phase 140 concludes with a sending design step 114. The sending design step 114 transmits the verified electromagnetic structure designs to manufacturing systems or design databases for subsequent physical implementation. The sending design step 114 packages the synthesized structure geometries, material specifications, and performance characteristics into formats suitable for manufacturing processes or further design integration. The method 100 transitions from the synthesis phase 140 to a manufacturing phase 150, where the synthesized electromagnetic structure designs are transferred to manufacturing facilities for physical fabrication. The manufacturing phase 150 enables the physical realization of the optimized electromagnetic structures that have been designed using the deep learning enabled synthesis approach. The manufacturing phase 150 begins with a receiving design step 116. The receiving design step 116 receives output from an optimization algorithm configured to synthesize a target electromagnetic structure having target scattering parameters. The optimization algorithm comprises an electromagnetic surrogate model trained according to a method that
Princeton - 102776 includes the generating training data step 102 and the training EM surrogate step 104. The receiving design step 116 processes the design specifications transmitted from the sending design step 114, including geometric layouts, material specifications, and performance requirements for the synthesized electromagnetic structures. The receiving design step 116 receives design data that includes pixelated structure representations, port configurations, and dimensional specifications that define the physical implementation requirements. The design data includes layer stackup information, metal thickness specifications, and dielectric material properties that correspond to the foundry process design kit used during the electromagnetic surrogate model training. The receiving design step 116 can receive designs for various electromagnetic structures including antennas, filters, power dividers, power combiners, diplexers, and quadrature hybrids that have been synthesized using the trained convolutional neural network. The manufacturing phase 150 continues with a manufacturing structure step 118. The manufacturing structure step 118 manufactures the target electromagnetic structure according to the design specifications received in the receiving design step 116. The manufacturing structure step 118 implements the pixelated electromagnetic structure designs using semiconductor fabrication processes, printed circuit board manufacturing, or other suitable manufacturing technologies depending on the target application and frequency range. The manufacturing structure step 118 translates the binary pixelated representations into physical conductive patterns using photolithography, etching, and metallization processes. The manufacturing structure step 118 can implement structures using various foundry processes including 90 nm SiGe BiCMOS, 65 nm CMOS, or other semiconductor technologies that provide the required metal layers and dielectric stackups. The manufacturing structure step 118 ensures that the physical implementation maintains the geometric fidelity of the synthesized designs while adhering to design rule constraints and manufacturing tolerances. The manufacturing structure step 118 can include integration of the synthesized electromagnetic structures with active circuit elements to create complete radio-frequency and millimeter-wave systems. The integration process combines the passive electromagnetic structures with transistors, resistors, capacitors, and other circuit components to implement power amplifiers, transceivers, or other electronic systems. The manufacturing structure step 118 can include packaging and assembly processes that provide external connections and environmental protection for the manufactured electromagnetic structures. The method 100 can include performing circuit electromagnetic co-simulations to verify end-to-end performance of the target electromagnetic structure after the manufacturing
Princeton - 102776 structure step 118. The circuit electromagnetic co-simulations validate that the manufactured structures exhibit the predicted electromagnetic behavior and meet the target performance specifications. The co-simulations can include measurements of scattering parameters, radiation patterns, and other electromagnetic characteristics to confirm that the physical implementation matches the design predictions from the electromagnetic surrogate model. The manufacturing phase 150 enables rapid prototyping and production of electromagnetic structures that have been optimized using the deep learning enabled synthesis approach. The manufacturing phase 150 reduces the time from design concept to physical implementation by eliminating the iterative design cycles typically required with conventional electromagnetic structure design approaches. The manufacturing phase 150 supports the production of electromagnetic structures with performance characteristics that may exceed those achievable through conventional template-based design methods. FIG.3 illustrates a system 300 for implementing deep learning enabled electromagnetic structure synthesis. The system 300 provides a distributed computing architecture that supports the computational demands of training convolutional neural networks for electromagnetic modeling while enabling user interaction and manufacturing integration. The system 300 comprises a device 302 and a device 312 that are interconnected to facilitate distributed processing and data sharing. The device 302 includes a processing unit 304, a processing unit 306, memory 308, and storage 310. The processing unit 304 and the processing unit 306 are configured to execute computational tasks related to electromagnetic modeling and synthesis processes. The processing unit 304 can handle convolutional neural network training operations, while the processing unit 306 can manage optimization algorithms and synthesis tasks. The memory 308 provides temporary storage for data and instructions during processing operations, including neural network parameters, training datasets, and intermediate computational results. The storage 310 provides persistent storage for training data, model parameters, and synthesized electromagnetic structures. The device 312 includes a processing unit 314, memory 316, and storage 318. The processing unit 314 executes computational operations related to the electromagnetic synthesis workflow, including verification and co-simulation tasks. The memory 316 provides temporary storage capabilities for processing operations, while the storage 318 provides persistent storage for design specifications, verification results, and manufacturing data. The device 302 is operably connected to a user 320, allowing for user interaction and control of the electromagnetic synthesis process. The user 320 can input design specifications, monitor training progress of the convolutional neural network, and review synthesized
Princeton - 102776 electromagnetic structures through the connection to the device 302. The user 320 interface enables specification of target scattering parameters, frequency ranges, and optimization constraints for the synthesis process. The device 312 is operably connected to a manufacturing location 330. The connection to the manufacturing location 330 enables transfer of synthesized electromagnetic structure designs from the computational system to manufacturing equipment for physical implementation. The manufacturing location 330 receives design specifications, geometric layouts, and material requirements for fabricating the synthesized electromagnetic structures. The system 300 comprises one or more processing units operably coupled to a non- transitory computer-readable storage medium. The storage 310 and the storage 318 function as non-transitory computer-readable storage media containing instructions that, when executed by the processing unit 304, the processing unit 306, and the processing unit 314, cause the processing units to perform the method 100 of creating an electromagnetic surrogate model. The instructions include code for the generating training data step 102, the training EM surrogate step 104, and the optimizing EM surrogate step 106. The processing unit 304, the processing unit 306, and the processing unit 314 are operably coupled to the storage medium containing instructions for electromagnetic surrogate model creation method. The distributed processing architecture allows the system 300 to handle the computational intensity of training convolutional neural networks on large datasets of electromagnetic simulations while maintaining responsiveness for user interactions and manufacturing communications. The interconnected configuration of the device 302 and the device 312 enables parallel processing of different aspects of the electromagnetic synthesis workflow. The device 302 can focus on neural network training and optimization tasks, while the device 312 can handle verification, co-simulation, and manufacturing preparation tasks. The system 300 supports the complete workflow from the training phase 130 through the synthesis phase 140 to the manufacturing phase 150, providing computational resources and data management capabilities for each phase of the electromagnetic structure synthesis process. FIG. 4 illustrates an inverse generative network training and synthesis overview 400 that provides an alternative approach to electromagnetic structure synthesis using neural network-based inverse design. The inverse generative network training and synthesis overview 400 demonstrates a methodology where target electromagnetic scattering parameters serve as inputs to directly generate electromagnetic structure geometries without requiring iterative optimization algorithms.
Princeton - 102776 The inverse generative network training and synthesis overview 400 begins with input data 402. The input data 402 comprises real and imaginary parts of S-matrix elements that define the target electromagnetic behavior for the desired structure. The input data 402 includes complex-valued scattering parameters across frequency ranges, providing the electromagnetic specifications that the inverse network will use to synthesize corresponding geometric structures. The input data 402 can include multi-port S-parameter matrices for structures with two, three, or four ports, depending on the target electromagnetic device configuration. The input data 402 flows into an inverse network 404. The inverse network 404 processes the electromagnetic scattering parameter specifications and generates geometric structure representations that can achieve the target electromagnetic behavior. The inverse network 404 comprises fully connected layers with leaky ReLU activation functions in intermediate layers and a tanh activation function in the final layer to perform binary thresholding for pixelated structure generation. The inverse network 404 includes six fully connected layers that map from the S-parameter input space to the geometric structure output space. The inverse network 404 produces an intermediate output 406. The intermediate output 406 represents a pixelated electromagnetic structure as a vector that is reshaped into a grid format suitable for electromagnetic analysis. The intermediate output 406 comprises a binary representation where each pixel indicates the presence or absence of conductive material in the synthesized structure. The intermediate output 406 undergoes binary thresholding to ensure that the generated structure consists of discrete conductive and non-conductive regions that can be physically manufactured. The intermediate output 406 is then processed by a forward predictor model 408. The forward predictor model 408 comprises the trained convolutional neural network from the training EM surrogate step 104 that predicts electromagnetic scattering parameters from the pixelated structure representation. The forward predictor model 408 evaluates the electromagnetic behavior of the structure generated by the inverse network 404, providing predicted S-parameters that can be compared against the target specifications from the input data 402. The forward predictor model 408 remains frozen during the inverse network training process, with only the inverse network 404 parameters being updated during backpropagation. The forward predictor model 408 generates a step 410. The step 410 represents the predicted electromagnetic scattering parameters for the structure generated by the inverse network 404. The step 410 enables calculation of the difference between predicted and target S-parameters, which serves as the loss function for training the inverse network 404. The step
Princeton - 102776 410 provides feedback that allows the inverse network 404 to learn the mapping from electromagnetic specifications to geometric structures through gradient-based optimization. The training EM surrogate step 104 can use ADAM optimizer with specific learning rate schedules and L2 regularization factors. The ADAM optimizer provides adaptive learning rate adjustments that enhance convergence stability during the training process. The learning rate schedules can include exponential decay or step-wise reduction to fine-tune the training dynamics as the network approaches convergence. The L2 regularization factors prevent overfitting by penalizing large network weights, thereby improving the generalization capability of the electromagnetic surrogate model across different structure configurations and frequency ranges. The inverse generative network training and synthesis overview 400 provides a direct synthesis pathway that bypasses the iterative optimization algorithms used in the synthesizing structures step 108. The inverse generative network approach can generate electromagnetic structures instantaneously from S-parameter specifications, though the approach may require more extensive training datasets and can be susceptible to mode collapse or training instabilities compared to the optimization-based synthesis methods. The system 300 and the method 100 work together to perform a complete electromagnetic structure synthesis function through coordinated interactions between the training phase 130, the synthesis phase 140, and the manufacturing phase 150. The integration of these components enables automated design and manufacturing of electromagnetic structures with target performance characteristics. The training phase 130 establishes the foundation for electromagnetic surrogate modeling through data flow between the generating training data step 102, the training EM surrogate step 104, and the optimizing EM surrogate step 106. The generating training data step 102 creates datasets by performing electromagnetic simulations on pixelated electromagnetic structures as shown in FIGS. 2A and 2B, where each structure comprises a grid of pixels representing presence or absence of conductive material. The training data associates each pixelated structure with corresponding electromagnetic scattering parameters as illustrated in FIG. 2B, which shows complex S-parameters across frequency ranges. The training EM surrogate step 104 receives the training data from the generating training data step 102 and trains a convolutional neural network to predict electromagnetic scattering parameters from pixelated structure representations. The optimizing EM surrogate step 106 receives the trained network from the training EM surrogate step 104 and refines the electromagnetic surrogate model parameters using cost functions that define desired electrical properties.
