WO2024044935A1

WO2024044935A1 - Method and apparatus for structural optimizition

Info

Publication number: WO2024044935A1
Application number: PCT/CN2022/115720
Authority: WO
Inventors: Jun Zhu; Songming LIU; Zhongkai HAO; Chengyang YING; Hang SU; Ze CHENG
Original assignee: Robert Bosch Gmbh; Tsinghua University
Priority date: 2022-08-30
Filing date: 2022-08-30
Publication date: 2024-03-07

Abstract

The present disclosure provides computer implemented method for optimizing structure of an object. The method comprises: receiving structural quantities by a hard-constraint neural network (NN) framework, wherein the structural quantities are used to describe boundaries of the object, wherein the hard-constraint NN framework comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system related to the object with physical quantities of the object; generating a first prediction of solutions of reformulated PDEs by the at least one primary NN; generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN; obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit; and updating the structural quantities representing the boundaries of the object based on the physical quantities of the final prediction.

Description

METHOD AND APPARATUS FOR STRUCTURAL OPTIMIZITION

FIELD

Aspects of the present disclosure relate generally to artificial intelligence (AI) , and more particularly, to optimizing structure of an object.

BACKGROUND

The structure of an object, such as its shape, size, or distribution of materials, influences the performance of the object. Examples of the object may be the wing of an airplane, the truss of a house, the pipes in a reactor, or the like. For example, the shape of an airfoil influences the pressure and velocity of the airflow with respect to the airfoil while the pressure and velocity influencing the performance of the physical system of the airfoil. The physical system can be described by partial differential equations (PDEs) . The physical quantities of the physical system such as the pressure and velocity can be obtained by solving the PDEs based on the structural quantities such as those describing the shape of the object.

It is desirable to design the structure of an object so as to improve its performance, this procedure may be referred to as structural optimization. The structural optimization is applicable in many areas including scientific area, engineering area, industrial area or the like. For example, the shape of chemical catalyst pellets, the shape of auto parts or the like may be optimized through the structural optimization procedure before manufacturing.

Since most physical systems are described by partial differential equations (PDEs) , structural optimization is typically carried out with the governing PDEs, which need to be solved to determine the state of the structure at each iteration of the optimization. The efficiency and accuracy of the solving of PDEs, especially for an object with complex structure, would be critical for improving the performance of the structural optimization of the object to be manufactured.

SUMMARY

In order to improve the performance of the structural optimization, the disclosure proposes a novel framework for structural optimization and solving of PDEs, by which the time consumption and computation requirement may be reduced and accuracy of the solution of PDEs and accordingly performance of structural optimization may be improved.

According to an embodiment, there provides a computer implemented method for optimizing structure of an object. The method comprises: receiving structural quantities by a hard-constraint neural network (NN) framework, wherein the structural quantities are used to describe boundaries of the object, wherein the hard-constraint NN framework comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system related to the object with physical quantities of the object; generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, the reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object; generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object; obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit, wherein the final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object; and updating the structural quantities representing the boundaries of the object based on the physical quantities of the final prediction.

According to an embodiment, there provides a computer implemented method for predicting physical quantities of an object. The method comprises: receiving structural quantities by a hard-constraint neural network (NN) framework, wherein the structural quantities are used to describe boundaries of the object, wherein the hard-constraint NN framework comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system related to the object with physical quantities of the object; generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, the reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object; generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object; obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit, wherein the final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object.

According to an embodiment, there provides a computer system, which comprises one or more processors and one or more storage devices storing computer- executable instructions that, when executed, cause the one or more processors to perform the operations of the method as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

According to an embodiment, there provides one or more computer readable storage media storing computer-executable instructions that, when executed, cause one or more processors to perform the operations of the method as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

According to an embodiment, there provides a computer program product comprising computer-executable instructions that, when executed, cause one or more processors to perform the operations of the method as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

By using the HCNN framework in the structural optimization framework, the efficiency of structural optimization can be improved while the bottleneck of solving PDEs being resolved. Moreover, by using the HCNN framework, most commonly used BCs (i.e., Dirichlet, Neumann and Robin BCs) can be automatically satisfied during the training of the NNs, therefore the model can be trained without the loss of these BCs, this alleviates the unbalanced competition between the loss terms of PDEs and BCs, and significantly improves the performance of solving geometrically complex PDEs, and accordingly significantly improves the performance of structure of an object. Other advantages and enhancements are explained in the description hereafter.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosed aspects will hereinafter be described in connection with the appended drawings that are provided to illustrate and not to limit the disclosed aspects.