Princeton - 102776 The synthesis phase 140 utilizes the trained electromagnetic surrogate model from the training phase 130 to generate optimized electromagnetic structures through coordinated execution of the synthesizing structures step 108, the verification step 110, the co-simulation step 112, and the sending design step 114. The synthesizing structures step 108 receives the optimized electromagnetic surrogate model from the optimizing EM surrogate step 106The method 100 integrates multiple phases and components to perform complete electromagnetic structure synthesis, from initial data generation through physical manufacturing. The training phase 130, synthesis phase 140, and manufacturing phase 150 work in concert to enable rapid design and production of optimized electromagnetic structures. The training phase 130 begins with the generating training data step 102, which performs electromagnetic simulations on pixelated electromagnetic structures as shown in FIGS. 2A and 2B. These simulations generate datasets associating pixelated structure representations with corresponding electromagnetic scattering parameters. The training EM surrogate step 104 then uses this data to train a convolutional neural network, creating an electromagnetic surrogate model capable of predicting scattering parameters from structure representations. The optimizing EM surrogate step 106 further refines this model using cost functions defining desired electrical properties. The synthesis phase 140 leverages the trained electromagnetic surrogate model within optimization algorithms during the synthesizing structures step 108. This step rapidly explores the design space to generate candidate structures meeting target specifications. The verification step 110 then verifies the functionality of these synthesized structures using electromagnetic solvers, ensuring the accuracy of the surrogate model predictions. The co-simulation step 112 performs circuit electromagnetic co-simulations to verify end-to-end performance of circuit designs incorporating the synthesized electromagnetic structures. This step combines the electromagnetic behavior of the passive structures with active circuit elements to evaluate complete system performance. The sending design step 114 then transmits verified designs for manufacturing. The manufacturing phase 150 begins with the receiving design step 116, which processes the transmitted design specifications. The manufacturing structure step 118 then physically fabricates the electromagnetic structures according to these specifications, translating the pixelated representations into physical conductive patterns. The system 300 shown in FIG. 3 provides the computational infrastructure supporting this integrated workflow. The device 302 and device 312 work in tandem, with the processing units 304, 306, and 314 handling different aspects of the synthesis process. The memory 308
Princeton - 102776 and 316 provide temporary storage for active computations, while the storage 310 and 318 maintain persistent data across the workflow phases. The user 320 interacts with the system 300 to input design specifications and review results, while the manufacturing location 330 receives final designs for physical implementation. This distributed architecture enables parallel processing of training, optimization, and verification tasks while maintaining responsiveness for user interactions and manufacturing communications. FIG.4 illustrates an alternative synthesis approach using the inverse generative network training and synthesis overview 400. In this method, the input data 402 comprising target scattering parameters flows directly into the inverse network 404, which generates candidate structure geometries. The intermediate output 406 from this network is then evaluated by the forward predictor model 408, producing predicted scattering parameters in step 410. This approach provides a direct pathway from electromagnetic specifications to structure geometries, complementing the optimization-based synthesis methods used in the synthesis phase 140. Through this integrated workflow, the method 100 and system 300 enable rapid design, optimization, and manufacturing of electromagnetic structures with performance characteristics that may exceed those achievable through conventional design approaches. The seamless interaction between computational modeling, optimization algorithms, and physical manufacturing processes supports efficient exploration of vast design spaces and realization of optimized electromagnetic structures. In various aspects, a method of manufacturing a circuit may be provided. The method may include receiving a circuit and/or electromagnetic structure that has been designed utilizing inverse synthesis with a forward model (an EM surrogate) created using a method as disclosed herein. The method may then include realizing the circuit and/or electromagnetic structure on a substrate (such as a printed circuit board, an integrated chip, etc.). The substrate may be a printed circuit board, the circuit and/or structure may be realized in a package or in an integrated chip, or a combination of these. The printed circuit boards may include any appropriate circuit board material including, e.g., flame retardant 4 (FR-4) epoxy laminate, a Rogers PCB, a Megtron PCB, a metal-core PCB (which may include, e.g., aluminum), a low-temperature co-fired ceramic (LTCC) PCB, or polytetrafluoroethylene (PTFE). The integrated chip could be include any appropriate chip material, and may be in, e.g., silicon, GaAs, GaN, InP, SiC, diamond, or other material.
Princeton - 102776 In various aspects, a method for designing circuits may be provided. Once the forward models are created, it may be possible to create a tool, such as a software tool, that allows users to select one or more appropriate models. The desired performance characteristics of the device may be provided (e.g., target scattering and/or radiative parameters), and then, via inverse design synthesis that incorporates the selected model(s), one or more circuits may be designed or co-designed to achieve the target performance. The one or more circuits that may be designed or co-designed may include may be coupled directly or indirectly to each other. There may be one or more active elements in the design. The active elements may be an active device, such as a semiconductor device, such as a transistor or diode. The one or more circuits may define an amplifier with a desired response across a frequency range. The one or more circuits may define a mixer with a desired response across frequency range. The one or more circuits may define a power amplifier with a desired response across frequency range. The circuit(s) may define a combination of low noise amplifier and power amplifier connected with a switch or a duplexer. The circuit(s) may define a phase shifter. The circuit(s) may define a variable gain amplifier. The circuit(s) may define an oscillator with or without a phase locked loop. The circuit(s) may realize a radio frequency transmitter or receiver or both. The one or more circuits may define, e.g., a 1-port multi-band antenna, a 2-port band pass filter, a 3-port divider with 90° phase difference, a 4-port frequency diplexer, a 4-port hybrid coupler, an unequal power divider, a filtering divider with even split, a mm-Wave antenna, or a mm-Wave amplifier. The circuit(s) may be realized in an integrated chip or with multiple chips in a packaged system. The circuit(s) may be realized on a printed circuit board or in combination with one or more chips. Example 1 In high-frequency systems, electromagnetic (EM) structures and their co-design with circuits play a critical role. Historically, the design of such EM structures (such as matching networks, combiners, splitters, quadrature hybrids, and couplers) has relied on intuition-based approaches stemming from a priori known physical effects. The design process starts from a parameterized model of such a structure with pre-selected topology that is constituted of lumped or distributed components. The unit elements are transmission lines, inductors, capacitors, and any combination of them, where each element is well understood. This is subsequently optimized for bandwidth and insertion loss with time-consuming parameter sweeps, population-based metaheuristic optimization methods including evolutionary algorithms such as genetic algorithm (GA), differential evolution algorithm, particle swarm optimization (PSO), and other population algorithm variants; or by exploiting machine learning
Princeton - 102776 enabled regression-based surrogate models. While such a bottom-up approach allows an insight into how the structure operates, any pre-selection of the EM topology already limits the structures to be modular, regular, and limited to a narrow set of tuning parameters. It is highly unlikely that these bottom-up geometries will lie close to being "global optimum" performance in the space of all possible EM structures occupying the same volume. These design methodologies, therefore, restrict the searchable space of design choices and limit achievable circuit performance. Given the proliferation of spectral bands in the millimeter-wave (mm- Wave) across 24 െ 100GHz for communication and sensing (and jointly both) and opportunities of spectrum sharing across licensed/unlicensed bands, future mm-Wave front ends may need to address multiple bands spread across the spectrum and potentially concurrently. Such design challenges need to be addressed with new design insights going beyond the traditional bottom-up design methodology. Inverse design approaches, as popular in scattering theory, quantum material synthesis, and nanophotonics, attempt to search for the near-optimal structure with a desired performance through a top-down approach. Such inverse-designed structures, as demonstrated by recent advances in nanophotonics, are often irregular, non-intuitive, and seemingly arbitrary but can exceed the performance limitation of traditional designs. For the power amplifier (PA), this would imply search of an EM structure (ideally 3-D) that allows close to the optimal distribution of electrical/magnetic energy storage across frequencies for lowest insertion loss broadband matching. Search of such structures in the space of all on-chip manufacturable structures within area constraints would be an insurmountable problem. As an example, consider an area on chip of size 300 x 300 µm that is discretized into 16 ൈ 16 pixels, where each pixel (25% larger than 18.75 µm on each side to allow diagonal overlaps) can be either have metal or no metal (dielectric). The size of the space of all possible structures is 2ଶହ^ ^ 10^^. Searching this highly non-convex space with population-based meta-heuristic optimization methods utilizing time- and resource-intensive EM simulations is generally infeasible and severely limits the search space. In recent years, deep learning has been applied in such inverse problems, utilizing their ability to extract the relevant features automatically and probe the highdimensional search space efficiently. In the field of circuits, such as PAs, machine learning-based surrogate model- based optimization methods have been shown for pre-selected topologies, thereby limiting the search space in the neighborhood of the defined geometries.
Princeton - 102776 To address a global search in the space of arbitrary designs, propose herein is an inverse design approach where time- and resource-intensive complex EM simulations are replaced with a deep convolutional neural network (CNN) surrogate model that takes the arbitrary planar EM structure geometry as an input and predicts its scattering parameters accurately and almost instantaneously. This allows the optimization algorithm (that itself can be a neural network such as a generative adversarial network or GAN) to expand the searchable space by orders of magnitude, potentially reaching toward the global optimum. While the method is presented in this example in the context of a broadband mm-Wave PA design covering 30 െ 94GHz, the approach is applicable to any EM structure, allowing co- design with other circuit elements and end-to-end transceivers. To allow rapid and generalizable training, a transfer learning approach was used to train an initial network with faster in-air simulations and then retrain with a smaller number of training samples with dielectric stackup, thereby reducing the required simulation time for dataset generation significantly. For example, in the case of 300 x 300 µm structures, 4.5 days of total simulation time was enough instead of the estimated 14 days. This generalization also allows one to create an easily transferable model for other EM structures across chip stacks, size, and frequency range. Once the network is trained, the algorithm generates the EM structure for the given circuit and desired performance in minutes without any direct human input or manual tuning. Deep-CNN as an EM Emulator. The overall approach in the inverse synthesis of the planar EM structures with desired scattering parameters is shown in FIG. 4. The input to the algorithm is the input and output impedances to be matched, and the output is the synthesized EM structure. In the iterative optimization process, as the structure evolves, we use a deep-CNN model, as shown in the figure, to accurately predict the S-parameters of the evolved structure and successfully converge to the synthesized structure in the enormous design space. A. Design Space This synthesis is described in the context of the output matching network (OMN) of a 30 െ 100GHz SiGe-based mm-Wave PA, though the general principle is applicable to a wide range of EM structures, including other matching networks, filters, splitters, couplers, combiners, and antennas. 1. Discretization: The 2-D planar structure realized over a ground plane is first discretized into a 2-D pixel array where different geometries are represented through a binary matrix. In this matrix, “1” represents the presence of metal and "0" represents the absence of metal (although it could easily be reversed).
Princeton - 102776 2. Pixel Size and Count: In this example, the structure is divided into 16 x 16 pixels. With a 300 x 300 µm structure, each pixel is approximately ^^/100 at the center frequency of 65 GHz , by taking the effective dielectric constant (^^Eff ) of chip stack up into account. Higher pixel count allows higher resolution of the structure, access to a larger design space, and potentially better performance at the cost of computational time, and similarly, lower pixel count provides lower resolution of the structure, with smaller design space, and worse, but faster, performance. It is worth noting that diagonal connections have a width of 6.6 µm to pixel overlaps. Consequently, dc resistance between PA and ^^େେ is on the order of milliohms. 3. Feed Locations: Similar to that shown in FIGS.2A and 2B, the feed location of the PA and the output pads (or the antenna) can be in either of the 16 pixels on the sides. Also, to allow a supply feed ( ^^େେ ), any pixel on the top and the bottom edge can be a supply location where an RF short can be implemented with a bypass capacitor array. This leads to a total search space of a 2ଶହ^ possible EM structures with 16ସ pin locations overall. 4. EM Structure Size: The size of the EM structure is fundamentally related to the wavelength of operation. Here, an area of 300 x 300 µm was chosen, that is approximately 0.12^^ at the center frequency of 65 GHz. This size is, of course, a function of the spectral range and the desired impedance characteristic and could be incorporated into the optimization algorithm. B. Deep-CNN Architecture Neural networks have extraordinary abilities to capture nonlinear relationships in a high-dimensional input space. Multi-layer perceptrons (MLPs) have been utilized in the past as surrogates for template-based designs. In the presented design space, the structures are more complex and can be thought as an image, where the coupling of the neighboring pixels needs to be captured by the surrogate model. This lends CNN as a natural choice as the multiple layers of varying convolutional filter sizes can capture the local spatial effects. CNNs can learn to extract necessary features by itself, in contrast to traditional regression where designer will provide a set of predetermined and extracted features as the input. The features imply how the distribution of various localized patterns within the 2-port network, such as exact positions of metals and holes, port locations, and ac shorts ( ^^େେ supply), can impact the predicted function (scattering parameters). Small changes in the local structure may or may not strongly impact S-parameters. Therefore, it is important for the CNN to effectively learn what structural signatures within the geometry are important and useful for the prediction of the scattering parameters.
Princeton - 102776 The deep-CNN architecture is shown in FIG. 5. The input to the CNN is the 18 ൈ 18 matrix that represents the EM structure. The output is the predicted 2-port complex S- parameters, where the two ports are the PA cell and the output load (terminated with 50Ω in parallel to pad capacitance). The two supply feeds are terminated as ac short to generate the 2- port S-parameters. We discretize the frequency space between 30 and 100 GHz into nine points. For PA matching, target load-pull contour traverses a limited portion of Smith chart. Subsequently, current frequency resolution can sufficiently characterize broadband matching. Furthermore, given the compact size of the network, sharp resonances can be avoided. The resolution can be efficiently improved with transfer learning. At a given frequency point, CNN predicts the real and imaginary parts of ^^^^, ^^ଶ^, and ^^ଶଶ. All the parameters of the two-port network are required to effectively capture power transfer degradation in the PA due to insertion loss and impedance mismatch from the ideal load pull. Since ^^ଶ^ ൌ ^^^ଶ for passive structures, predicting six terms for every frequency point is sufficient. Therefore, the output layer is 54 elements long. The deep-CNN architecture consists of 12 convolutional and five fully connected (FC) layers, and the layer configurations are optimized to minimize the prediction error, as we will elaborate later. Convolutional filters, through their progressively reduced sizes along the network, extract the necessary local and global features. At each convolutional layer, a network has 64 learnable filters. Interconnect layers and certain hyper-parameters (such as layer widths) are decided in order to enhance the generalization ability of the network. Leaky rectified linear unit (ReLU) are used here as the non-linear activation due to its non-zero gradient unlike many other alternative activation functions such as hyperbolic tangent (tanh) or sigmoid. Inputs to the leaky ReLU undergo batch normalization operation. This allows the network to be trained faster and allows cascading of more layers. To promote generalization, the network has dropout layers at FC levels. During training, dropout layers randomly shut down a fixed percentage of neurons with a dropout ratio of 50%. In addition, weight regularization is used for all learnable coefficients as an additional method to avoid overfitting. Due to the passive nature of the structure, the real and imaginary parts of predicted S-parameters are limited to the range of [- 1, 1] by using the "tanh" layer at the output. C. Dataset Generation and Augmentation To expedite the process of training, six different 2-port EM structures are generated from a single 4-port EM simulation. As shown in FIG. 6, by choosing two locations as ^^େେ supply ports, rotation, translation, and mirror reflection can be applied to create two-port EM
Princeton - 102776 structures, all of whose S-parameters can be derived from the original 4-port structure. This reduces the dataset creation time significantly. D. Training Process and Transfer Learning for Generalizability Across Dielectric Stack and Box Size To allow a training flow that is adaptable across different dielectric stacks, different structural sizes, and across different frequencies, transfer learning is used to avoid re-training from the beginning when some of these parameters change. In addition, to expedite the training phase, transfer learning is used to finally predict the performance of the EM structure located within the chip dielectric stack. Here, an initial network is trained with simulation data using air as a dielectric. To make this a reasonable approximation for the final EM structure with chip dielectrics, the EM structure size for air was upscaled by a factor of ^^^Eff ^^/ଶ. This approach follows from the equivalent transverse electro-magnetic (TEM) problem of microstrip transmission lines where the electrical length of the line can be accurately determined by using the effective dielectric constant. This simplified simulation setup in air typically takes ten times less resources than simulating with the dielectric stackup. As a result, resource requirements are reduced considerably. If scaled structures embedded in dielectric can be transfer learned, then logically, scattering parameters with size scaling (keeping dielectric same) and different frequency ranges can potentially be learned as well, generalizing the use of the one-time training process for other circuits and/or frequency ranges. Training is done with MATLAB's deep learning toolbox. An adaptive moment estimation (ADAM) optimizer is used for stochastic gradient descent algorithm. The initial learning rate is 0.001 and it is multiplied by 0.8 at every four epochs. The ^^2 regularization factor of 0.0001 is used. The output cost function is the mean absolute error. As shown in FiG. 7, EM simulations are performed of 250k randomized structures in air (augmented to 600k 2-port structures using the method in Fig. 5) and 75 k simulations in dielectric stack (augmented to 204k 2-port structures). Initial training runs (with in-air simulations) are done with a training-validation-test split of 500 െ 50 െ 50k. The batch size of 5000 and the validation frequency of 100 are used. Periodic validations are essential to detect and avoid overfitting. Training is limited to 40 epochs. An important distinction is the activation function of the output layer. Since the "tanh" function suffers from vanishing gradients, there is no output activation function for the initial