Fig. 1 illustrates an exemplary framework for optimizing structure of an object according to aspects of the disclosure.

Fig. 2 illustrates three types of most commonly used BCs according to aspects of the disclosure.

Fig. 3 illustrates an exemplary transformation of coordinates systems according to aspects of the disclosure.

Fig. 4 illustrates an exemplary structural optimization of 2D battery pack according to aspects of the disclosure.

Fig. 5 illustrates an exemplary architecture of HCNN according to aspects of the disclosure.

Fig. 6 illustrates an exemplary structural optimization of airfoil according to aspects of the disclosure.

Fig. 7 illustrates an exemplary architecture of HCNN according to aspects of the disclosure.

Fig. 8 illustrates an exemplary architecture of HCNN according to aspects of the disclosure.

Fig. 9 illustrates an exemplary process for optimizing structure of an object according to aspects of the disclosure.

Fig. 10 illustrates an exemplary process for predicting physical quantities of an object according to aspects of the disclosure

Fig. 11 illustrates an exemplary computing system according to aspects of the disclosure.

DETAILED DESCRIPTION

The present disclosure will now be discussed with reference to several example implementations. It is to be understood that these implementations are discussed only for enabling those skilled in the art to better understand and thus implement the embodiments of the present disclosure, rather than suggesting any limitations on the scope of the present disclosure.

Various embodiments will be described in detail with reference to the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. References made to particular examples and embodiments are for illustrative purposes, and are not intended to limit the scope of the disclosure.

The object to be structurally optimized in the example shown in Fig. 1 is an airfoil. As shown in block 120 of Fig. 1, the shape of the airfoil is presented by a boundary γ, which is parameterized by structural quantities θ. In this example, the structural quantities θ may be a set of control points which consist of splines representing the shape of the airfoil. In other words, the structure to be optimized is the shape of the airfoil represented by splines with a set of control points θ. The label Ω denotes the problem domain or particularly denotes the domain of the physical system related to the airfoil.

As shown in block 110 of Fig. 1, the arrowed lines denote airflows on the airfoil. The physical state of the airflow on the airfoil may be represented by physical quantities of the physical system related to the airfoil. In this example, the physical quantities may be the velocity u=u (x) and the pressure p=p (x) . The label x denotes spatial coordinates in the domain Ω.

The velocity u=u (x) and the pressure p=p (x) may be characterized by nonlinear PDEs of Navier–Stokes (NS) equations, with the constraint of the shape of the airfoil represented by the boundary γ. It is appreciated that NS equations are well known physical equations that can be used to describe the three-dimensional motion of viscous fluid substances. The NS equations are second-order nonlinear PDEs, and may be used to model the weather, ocean currents, heat conducting, air flow around an airfoil, water flow in a pipe or in a reactor and many other applications. The PDEs of the NS equations for a physical system of an object may be formulated according to physical laws related to the physical system of the object. In the illustrated example, the PDEs of the physical system of the airfoil may be established according to the related physical law as shown by label 140 of Fig. 1. As the PDEs are used to govern the optimization of the structure of the airfoil, they may be referred to as governing PDEs. It is appreciated that the formulation of the governing PDEs may be implemented with well-known physical knowledge, and other kinds of PDEs used to describe a physical system of an object may also be used as the governing PDEs in aspects of the disclosure.

As shown by label 130 of Fig. 1, the objective or goal of the structural optimization of the airfoil is to reach the desired pressure distribution p _ref on the airfoil surface by changing the structural parameter θ. The objective function J may be formulated as shown by label 130, and the aim of the structural optimization is to minimize the objective function J, which is a functional of the pressure p. The shape of the airfoil corresponding to the boundary γ parameterized by structural quantities θ may be iteratively optimized to minimize the objective function J, so as to reach the desired pressure distribution p _ref on the airfoil surface.