Princeton - 102776 network. On the second stage of training, an accurate simulation dataset is divided into 170- 17-17k training-validation-test split. While all the weights of the initial model are transferred without change, the final layer is modified by adding the "tanh" activation function. In this case, the batch size is 1700, whereas the validation frequency is still 100. To allow for portability across EM structure size (or frequency), we can re-train the final network for a size of 200 ൈ 200^^ m with only 15 k simulation (augmented to 50k) (FIG. 7). The resulting network achieves a validation rms error of 0.4, indicating good convergence. In addition, the frequency resolution of the original network can be improved by re-tuning it with a small number of finer frequency resolution simulations. The second stage of the training takes around 1 h with A100 GPU over 120 epochs. With a GPU, predicting the scattering parameters (or inference) is very fast and takes a fraction of a second for thousands of structures. Utilizing the described training flow, the appropriate choice of the number of layers allows us to create robust and accurate EM predictors, as shown in FIG.8. It is observed that the combination of 12 convolutional and five FC layers is preferable over the other combinations due to low absolute error and small difference between training and test set rms error. This indicates less overfitting and higher adaptability. The detailed description of the neural network is given in Table I. The distribution of the absolute value of the error is heavily skewed to 0, indicating good prediction accuracy. Table 1. Details of the Deep-CNN Architecture and Its Various Layers Block Conv layer filter size Number of filters
Princeton - 102776 ^^^^ 3 ൈ 3 64 ^^ 3 3 64 le)
alues and frequency; in this example, the error is typically very small throughout the grid, except for low values of ^^^^ near 30 GHz. This is primarily because the randomized generation of these pixelated networks does not provide a good input match close to 30 GHz due to the compactness of the network (where box size is close to 0.05^^ ). The EM simulation and CNN prediction results for two test set samples with two different sizes were compared, showing a near-perfect match between predicted and EM simulated results. The computational requirements for the described inverse synthesis method can be compared to classical meta-heuristic methods with iterative EM simulations. Once the model is trained, it can be utilized for rapid synthesis over an enormous amount of possible structures, thereby allowing one to progress toward the global optimal. Compared to the classical EM- based optimization, deep learning-based inverse design takes the order of magnitudes less time (400 s versus estimated 21 days). At the same time, memory requirements for CPU-intensive tasks per prediction are a lot smaller than that of EM simulations. As an example, the total memory of 128 GB is utilized for a population size of 4096, resulting in 32 MB per population member. Also, the generalizable model makes it useful for easy portability for the synthesis of other EM structures such as filters, couplers, antennas, and co-design with circuits, which will otherwise require separate EM-based optimization every time. While higher degree of freedom will allow more precise control over the EM characteristics of a structure, it also does increase the training cost. As can be seen in FIG.9, a
Princeton - 102776 larger number of pixels open up to a larger design space reducing the loss in the example of a broadband matching network for the PA, but at the cost of an increased simulation time. The tradeoff is, therefore, application-dependent. Also, very fine discretization will necessitate considering dc resistance and design rules during the optimization phase. Deep Learning-Aided Inverse Design FOR PA Synthesis Inverse design of a PA with given specifications (such as output power, small-signal performance, bandwidth, compression points, and linearity) essentially implies the algorithmic synthesis of the end-to-end circuit without the manual tuning and optimization processes of the circuits and matching networks. A hybrid approach that relies on a judicious choice of the PA cell with the automated synthesis of the EM structures with designer S-parameters can allow us to create such circuits with a co-design methodology. Here, the inverse design methodology is illustrated with the design example of mm-Wave PAs in a 90 െ nm SiGe BiCMOS process highlighting the synthesis of OMNs across power levels, dual-band operation, enforcing harmonic terminations, and for broadband performance. Similarly, it is possible to apply this method for other matching networks to ensure stability and flat small-signal gain across frequency. A. Choice of PA Cell Given the PA specifications, the optimal choice of the mm-Wave PA cell, output and drive cells, and biasing for the desired performance is not unique. First, for the optimal voltage swing for reliable and efficient operation of PAs, safe operating regions need to be characterized. For a common-emitter (CE) topology with the presence of a bias resistance at the base terminal, breakdown voltage lies between BVCeo and BVCBO due to reduced injection efficiency. With appropriate voltage limits estimated, ^^େେ ൌ ^^^ୟ^/2 can be chosen as dc operating point. It is worth noting that the choice of supply voltage is not independent of the device configuration. The commonbase (CB) configuration has a breakdown limit of ^ BVCBO , which is roughly three times of BVCEO in contemporary SiGe transistors. Alternatively, stacked PA cores can be utilized to alleviate breakdown problem. Of the widely used PA stages, stacked PA cores have improved ^^SAT , gain, and stability. However, it is challenging to provide wideband intra-stage load-pull matching between devices. Hence, power added efficiency (PAE) performance usually falls short of single-device PAs. Between CE and CB, CB draws attention with a number of advantages such as high linearity, high breakdown, and higher available gain. However, device stability needs to be carefully maintained.
Princeton - 102776 B. Inverse Synthesis of OMN Given the desired input-output impedances of the 2-port or multi-port OMN, there are several approaches for inverse synthesis of the EM structure. It is possible to use a neural network itself in a reverse fashion for this purpose (tandem approach). Here, we use GA with discrete variables and tournament parent selection in combination with the deep learning-based surrogate model for the synthesis. 1. Cost Function: In contrast to time-intensive EM simulations, the deep learning-based modeling approach allows us to predict S-parameters corresponding to a population of thousands of structures within a fraction of seconds in each step of the optimization. During the optimization, using the predicted S-parameters of population members, a cost function considering impedance mismatch and insertion losses as follows is minimized: ே ^ ^^^^^^^ ൈ ห^^^^^^^ െ ^^^୮^^^^^ห ^ ^^ଶ^^^^ ൈ ^1 െ |^^ଶ^^^^^|^ଶ where the points (across 30 െ
100GHz here) ^^ଶ^ is calculated by assuming a source impedance equal to the complex conjugate of the optimal load-pull impedance ( ^^Opt ), while the output load takes the pad capacitance into account. Therefore, ^^ଶ^ quantifies both the insertion loss of the matching network and mismatch to the optimal load-pull impedance. The first term accounts for power reduction in the nonlinear power and efficiency contours. Since reaching the actual load-pull point can incur more losses, the coefficients ^^^^^^^ do not disappear. One can also consider dc supply resistance and design rule check (DRC) considerations such as local metal density into the cost function. C. OMN Design Examples In addition to the RF performance, a dc supply path is required for feeding the PA cell (Fig.13). To ensure the presence of a direct route between shorting location (which can be used as a ^^େେ feed point), population members with no dc path are penalized heavily. A simple path finding algorithm is utilized to check the existence of the connection [67]. Also, the network might include other components such as a dc blocking capacitor as shown in FIG.10. The co- optimization of this component with the pixelated network is done through a restricted sweep of the capacitor values considering the selfresonance frequency high-density MIM capacitor. In particular, 2-port S-parameters of series capacitors are stored in a lookup table and cascaded with the predicted S-parameters of population members. For each member, the minimum
Princeton - 102776 achieved cost value over the capacitor values is stored. FIG. 11 summarizes these joint objectives and demonstrates the output network topology. 1. Population Update Rules: Upon evaluating the cost values of the current population, we perform parent selection for each member of the next population. First, " ^^ " members were randomly picked from the current population and two best performing members of this subset were assigned as parents. This scheme is named tournament selection where " ^^ " denotes the tournament size. Rows of the offspring matrix are reconstructed from each parent through a random cut line, as shown in FIG.11. On the other hand, port and shorting locations are transferred from either of the parents. After this step, random mutations that flip metal-no- metal locations are added to the offspring, along with mutations that perturb port and shorting locations. This is repeated " ^^ " times for a population size of " ^^." In addition to the creation of new offspring, " ^^ " best performing members from the current population are directly transferred to the next population. In this way, a monotonic decrease of the best cost value is guaranteed. 2. Computational Resources and Timing: A typical set of parameters for GA runs is adopted as a population size of ^^ ൌ 4096 and tournament size of ^^ ൌ 256, for a total of 100 generations. Due to the relatively small tournament size, a diverse set of parents is achieved. This helps avoiding local optimums around starting point. In each iteration, ^^ ൌ 8 best performing members of the previous generation are directly transferred to the offspring. The mutation probability of each pixel starts from 0.1 and linearly decreases to 0 toward the final iteration. With these settings, total optimization time is around ^ 400 s with an A100 GPU and 32 AMD EPYC CPU cores. Average time spent on different elements of the optimization for a single generation is shown in FIG.12. It can be noted that a large portion of the computational time is spent on the dc block capacitor optimization as opposed to the prediction of the S-parameters, as shown in FIG.12.. While S-parameter prediction is carried out with a GPU, the rest of the tasks heavily relies on multi-threaded CPU-intensive calculations. Due to effective parallelization of the code, we can accurately estimate time required for different population sizes or computational resources. The optimization process can be accelerated by removing some of the overheads intermittently such as passivation of the predicted S-parameters (which aims to avoid physically impossible results arising from numerical errors), checking connections between shorting locations and input port for dc feed, and dc block capacitor optimization.
Princeton - 102776 Here, OMN designs are illustrated for various functionalities. The CE configuration of the PA cell with ^^େେ ൌ 1.8 V and ^^^^ ൌ 0.85 V. 3. Broadband OMN for Varying Output Power: Using the optimization scheme and GA parameters described, a broadband OMN design can be performed for different output stages for varying power generation capabilities. PA cells of varying sizes and optimized dc block capacitor and output network can be synthesized that achieve close to the optimal load-pull impedances across 30 െ 100GHz. Here, ^^Opt target is roughly found as 22Ω//80fF for a PA cell of 8 µm x 4 and scaled for other PA cell sizes using equivalent ^^^^ model of load-pull impedance. Cost function terms were adjusted to emphasize edge frequencies for bandwidth extension and high frequency performance. 4. Dual-Band OMN Across 5 G mm-Wave and 70 െ GHz Unlicensed Bands: To demonstrate the flexibility of the proposed method, one can also create dual-band PAs that can address 5G mm-Wave bands (37 GHz) and the 70 െ GHz unlicensed band for joint sensing and communication. In some examples to achieve flat response around target frequencies, matching efficiency was optimized at 37,40,60, and 70 GHz. 5. Narrowband OMN With Harmonic Termination: In addition to the fundamental operation, one can also impose constraints on harmonic termination. Here, one can synthesize the network for optimal impedance across 37 െ 40GHz while ensuring second harmonic termination for improved power efficiency. In these examples, the realized impedance follows closely to the desired one, with the second harmonic impedance lying close to the expected shorted termination. To demonstrate the effect of the random starting point in the optimization, the achieved matching conditions with ten different runs for the 8 µm x 4 PA cell dual-band designs. The algorithm converges to EM structures that realize good impedance matching even with different initial conditions at the beginning of the optimization runs. This can be attributed to the exploratory strategy of GA through the diverse set of parent selection and high initial mutation rate. Prototype 30-94 GHz PA Design and Measurement Results A prototype proof-of-concept mm-Wave PA with an inverse designed OMN was designed for ^^sat ,ଷ ^^ bandwidth of 30 െ 100GHz in a 90 െ nm SiGe process with ^^max of 360 GHz and rated for BVCEO of 1.68 V , while BVCBO is 5.3 V. The PA is based on a two-stage CB topology as CB provides higher available gain, higher breakdown voltage, sharper compression behavior, and superior back-off efficiency enhancement due to the current
Princeton - 102776 clamping effect. Based on the targeted output power of 18 dBm, 8µm x 4 was chosen for the PA stage. The PA schematic with the pixelated output network is shown in FIG.13. Conventional transmission line-based matching using a combination of open and shorted stub lines is used in the input and interstage matching for broadband design, while the OMN is obtained through an inverse design algorithm. For the final verification, OMN is simulated from dc to 300 GHz with HFSS. As can be seen from FIG. 13, the output network achieves a broadband performance with insertion losses varying between 0.55 and 1 dB. The chip micrograph is shown in FIG.14. FIG. 15 shows the measured and simulated small signal performance. The peak ^^ଶ^ is 12.5 dB at 66 GHz ( ^^ଶ^,ଷ ^^ bandwidth of 21 GHz ), and the measured ^^^^ is below -10 dB from 50 to 105 GHz. The measured group delay shows a flat 25 െ 35 across 37 െ
94GHz. While in this example, the optimization was on PA saturated power flatness across frequency, broadband small-signal gain and flat group delay can also be incorporated in multi- objective optimization across multiple stages. The measured large-signal continuouswave (CW) performance between 30 and 102 GHz was considered. Between 30 and 94 GHz , the PA achieves between 16.5 and 19.5 dBm ^^sat , 16% െ 24.7% PAE, and 19% െ 27.5% total collector efficiency ^^^^, including driver and output. High linearity can be observed with sharp compression behavior where ^^sat െ ^^^ dB varies between 0.6 and 1.9 dB. The PA achieves a 6 െ dB backoff efficiency enhancement of 1.75-2.45 times class-A across 30 െ 102GHz. The PA was tested with high-speed single-carrier modulation signals between 36 and 88 GHz. While no digital pre-distortion was used, channel equalization was applied for calibrating the measurement software. PA demonstrates ^^avg of 12.6/11.5/10.55dBm, PAEavg of 10.7%/10.5%/8.3%, error vector magnitude (EVM) of െ26.4/െ23.6/െ27 dB, and adjacent channel leakage ratio (ACLR) of െ32.8/െ28.8/െ30.6dBc for 64 QAM modulation with 4.5 െ/4.5 െ/2.4 െ Gb/s data rate at 40/66/88GHz, respectively. At higher frequencies, a data rate is limited by the measurement setup. To test the ability of the PA across 30 െ 94GHz, the performance was measured under large-signal concurrent multiband (dual/triple) signals. Such concurrent multi-band operation can achieve higher data rates, efficient usage of the spectrum, dynamic spectrum management, and quality of service ensuring always-connected capability. Of course, this also needs antennas and filters capable of handling such signals and cleaning off inter-modulation products, but this example is concentrated on the PA performance. First, the PA was tested
Princeton - 102776 with two tones in a CW setup. Two tones were injected in the 5G licensed mm-Wave band ^^^ ൌ 40GHz and in the unlicensed spectrum at ^^ଶ ൌ 66GHz. The input power was varied across ^^^ and ^^ଶ (over the 2-D plane) and the output power was measured at ^^^ and ^^ଶ, as well as concurrent PAE. The PA can support 13.1-dBm power at both ^^^ and ^^ଶ with a concurrent PAE of 15%. The intermodulation products in the broadband PA
-10 dBc and below (maximum at 2^^^ ). Concurrent triple-band operation at 36,39 , and 66 GHz each with 2.5 െ Gb/s32 QAM modulation and aggregate data rate of 7.5 Gb/s was shown using the well-calibrated measurement setup FIG.16. In this setup, two data streams with differing IF ( 1 and 4 GHz ) are combined and upconverted via 35 െ GHz local oscillator (LO) to 36 and 39 GHz. Image frequencies located at 31 and 34 GHz are suppressed through a bandpass filter. A third data stream is upconverted to 66 GHz. These two signal branches are connected to driver PAs to deliver enough power to the chip through a broadband combiner. The output signal from the chip is divided into frequencies and measured separately. Across the three carrier frequencies, the PA maintains superior linearity. In concurrent operation, the PA reaches ^^avg of 6.1/6.9/9.1dBm, EVM of െ27.5/െ25.4/െ23.8 dB, and ACLR of െ31.9/െ30.2/െ29.9dBc at 36/39/66GHz, respectively, with a concurrent PAE of 10.7%. In this example, an inverse design is for synthesis of EM structures with the desired scattering properties in the global space of arbitrary fabricable EM structures on-chip and co- design with circuits. The design is aided by a deep learning-based forward model, which acts as a robust and accurate predictor of scattering parameters of arbitrary structures, thereby eliminating time- and resource-intensive EM simulations for inverse synthesis. The tradeoff space is discussed, along with several design examples for mm-Wave PA designs. A 30–94- GHz Psat, 3 dB bandwidth PA with an inverse-designed OMNis is demonstrated, capable of concurrent multi-band modulation. As shown at FIG. 17, this example achieves one of the highest Psat, 3 dB bandwidth. While there cannot be a claim of global optimality in such a highly non-convex space without exhaustive search, such algorithmic approaches can open up: 1) an untapped design space that can potentially exceed previous performance thanks to exponentially rich possible designs and 2) allow rapid and automated synthesis of complex RF- to-terahertz (THz) circuits and systems reducing labor-intensive optimization and human expertise. Example 2
Princeton - 102776 Millimeter-wave and terahertz integrated circuits and chips are expected to serve as the backbone for future wireless networks and high resolution sensing. However, design of these integrated circuits and chips can be quite complex, requiring years of human expertise, careful tailoring of hand crafted circuit topologies and co-design with parameterized and preselected templates of electromagnetic structures. These structures (radiative and non-radiative, single- port and multi-ports) are subsequently optimized through ad-hoc methods and parameter sweeps. Such bottom-up approaches with pre-selected regular topologies also fundamentally limit the design space. Here, a universal inverse design approach for arbitrary-shaped complex multi-port electromagnetic structures with designer radiative and scattering properties is demonstrated, co-designed with active circuits. To allow such universalization, deep learning based models are utilized, and synthesis is demonstrated with several examples of complex mm-Wave passive structures and end-to-end integrated mm-Wave broadband circuits. The presented inverse design methodology, that produces the designs in minutes, can be transformative in opening up a new, previously inaccessible design space. In the last two decades, there has been tremendous progress in radiofrequency, millimeter-wave, and sub-terahertz integrated circuits and systems, demonstrating complex phased arrays and multi-input, multioutput (MIMO) arrays with chip-scale systems, enabling new applications in radar, autonomous driving, 5 G , gesture recognition, localization and more. Design of these chips involves a series of complex iterative design processes constituting of co-design and optimization of integrated active circuit elements and passive electromagnetic (EM) structures. Every circuit in the high-frequency path of RF/sub-THz integrated chips (such as low noise and variable gain amplifiers, mixers for frequency translation, filters, signal distribution networks, quadrature signal generation, power amplifiers) is a combination of active and passive circuits. The passive structures include one-port structures such as antennas; two-port structures such as matching networks, filters; three-port structures such as diplexers, combiners, splitters, switches; four-port structures such as couplers for quadrature signal generation, baluns for differential signal generations; and higher order networks such as beamforming structures with lens-like properties. Design processes of these circuits, active and passive, radiative and non-radiative, single-port or multi-port, are quite distinct from each other as they have different pre-selected templates with regular geometries. These design approaches have historically relied on intuition-based standard templates with a finite set of parameters. These are subsequently optimized with time-consuming parameter sweeps, ad-hoc population- based meta heuristic optimization methods, or with machine-learning-based surrogate models. The co-design process essentially aims to tailor the electromagnetic structures and the active
Princeton - 102776 circuit topologies to optimize the collective response of the end-to-end circuit. The complexity of the design space makes exhaustive optimization non-tenable. Currently, there exists no universal and generalized approach to synthesis for such complex high-frequency circuits. The bottom-up iterative design approaches with pre-fixed templates, hand-crafted designs, and ad-hoc parameter sweeps have several limitations. First, by choosing a template, we have already narrowed down the design and trade-off space. It is not clear at all that the end result could be close to being optimal in terms of efficiency, compactness, spectral range coverage, or other performance parameters. It can be noted that most of these designed structures are typically far from the fundamental limits (such as Bode- Fano) due to the losses associated with energy storage elements. Therefore, it is quite possible that an entire design space that lies outside the scope of these regular template-based geometries can yield significant improvement over traditional solutions. Second, the design process is time and resource intensive. Sometimes design is left to the domain expert, dominated by rules of thumb and trial-error. Otherwise, iterative optimization with complex electromagnetic simulations is needed. Moreover, classical approaches with machine-learning-based surrogates, are only limited to one particular type of functional block with a pre-defined topology. Third, due to the nature of simplicity, these geometries have limited functionalities. For example, design of symmetrical power dividers with amplitude and phase equality operating at a given frequency is known with a rule-based approach. However, there are no set rules to engineer broader bandwidth or allowing unequal power division with spectrally dependent phase relationship across frequencies. Nevertheless, as an example, for the same number of reactive elements, it has been shown that asymmetric circuits relax the trade-offs of classical symmetrical topologies, allowing higher efficiency operation over much broader bandwidths when combined with power amplifiers. In this regard, new sub-THz/THz sensing and communication architectures emerge through such innovative design methods, that require new insights into particular applications. Here, a generalized approach for synthesis of arbitrary-shaped planar multi-port RF/sub-THz EM structures in minutes, with arbitrary scattering and radiative properties, integrated on chip or realized off-chip in package, is presented. The approach to search this extensively large design space is made possible by deep learning-based forward modeling that allows us to eliminate time and resource intensive EM simulation runs, but is also robust enough to capture the complexities of scattering and radiating properties of arbitrary multiport geometries. When co-designed with circuits, this can open up a path towards automated