In each loop of the structural optimization, the PDEs 140 need to be solved to obtain the physical quantities such as the velocity u and pressure p, and then the objective function J may be evaluated based on the physical quantities such as the pressure p. Then the structural quantities θ may be updated or optimized based on the calculated objective function J as shown by label 160.

In an example, as shown by label 150, a NN model may be used to obtain the velocity u and pressure p by solving the PDEs 140. For example, Physical-informed neural network (PINN) (Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378: 686–707, 2019) is one of the most influential works for predicting the solutions of the PDEs and may be employed to implement the NN model 150. The PINN is trained in the way of taking the residuals of both the PDEs and the BCs as multiple terms of the loss function. However, there exists an unbalanced competition between the terms of PDEs and BCs, causing the computational requirement of the PINNs to be large and the accuracy of the PINNs to be decreased for solving PDEs with complex BCs. One way to address this problem is to embed BCs into the ansatz so as to enable the neural networks to automatically satisfy the BCs and no longer require adding corresponding loss terms. However, the challenge is that the equation forms of complex BCs such as the Neumann BCs and Robin BCs as illustrated in Fig. 2 have no analytical solutions in general and are thus difficult to be embedded into the ansatz.

Fig. 2 illustrates three types of most commonly used BCs according to aspects of the disclosure. In the illustrated example of heat transfer, T = T (x) is the temperature, k is the thermal conductivity, and h is the heat transfer coefficient. As illustrated in Fig. 2 (a) , the Dirichlet BC specifies the value of the solution of PDEs at the boundary. It is assumed a constant temperature T _H at the heat source (x = x ₀) . As illustrated in Fig. 2 (b) the Neumann BC specifies the value of the derivative at the boundary. It is assumed that the right wall (x = x ₁) is adiabatic and forces the derivative to be zero. As illustrated in Fig. 2 (c) the Robin BC is a combination of the Dirichlet BC and the Neumann BC. And it is used to describe the heat convection between the heat source and the environment (T _∞) at the surface (x = x ₂) .

In an embodiment, as illustrated in Fig. 1, a hard-constraint neural network (HCNN) framework 150 is employed in the structural optimization frame 100 to predict the solutions of the PDEs 140 with various boundary shapes represented by structural quantities θ. The HCNN 150 is a unified hard-constraint framework for all the three most commonly used BCs, i.e., Dirichlet, Neumann and Robin BCs. With HCNN, an ansatz can be constructed to automatically satisfy the three types of BCs. Therefore, the model can be trained without the losses of these BCs, which alleviates the unbalanced competition and significantly improves the accuracy of solving geometrically complex PDEs, and accordingly significantly improves the efficiency and performance of the structural optimization for the object such as the illustrated foil.

The PINN is introduced briefly in order to better understand the HCNN according to aspects of the disclosure. The following Laplace’s equation may be considered as an example,

Δu (x ₁, x ₂) =0, x ₁∈ (0, 1] , x ₂∈ [0, 1] (1a)

u (x ₁, x ₂) =g (x ₂) , x ₁=0, x ₂∈ [0, 1] (1b)

where Eq. (1a) gives the form of the PDE, and Eq. (1b) is a Dirichlet boundary condition (BC) . A solution to the above problem is a solution to Eq. (1a) which also satisfies Eq. (1b) .

PINNs employ a neural network NN (x ₁, x ₂; w) to approximate or predict the solution of the PDE, i.e.,

where w denotes the trainable parameters of the neural network. And the parameters w are learned by minimizing the following loss function:

where the first loss term

is configured to restrict the prediction

of the solution to satisfy the PDE (Eq. (1a) ) while the second loss term

is configured to restrict the prediction

of the solution to satisfy the BC (Eq. (1b) ) ,

is a set of N _f collocation points sampled from the domain [0, 1] ², and

is a set of N _b collocation points sampled from the boundary x ₁=0, x ₂∈ [0, 1] .