Princeton - 102776 synthesis of end-to-end RF/sub-THz ICs, potentially achieving performance beyond the capabilities of state of the art template-based circuits. Prior works in nanophotonics have demonstrated the class of inverse methods for specific dielectric-based passive structures through gradient based optimizations such as adjoint method. In the photonics domain, the inverse-designed devices are mostly passive structures with standard waveguide interfaces that achieve a certain functionality with a tailored frequency response. The key difference in mm-Wave systems is the presence of active devices. Multiple active devices interact closely with passive structures, and also with each other to create functionalities such as impedance modulation, frequency synthesis, up/down- conversion, and so on. The complex impedances of these devices, their ability to amplify, and their mutual interactions need to be taken into account to realize an optimized end-to-end design process. Therefore, the passives are not designed separately as functional blocks, but collectively with the integrated devices-the latter may number from tens to tens of thousands depending on the complexity of circuits that are being realized, all of which can still be integrated on-chip. Second, the RF/sub-THz passive structures mostly support quasi-TEM modes with metal surfaces (realized in the back-end-of-the-line metal layers inside the chip) sandwiched within the chip dielectric. The losses per wavelength are much higher compared to photonics, and area minimization is a critically important consideration. Building robust model that captures the loss factors accurately, at the resolution of each pixel ( ∼ ^^/100 ) is also very challenging. When compared to low-frequency RF, on the other hand, it can be pointed out that at sub-THz and THz frequencies, the passives are small enough that they can be integrated on-chip directly with the active devices, allowing these complex systems to be fully integrated. These new opportunities as well as new challenges, can be addressed through Al-aided design tools. When compared to more restrictive cases in prior works, this work presents a generalized method to explore the large design space of arbitrary multi-port EM structures and circuits, that allows curated functionalities and spectral properties. To navigate the space of arbitrary-shaped multi-port RF/sub-THz EM structures and designer specified properties, we utilize a set of deep learning-based models that predict accurately both scattering and radiative properties of arbitrary planar EM structures. This is a non-trivial exercise given that for an arbitrary pixelated structure, as represented in FIGS. 18 and 19 with a 25 ൈ 25 grid, the design space covers 2^ଶହ ^ 10^଼଼ possible designs. Utilizing these models, a generalized synthesis approach is explored across several single and multi-port structures (including filters, resonators, power splitters, combiners, quadrature hybrids,
Princeton - 102776 diplexers, and antennas) and end-to-end mm-Wave circuits. This is the first step towards AI- enabled inverse design approach of complex RF/sub-THz circuits, which would open up a new design space by removing the constraints of template geometries, designer experience, trial and error, and ad-hoc optimization methods. Deep convolutional neural network as arbitrary scattering and radiative multi-port EM emulator The conceptual EM emulator of the N-port arbitrary EM structure is represented in FIG. 20. The input is the arbitrary geometry with arbitrary port placements, and the output is the predicted multi-port scattering and radiative properties, including complex S-parameters and radiation patterns. The relationship between the arbitrary planar geometry to the EM behavior is complex and nonlinear, and while multi layer perceptrons can act as a universal nonlinear mapper, simple application of such networks may result in overfitting due to the vastly under- sampled training data set. To this end, in order to justify the selection of convolutional neural networks, we rely on physics-based intuition. The effective scattering and radiating properties of these EM structures are effectively determined by the spatial distribution of electrical and magnetic near-fields and the boundary conditions. To capture these spatially distributed properties within the EM structure, several layers of convolutional filters of varying sizes, are applied to the image of the structure. Since convolution operation is shift invariant, relative location of features are captured within the structure. An example feature would be the distance between excitation ports and AC shorting locations. Even if ports are placed at different locations within the structure, sliding convolutional filters can still extract relative placement of other features. Such geometric properties of convolutional networks supplement their applicability to physics problems where structures with equivalent geometries are frequently encountered. In addition, through training, the network can learn to extract the relevant features, which is in contrast to classical machine learning tools where a set of pre-defined and pre-extracted features are utilized. It should be noted that since deep learning-based synthesis approach aims to move away from intuition-based parameterized designs, providing a set of pre-determined features would be counteracting this goal. To enable a widely generalizing model of such arbitrary structures, we pay close attention to the training dataset and optimal CNN architecture. CNN architecture and technology portability The goal here is to perform efficient training through the use of transfer learning which is applicable to multiple different scenarios such as different process nodes in chips, frequency sets, different dielectric stacks and packaging. Generalizability and model accuracy are key
Princeton - 102776 training targets to ensure robust inverse synthesis. As different CNNs are trained for distinct datasets (across process nodes, bounding box size, and pixel number), an architecture optimization would naturally give rise to different CNN hyper-parameters. See Table 2. Table 2. Training sets corresponding to distinct CNNs Freque Num Aft Traini Num ncy Range ber of er Data ng- ber of
Princeton - 102776 Quadra ture is
his, the first 10 to 15 layers of the CNN are convolutional layers with progressively reducing filter sizes. Each convolutional layer has 64 learnable filters followed by batch normalization (batch- norm) and leaky Rectified Linear Unit (ReLU). While batch-norm allows for smoother training for deep neural networks, leaky ReLU avoid vanishing gradients. The output of final convolutional layer is flattened and fed to fully connected (FC) layers. The number of FC layers are considerably less than convolutional layers (5-6). FC layers are followed by batch-norm, leaky ReLU, and dropout layers. During the training, dropout layers randomly shut down a fixed percentage of neurons, which mitigates overfitting. Moreover, during these training runs, periodic validations (after each epoch) were performed to end the training early if the validation error is no longer improving. ADAM optimizer was used for conducting training runs. Here, it is desired to limit real and imaginary part of predicted S-Parameters to the range of ^െ1,1^, as passive structures are being considered. Even then, small numerical errors in the prediction results can violate passivity condition. After the prediction, passivity violations are corrected with minimum perturbations. Finally, test set error and training set error was compared to ensure that generalization ability of the CNN is not impaired. The final architecture is then picked based on the test set error. Since this example is interested in integrated RF/sub-THz circuits, A CNN is trained that predict the spectral response of electromagnetic structures realized on-chip, across 24 െ 100GHz. This predictor model is the critical component that allows us to engineer EM structure spectral response. To allow efficient training, transfer learning is utilized to optimize prediction accuracy by using a large number of inexpensive, low accuracy simulations as the initial training step. This initial training result is then re-trained (or tuned) with a smaller number of higher accuracy simulations. The scalability of the model in predicting performance across varying bounding box sizes and frequency can also be ensured efficiently through transfer learning. In addition, data augmentation is applied, using geometric transformations (such as flipping the EM structure upside down) to reduce required number of simulations. Once the data has been generated, training takes a few hours (2-3 with contemporary GPUs), and inverse synthesis takes a few minutes starting from the desired parameters.
Princeton - 102776 Antenna Datasets and Training Strategy The information of antenna performance in air dielectric can be utilized to create robust models for scaled antennas embedded in dielectric through transfer learning. Intuitive reasoning for the transfer learning flow was described in the previous section. This is critically important since, simulations with high accuracy can require 10 times more resources (due to required mesh density). Moreover, a one-time air simulation model can be utilized to create rapid antenna predictor models for different dielectrics through transfer learning. Due to its rich representation, CNN trained with mm-Wave probe-fed antenna dataset was used as the initial point for transfer learning of other cases. In parallel to this, we refer to this CNN as 'framework network'. The initial training time for the proposed framework model is around 3 hours on a single A100 GPU. Model was trained for 60 epochs. Details for the training was provided at Table 2. Since other s^^ models were initialized from an already converged CNN, their training requires less data. As it can be seen, the input to the network is the 12 x 12 matrix representing the antenna structure and feed location. The output of the forward models is the return loss sampled at 81 frequency points for the mm-Wave antennas. Radiation patterns of mm-Wave antennas were sampled at 11 frequency points. Models were trained to predict ^^ ൌ 0 and ^^ ൌ 90 cuts. Each cut sampled from ^^ ൌ 90 to ^^ ൌ െ90 over 72 points. This totals to 72 x 2 for each frequency point and 1584 points over the entire range of frequencies. Frequency responses predicted by the emulator will depend on the statistical distribution of frequency bins. Contradicting this point, emulator is expected to perform well at predicting minority cases, which is impedance matching below -10 dB at multiple frequency bins. Due to randomized dataset generation, good matching and radiation pattern properties at lower frequencies (where the antenna is more compact) are a lot less common. This could result in CNN getting stuck at a local optimum during training. To be precise, CNN model can converge while ignoring the minority samples. This necessitates to construct a dataset that has good representation for minority cases. For this purpose, informed dataset generation was used to skew statistical properties of s^^ values towards well-matched cases. A semi-optimization during the dataset generation can populate the dataset with larger number of optimum samples. Hereby, search rules similar to genetic algorithm were applied to search for more optimal antenna structures. Description of this search algorithm can be summarized as following. 1. Step I: Calculate the costs of the entire dataset for the frequency bins of interest, i.e. 20 to 25 GHz where antenna structures are compact.
Princeton - 102776 2. Step II: Perform an exponentially distributed sampling by giving samples with lower cost values more probability of being chosen. 3. Step III: Explore the vicinity of selected samples (by introducing random mutations) or perform cross over between different samples to generate a new sample. The framework network for probe feed antenna dataset consists of a total of 560 K random samples and 500 K informed samples. After geometric transformations, the dataset is split as 10: 1: 1 between the training, validation, and testing samples, respectively. To obtain the optimal network with the appropriate number of convolutional and fully connected layers, we sweep the hyper-parameter space. This finalized network is then utilized as the starting point for the remaining antenna predictors. CNN based forward model is demonstrated for prediction of scattering parameters of arbitrary shaped passive EM structures, and an approach to inverse synthesis utilizing this forward model. One can use this forward model with a generative AI framework or with RL approach for synthesis. Both of these methods relate to the design, and not to the modeling aspect of it, and therefore, can be used with the CNN-based forward model. In order to provide a benchmark, an inverse neural network was created that aims to instantly synthesize a structure that can approximate inputted S-Parameters. As the test case, 2-port structures within the frequency range of 30 – 100 GHz is considered. Training of inverse network is done with the help of forward predictor model. Input to the inverse network is real and imaginary parts of S-Matrix elements. Input layer is followed by 6 fully connected layers. While activation function is leaky ReLU in other layers, in order to perform binary thresholding, last layer activation function is tanh ^^^^^^ ^ 0.5. Here, parameter ^^ is increased after each 10 epochs. Output of this layer (a vector of length 324) is reshaped to 18 x 18 matrix, which is the proper input format to the forward model. Forward model prediction is carried out with respect to the 18 x 18 input. Difference between predicted S-Parameters and targeted S- Parameters is used in back-propagation. It should be noted that only the inverse network is updated while weights of the forward model is kept frozen. An important challenge is to avoid forward model getting 'tricked' by the generative inverse network. For example, generative network can output an 18 x 18 matrix consisting of non-discrete values, which is not representative of a pixelated grid type of structure. While a hard thresholding function can ensure that output of the inverse network is solely 1 s and 0 s, performing back-propagation proved to be unstable due to challenges of defining derivatives in such cases. Moreover, during the initial training steps, it is plausible to relax the binary output requirement in order to boost inverse network training. This was approached by
Princeton - 102776 changing the value of ^^ in the thresholding function at every 10 epochs, ^^ was assigned as 1 for the first 10 epoch and adjusted to 4 and 8 in later steps. On the other hand, during the deployment of inverse network, hard thresholding was performed at the intermediate output. While inverse network tries to generate a structure that attempts to approximate the desired S-Parameter input, the achieved results often fall short of the target values. In some examples S-Parameters of an actual synthesized structure with the deep learning enabled GA methods are used as input to the inverse network. The first example is a 70 GHz band stop filter, and latter two are 70 GHz band pass filters. In most of these synthesis methods, the achieved results fail to approximate the targeted values. These examples, while not being comprehensive, demonstrate that the inverse network may not be able to generalize well. In general, generative networks are a lot more data intensive to train and prone to mode collapses and training stagnation. It should be noted that generative AI models, such as transformers and diffusion models are notoriously more data and training intensive, but can generalize better and can be explored further. Genetic Algorithm Heuristic Mechanisms It may not be immediately evident that heuristic approach implemented with the GA can result in cost minimization. In fact, due to non-convex nature of the design space, one cannot guarantee that each and every offspring are going to have better cost value than the parents. However, crossover and mutation operations resemble the evolution process in nature. Thanks to this heuristic, the cost function gets better asymptotically. To give better justification, FIG. 26A shows a snapshot from the 50th iteration of an optimization loop (for a 2-port network). FIG. 26A plots cost value versus number of times a sample was chosen as parent.8 inherited members from the 49th iteration were also annotated. We can observe that there is a diverse parent selection. In addition, we see that from generation 49 to generation 50, several population members that outperform best 8 members of generation 49 were obtained and generation 51 will still have 4 members from generation 49 thanks to their success. FIG. 26B plots the distribution of the cost function for the 51st iteration with respect to the cost values of parents. Compared the 5.02 value reported at (a), minimum cost is now 4.81, indicating the success of heuristic approach. Interestingly, minimum cost function is achieved at the 2nd bin even though average cost of parents is not minimum. This showcases the importance of exploratory strategy. Example Cost Functions A set of example cost functions are detailed as the following.
Princeton - 102776 Filters ^^ ൌ ∑ே ^ୀ^ ^^^^^^^ ൈ |^^ଶ^^^^^| ^ ^^ଶ^^^^ ൈ ^1 െ |^^ଶ^^^^^|^ ଶ (1) For the filter design problem pass-band insertion loss and stop-band rejection are
. band pass and band stop behaviors can be implemented. Also, to allow for a transition band, band edges can be ignored by assigning ^^^^^^^ and ^^ଶ^^^^ to 0. Power Divider ^^ ൌ ∑ெ ^ୀ^ ^^^^^^^ ൈ ห^^∼ ଷଶ ^^^^ െ |^^∗ ଷଶ ^^^^|ห ^ ^^ଶ^^^^ ൈ ห^^∼ ଷ^ ^^^^ െ |^^∗ ଷ^ ^^^^|ห ^ ^^ଷ^^^^ ൈ ^Arg ^^య ∗ భ ^^^ ^య ∗ మ ^^^^ െ ^^^^^^^ ^ ^∼ ^^^ ∗ ^^^ ∼ ^^^ ^∗ ^^^
are assigned for the absolute value of sଷଶ and sଷ^ (denoted as ^^ଷ ∼ ^ and ^^ଷ ∼ ଶ ). Hence, the first 2 terms of the equation penalize the deviation from target The second consideration is the
phase difference between 2 output ports. The third term in penalizes the deviation from the frequency dependent target phase value, which is shown as ^^^^^^. Lastly, it is observed that adding the 4th term improves the amplitude balance since it is minimized by ensuring the ideal amplitude ratio. Any of the a^^i^ െ aସ^i^ could be assigned as 0 to ignore the corresponding specification or a range of frequencies. For example, in the case of filtering divider, amplitude and phase balance terms were ignored outside of the pass-band. FIGS. 18 and 19 demonstrate the robustness of the deep learning models in predicting the scattering parameters of arbitrary planar multi-port EM structures across the broad range of frequencies, either integrated or realized on package, radiative and nonradiative. Once trained, the model accurately predicts the properties of a one-port classical corner cut antenna realized on off-chip package (12 x 12 pixels within 2.4 x 2.4 mm). The figures also show accurate prediction of classical two-port filters realized with transmission lines with multiple stubs (16 x 16 pixels within 300 x 300 µm), and an on-chip randomly generated 3-port pixelated network (25 x 25 pixels within 400 x 400 µm). For antennas, the predicted radiation patterns also match closely to the EM simulations. See FIG. 21. Considering the average prediction errors for three-port EM structures (25 x 25 pixels for 400 x 400 µm structures) across 24, 50, and 80 GHz demonstrates robust predictions. Here, prediction error was calculated as |s ᇱ െ s| where s ' is the predicted S-Parameter. Scatter plots of the simulated and predicted S-
Princeton - 102776 parameters (real and imaginary) at the selected frequency of 50 GHz show that, for different scattering parameters terms, the heat map is tightly concentrated around the ^^ ൌ ^^ diagonal, indicating good correlation between prediction and simulation results. These results show a remarkably low average error in prediction ability in the design space of arbitrary structures. Generalized inverse synthesis Utilizing the deep learning enabled robust EM emulator that eliminates the need for time and resource intensive EM simulations, one can now conceive of rapid synthesis in the large design space of arbitrary structures and circuits with optimization algorithms. See FIG. 20. This can either be achieved through heuristic algorithms such as genetic algorithms (GA), simulated annealing or generative AI tools such as auto-encoders or tandem neural networks. One can use the predictive model with a generative AI framework or with RL. Starting from a random structure, quick convergence to candidate solutions can be achieved through the evolution process. In this aspect, hyper-parameter choice for the GA is essential for ensuring exploratory strategy of the optimization loop. Firstly, we use a tournament selection method to determine the parents of each member of the next generation. For a typical population size of M ൌ 4096, a tournament size of N ൌ 256 is adopted. As a result, population members with above average success have a decent chance for being selected as the parent. Upon parent selection and crossing over, each pixel has a mutation probability of flipping. This probability starts from 0.1 and vanishes to 0 towards the end of total number of iterations (100 in this example). In addition, the best A ൌ 8 members of the current population are directly transferred to the next generation, guaranteeing a monotonous cost value reduction trajectory. FIGS. 22A-22E illustrate with several examples robust generalized inverse synthesis approach performed from single port antennas to multi-port EM structures based on the targeted scattering parameters; including one-port multi-band antennas, two-port bandpass filters, three-port power dividers with unique phase relationship, four-port resonant quadrature hybrid and three-port frequency diplexers. FIG. 22A shows a packaged multi-band antenna (matched at 25 and 28 GHz) implemented in a compact area. FIG.22B shows the synthesis of a band-pass two-port filter designed for 50 െ 60GHz, compressed to a size of ^^/10 ൈ ^^/10 synthesized in a few minutes. For three ports, two complex design examples are demonstrated. In FIG. 22C, the input signal gets equally power split between two paths and maintain a 90∘ phase difference between the two paths at 60 GHz. The evolved EM structure achieves this almost exactly with nearly 0 dB amplitude difference, 1 dB insertion loss and 88∘ phase
Princeton - 102776 difference. In the second 3-port structure (FIG. 22E), a diplexer is synthesized that splits the single input into two paths depending on the frequency (similar to a wavelength division multiplexer). The lower band is 24 െ 40GHz and the upper band is 60 െ 80GHz. It can be seen that the method successfully achieves a diplexer with 2 െ 4 dB insertion loss, and adjacent channel leakage less than -10 dB between 30 െ 80GHz, within an implemented area of ^^/6 ൈ ^^/6 at the crossover frequency of 50 GHz. FIG. 22D shows a 4-port EM structure with quadrature symmetry implementing a hybrid coupler. The initial structure evolves to the final one to approach the desired performance. For example, the frequency diplexer circuit evolves across optimization steps. The two frequency slices show the desirable performance where the signal flows to port 1 at 37GHz^^^ଷ^ ൌ 1, ^^ଷଶ ൌ 0^, and it flips at 80GHz^^^ଷ^ ൌ 0, ^^ଷଶ ൌ 1 ). One can observe that initially most of the population is spread randomly, and as the generations evolve, the structures start to move towards the target points, finally leading to the structure that approaches the desired performance. Generalization ability of CNN The EM emulator model needs to be able to predict structures that are not present in the dataset, such as those that can outperform structures present in the training data set. This is necessary for generalized inverse synthesis method. The inverse synthesis can tailor the structure to engineer the spectral response, distinctly different from what it has been trained with. The figure compares the results of structure that emerged through inverse synthesis and the best results from the training data for 2 different targets. For example, the method can be used to synthesize a broadband uneven power divider that aims a power splitting ratio of 75% െ 25% across the bandwidth from 24 to 80 GHz. The inverse design solution achieves better bandwidth, amplitude balance, and total transmission across 24 െ 80GHz, when compared to the best result from the training data. This demonstrates that the CNN model is not a lookup table, but is able to extract the relevant features that can robustly predict the scattering parameters of arbitrary structures, allowing the ability to search through this unexplored space during synthesis. The method was also used to synthesize a filtering power divider that performs equal power split and band pass filtering in each of the signal paths. Compared to the sample from the dataset, inverse design solution can provide a better trade- off between passband and stop-band requirements, while achieving near perfect amplitude and phase balance.