PINNs have a wide range of applications, including heat, flow, and atmosphere. However, PINNs are struggling with some issues on the performance. Previous analysis has demonstrated that the convergence of first loss term

can be significantly faster than that of the second loss term

This pathology may lead to nonphysical solutions which does not satisfy the BCs or initial conditions (ICs) . Moreover, for geometrically complex PDEs where the number of BCs is large, this problem is exacerbated and can seriously affect accuracy.

One potential approach to overcome this pathology is to embed the BCs into the ansatz in a way that any instance from the ansatz can automatically satisfy the BCs. In this way, the loss terms corresponding to the BCs are no longer needed, and thus the above pathology is alleviated. Such methods are called hard-constraint methods, and they share a similar formula of the ansatz as

where x is the coordinate, Ω is the domain of interest,

is the general solution at the boundary

and

is an extended distance function which satisfies

In the case of Eq. (1) (where x= (x ₁, x ₂) , Ω= [0, 1] ²) , the general solution is exactly g (x ₂) , and the following ansatz, which automatically satisfies the BC in Eq. (1b) , can be used.

However, it is hard to directly extend this method to more general cases of Robin BCs (see Eq. (7) ) , since it is difficult to obtain the general solution

analytically. Existing attempts are either mesh-dependent method, which are time-consuming for high-dimensional and geometrically complex PDEs, or ad hoc methods for specific (geometrically simple) physical systems. It is still lacking a unified hard-constraint framework for both geometrically complex PDEs and the most commonly used Dirichlet, Neumann, and Robin BCs.

A physical system governed by the following PDEs defined on a geometrically complex domain

is considered.

where

includes N PDE operators which map u to a function of x, u and its derivatives. Here, u (x) = (u ₁ (x) , …, u _n (x) ) is the solution to the PDEs, which represents physical quantities of interest. For each u _j, j=1, …, n, the suitable boundary conditions (BCs) may be posed as

where

are subsets of the boundary

whose closures are disjoint, a _i (x) and b _i (x) are two functions satisfying that

holds for

and n (x) = (n ₁ (x) , …, n _d (x) ) is the (outward facing) unit normal of corresponding boundary γ _i at point x. It is noted that Eq. (7) represents a Dirichlet BC if a _i (x) ≡1, b _i (x) ≡ 0, a Neumann BC if a _i (x) ≡0, b _i (x) ≡ 1, and a Robin BC otherwise.

For such geometrically complex PDEs, if PINNs are to be employed, there would be a difficult multi-task learning with at least

terms in the loss function. It will severely affect the convergence of the training due to the unbalanced competition between those loss terms. If the BCs can be embedded into the ansatz, every instance automatically satisfies the BCs. However, it is infeasible to directly follow the pipeline of hard-constraint methods (see Eq. (3) ) since Eq. (7) does not have a general solution of analytical form. Therefore, a new approach is needed to address this intractable problem.

According to aspects of the disclosure, the general solutions of the BCs are presented and used to construct the hard-constraint ansatz. Extra fields may be introduced to equivalently reformulate the PDEs. Let

which are substituted into Eq. (6) and Eq. (7) to obtain the equivalent PDEs

where (u (x) , p ₁ (x) , …, p _n (x) ) is the solution of the new PDEs,

are PDE operators after the reformulation. And for j=1, …, n, the corresponding BCs are as the following

By adding the extra fields shown in Eq. (8b) , Eq. (7) has been transformed into linear equations with respect to (u _j, p _j) , which are much easier to derive general solutions. Hereinafter, (u _j, p _j) is denoted by

and their general solutions at boundary γ _i (i.e.,

) are denoted by

According to aspects of the disclosure, a basis B (x) of the null space may be used to obtain the general solution of Eq. (9) , dimension of the null space is d. B (x) should be carefully chosen. Since Eq. (9) is parameterized by x, for any x∈γ _i, B (x) should always be a basis of the null space, that is, its columns cannot degenerate into linearly dependent vectors, otherwise it will not be able to represent all possible solutions.