Princeton - 102776 Furthermore, it can be seen that the initial computational cost for training dataset generation can be amortized very quickly by utilizing inverse design for different design goals. In one example, when compared to the traditional EM simulation-based metaheuristic methods, CNN aided inverse design can reduce design time to minutes from weeks, significantly reducing time and resource for synthesis. Moreover, once a model is trained it could be re- utilized for different design targets, which is in contrast to EM-based optimization where a new set of simulations needs to be performed at for different set of goals. Measurement results To validate the generalized synthesis approach, several examples of electromagnetic structures for mm-Wave frequencies were designed, fabricated, and measured as well as an end-to-end broadband mm-Wave amplifier. Chips are fabricated using a 90 െ nm BiCMOS process. Verifications of inverse synthesis results were carried out via HFSS. For the example amplifier circuit, a full die EM simulation was performed with EMX. On-wafer S-Parameter measurements are carried out with 67 GHz Anritsu VNA (with 110 GHz extension modules) using 110 GHz and 67 GHz GSG probes (Infinity and Z-Probe respectively). VNA calibrations were performed with calibration substrates of individual probe pairs. Antenna impedances were measured in a similar manner, albeit with a coaxial cable compatible calibration kit. Antenna radiation pattern verification was performed through over the air far field measurement using a 2-axis rotational hardware. For the multi-port measurement of quadrature coupler, 50Ω on chip termination loads were de-embedded. The results of two-port band-pass filter structures are first presented, and the effect of the size of the structure, number of pixels, and the results with different optimization algorithms, such as genetic algorithm (GA) and binary particle swam optimization (BPSO) are explored experimentally. Size. The inverse synthesis can be guided to search for a compact structure that achieves a desired spectral response. Ultimately, this is limited by Maxwell's laws. An electromagnetic structure effectively distributes electrical and magnetic energy within the structure to synthesize any spectral response. Therefore, the structure needs to be comparable to the wavelength to allow resonant-type behavior. This was shown experimentally, scaling the structure from ^^/12 ൈ ^^/12 (200 µm in each side) to ^^/5 ൈ ^^/5 (500 µm in each side) across three step sizes, all implemented in the BiCMOS process. The spectral response was seen to
Princeton - 102776 make broadband (covering 50 െ 70GHz ) with flatter passband responses, due to more effective containment of the energy at 60 GHz with less leakage. Number of pixels. To approach the nearly arbitrary dimensions (that allow maximum design freedom), the number of pixels has to be increased. However, this creates a trade-off between the training time and achieved performance. Here, the EM structure size was fixed and the cost minimization was observed for different pixel combinations. A 300 x 300 µm area was divided into 10 x 10, 12 x 12, and 16 x 16 pixels and optimized for 50-70 GHz band-pass filter response. While in this example, all of the three levels of discretization produce generally similar spectral response, in other more complex circuits, it is understandable that the additional degrees of freedom achieved via larger number of pixels can provide better performance. Synthesis algorithm. The Inverse synthesis approach that is presented here is best suited to discrete valued optimization algorithms. FIG. 23 illustrates one such comparison for an 80 GHz band-pass filter with 200 ൈ 200^^ m area, synthesized through a GA and a BPSO algorithm. Due to the non-convexity of the space, the different spectral response can be expected as seen in the measurement results. Multi-port structures. Inverse synthesis of multi-port electromagnetic structure was shown through a four-port quadrature hybrid realized on-chip. It is noteworthy that, by imposing quadrature symmetry on 4-port networks, a hybrid coupler can be designed by just satisfying |^^ଶ^| ൌ |^^ଷ^| ൌ 0.707. As can be seen in the figures, the hybrid demonstrates nearly equal power through the coupled and through ports ( ^ െ4 dB ) between 70-80 GHz. Antennas. mm-Wave antennas were synthesized with edge feed and probe feed configurations. These antennas are realized with standard PCB manufacturing. The edge feed configuration is a highly compact antenna occupying an area of ^^ଶ/9, compared to a typical patch antenna area of ^^ଶ/4, resulting in a 2.25x smaller area. The probe feed configuration demonstrates a dual-band design operating at 25 and 28 GHz. In addition, antenna radiation patterns are characterized at resonance frequency for the dominant polarization. Generalized multi-port inverse synthesis for co-design with millimeter-wave circuits. The examples discussed previously in this example assume standard 50Ω input and output impedance. However, for integrated RF/sub-THz circuits, when these multi-port structures are driven by active devices, the port impedances are frequency dependent and complex valued. An example of a multistage broadband mm-Wave amplifier in 90 െ nm SiGe-based BiCMOS process is provided that exploits the asymmetrical irregular 3-port structures (synthesized through the generalized inverse synthesis method).
Princeton - 102776 FIG.24. shows the schematic and overall principle of operation of the broadband mm- Wave amplifier. Unlike a traditional symmetrical amplifier, the input signal splits asymmetrically into two paths with a frequency-dependent non-zero phase difference. The signal in each branch passes through two stages of amplifier cells, and then combines in phase through a second 3-port asymmetrical structure. To compensate for the phase difference between the input paths ( ^Φ ), the output combiner is also designed to create an opposite phase relationship ( െΦ ), such that the signals combine in phase optimally at the output. It was shown that ideal phase difference for the output combiner between each branch is
provided by the Furthermore, the synthesis process aims to minimize the insertion loss and amplitude imbalances. Insertion loss at the output is between 0.7 - 1.2 dB across 24- 40 GHz. In the same frequency range, input matching network shows 1.5 - 4 dB insertion loss with an amplitude imbalance less than 1.3 dB. Lastly, since the ports are not isolated, active elements do interact with each other. This frequency-dependent interaction is the key to broadband impedance synthesis. As a result, amplifier cells at stage-2 are provided with optimal impedances for high gain. FIG. 25 shows measured and simulated S-parameters. A measured gain of 17.5-14.5 dB is obtained between 23.6 and 37.3 GHz, which covers most of the commercial 5G bands. The asymmetrical paths through the inverse-designed divider and combiner are shown in the figure. The RF short location of the output combiner provides the Vେେ connection via a bypass capacitor array. As is evident, the inverse synthesis approach allows us to explore this class of complex structures with the tailored scattering parameters necessary for the circuit functionality. Going beyond the traditional matching networks, the proposed method provides access to a new class of complex multi-port electromagnetic structures. Co-designing these EM structures with the circuits can allow untapped performance metrics and functionalities. Here, a generalized multi-port inverse design method for synthesis of complex integrated electromagnetic structures codesigned with RF/sub-THz circuits is presented. This approach that captures both radiative and non-radiative behavior, and multiple ports that can be excited or tapped with active devices, varying excitations, and boundary conditions, is critical for tapping a new design space of arbitrary geometric structures and circuits. This is a significant advancement to inverse synthesis of passive or simple two-port structures. This is enabled through a deep learning-based forward model that acts as a robust and accurate predictor of scattering parameters of arbitrary structures, thereby eliminating time and resource intensive EM simulations for inverse synthesis. Furthermore, it is illustrated that the forward
Princeton - 102776 model can demonstrate generalization and learning ability. The transfer learning approach allows quick portability to different dielectric stacks, bounding box size or frequency range. In this example, the AI-model and the synthesis approach determines this by itself. As shown in FIG. 27, the synthesis generates the asymmetric input and output networks while matching the total collective phase shifts between the paths. FIG. 27 provides detailed schematic of the multi-port mm-Wave amplifier that was implemented in 90 nm BiCMOS process. For the optimization of output combiner, frequency dependent RC values corresponding to optimum termination impedances were calculated and stored in a lookup table. During the optimization, these parasitic capacitances (arising from the device and output pad) were lumped to the CNN predicted S-Parameters of the pixelated structure. Total transmissivity across the target frequencies were then calculated by assuming optimum phase and equal amplitude excitation. As a secondary objective, presence of a DC path between the input ports and AC shorting location was ensured. Features of any of the examples or embodiments outlined above may be combined to create additional examples or embodiments without losing the intended effect. It should be understood that the description of an embodiment or example provided above is by way of example only, and various modifications could be made by one skilled in the art. Furthermore, one skilled in the art will recognize that numerous further modifications and combinations of various aspects are possible. Accordingly, the described aspects are intended to encompass all such alterations, modifications, and variations that fall within the scope of the appended claims.