Generally, for any dimension

it is non-trivial to find a simple expression for the basis. According to aspects of the disclosure, it is preferrable to find (d + 1) vectors in the null space, d of which are linearly independent (that way,

Then the general solution of

at the boundary γ _i may be constructed as

where

is a neural network with trainable parameters

and B (x) is given by

According to aspects of the disclosure, in the case of d = 1 or d = 2, a simpler expression for B (x) can be found. In an embodiment, in the case of d = 1, the basis of null space may be found as

where

In an embodiment, in the case of d = 2, the basis of null space may be found as

B (x) = [β ₁ (x) , β ₂ (x) , β ₃ (x) ] ,

where

With the parameterization of a neural network, Eq. (10) can represent any function in boundary γ _i, as long as the function satisfies the BC (see Eq. (9) ) . Since the problem domain contains multiple boundaries, the general solutions corresponding to each boundary γ _i may be combined to achieve an overall approximation or prediction. Hence, the ansatz may be constructed as follows

where NN _main:

is the main neural network with trainable parameters w _main.

are continuous extended distance functions (see Eq. (4) ) , and α _i (i=1, …, m _j) are determined by

where

is a hyper-parameter of the “hardness” in the spatial domain.

stands for the boundaries except γ _i. In Eq. (11) , extended distance functions are utilized to divide the problem domain into several parts, where

(

is its learnable part) is responsible for the approximation on the boundaries γ _i while NN _mainis responsible for the approximation of the internal apart from the boundaries. Furthermore, Eq. (12) ensures that the weight of

decays to

at the nearest neighbor of boundary γ _i, so that

does not interfere with the approximation on other boundaries.

According to aspects of the disclosure, if the parameters a _i, b _i, n or g _i in Eq. (10) and Eq. (11) are only defined at boundary γ _i, their definition can be extended to the domain

using interpolation or approximation via neural networks. The airfoil boundary (i.e, γ _af) is considered as an example. Supposing f (x) is only defined in airfoil boundary γ _af, the task is to extend its definition to

As shown in Fig. 3, two reference points (i.e., x ₀ and x ₁) are placed on the front and rear half of the airfoil. For any

it can be expressed as polar coordinates with respect to x ₀ and x ₁, respectively. The two polar coordinates are concatenated to form a new space. Next interpolation and approximation are performed under the new space. This is because in the new space it can better characterize the shape of the airfoil. It is appreciated that there are many ways for coordinate transformations, not limited to the example here. As for the interpolation, several points are sampled at the airfoil boundary γ _af to obtain the dataset

For any

the corresponding extended f (x) is generated by interpolating in the dataset. The interpolation method used here depends on the smoothness requirements of the ansatz. In addition, the number of reference points can also be changed, and in experiments it is found that only one reference point is enough.

Approximation via neural networks is a general method that does not require manual design. In this case, several points can be sampled at the airfoil boundary γ _af to construct the dataset

followed by training a neural network on the dataset, i.e. NN (φ ₀ (x ⁽ⁱ⁾ ) , φ ₁ (x ⁽ⁱ⁾ ) ) ≈f ⁽ⁱ⁾ . For any

NN(φ ₀ (x) , φ ₁ (x) ) is taken as the corresponding extended f (x) . According to aspects of the disclosure, the neural network can also be trained in the original space. However, experimental results show that training on the new space can achieve better results. The reason may be that the complex geometry become smoother and easier to learn in the new space.

It is worth noting that, in addition to the cases mentioned above, the extended distance functions l (x) (here the superscript is omitted and see Eq. (4) for its definition) may also be handled similarly. Because for the complex geometry, the distance function can be complex and it may be replaced with a cheap surrogate model. The methods are similar, including approximating the distance function with a neural network, or constructing splines function (Hailong Sheng and Chao Yang. Pfnn: A penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. Journal of Computational Physics, 428: 110085, 2021) .

Finally, the HCNN model can be trained with the following loss function

where

is defined in Eq. (11) and

is a set of collocation points sampled in the domain Ω. For neatness, the trainable parameters of neural networks are omitted here. The loss function of Eq. (13) measures the discrepancy of both the PDEs (i.e.,

) and the equilibrium equations introduced by the extra fields (i.e., Eq. (8b) ) at N _f collocation points.

According to Eq. (13) , BCs are successfully embedded into the ansatz, and it no longer needs to take the residuals of BCs as extra terms in the loss function as performed by PINN. That is, the proposed ansatz strictly conforms to BCs throughout the training process, greatly reducing the possibility of generating nonphysical solutions. Nevertheless, this comes at the cost of introducing (nd) additional equilibrium equations (see Eq. (8b) ) . But in many physical systems, especially those with complex geometries, the number of BCs (cnt (BCs) ) is far larger than (nd) (e.g., n = 3, d = 2, cnt (BCs) = 1260 for a classical physical system, a heat exchanger) . So aspects of the disclosure actually reduce the number of loss terms by an order of magnitude or two (Δcnt (losses) = nd -cnt (BCs) ＜＜ 0) , alleviating the unbalanced competition between loss terms.

According to aspects of the disclosure, the HCNN framework detailed above can be extended to the spatial-temporal domain. A physical system governed by the following time-dependent PDEs defined on a geometrically complex domain

is considered.

where t is the temporal coordinate, and the other notations are the same as those of Eq. (6) and Eq. (7) . For each u _j, j=1, …, n, the suitable boundary conditions (BCs) may be posed as

and an initial condition (IC)

u _j (x, 0) =f _j (x) , x∈Ω (16)

Following the pipeline described above with reference to Eq. (9) to Eq. (13) , the general solution of

at the boundary γ _i may be constructed as

where

is a neural network, and

The trainable parameters of neural networks are omitted for neatness.

Finally, the ansatz may be constructed as follows

where

is an intermediate variable that incorporates hard constraints in spatial dimensions, NN _main:

is the main neural network,

are extended distance functions (see Eq. (4) ) , α _i (i=1, …, m _j) is determined in Eq. (12) , and

is a hyper-parameter of the “hardness” in the temporal domain.

Fig. 4 is a schematic diagram illustrating structural optimization of a 2-dimentional (2D) battery pack according to aspects of the disclosure.

As shown in Fig. 4, the cell boundaries γ _c, i and cooling pipe boundaries γ _p, i located in domain Ω are the structure to be optimized. The goal of the structural optimization is to optimize the shape and position of the cells and cooling pipes so as to obtain an even distribution of the temperature over time.

The structural optimization problem may be represented by:

where θ denotes the structural parameters of the structure of the object to be optimized, which are also the parameters of the boundary shapes, Θ denotes the space of θ. The structural parameters θ is changing during the optimization of the structure but is within the design space Θ. J is the objective function whose value measures how good a given structure is, W stands for the constraints on both the structural parameters θ and the state variable s = s (x) which denotes the state of the physical system, for example, velocity, pressure, temperature or the like, where x denotes spatial coordinates. And the symbol “≤°” is an elementwise comparison.

The structural optimization problem for the 2D battery pack shown in Fig. 4 may be represented by

J (θ) =∫ _t∫ _Ω|T (x) -T _ref (x) |dx dt (20)

where T is the temperature, T _ref is the reference temperature.

The governing PDEs describing the physical system related to the 2D battery pack may be formulated according to physical laws, for example, the PDEs may be given by:

T (x, 0) =T ₀, x∈Ω, (21e)

where x= (x ₁, x ₂) is the spatial coordinate, t is the temporal coordinate. T (x, t) is the temperature over time, k is the thermal conductivity and is set to k = 1 in this example,

h is the heat transfer coefficient and is set to h = 1 in this example.

T _a, T _c, T _w are respectively the temperature of the air, the cells (n _c= 11 cells of radius r _c) , the cooling pipes (n _w = 6 pipes of radius r _c, ) , and are set to T _a = 0.1, T _c = 5, T _w = 1. T ₀ is the initial temperature and is set to T ₀ = 0.1 in this example. In Fig. 4, γ _ou stands for the outer boundaries of the battery packet, γ _c, i stands for the boundaries of the cells, γ _p, i stands for the boundaries of the cooling pipes.

The structural quantities include the center and the radius of cells and the cooling pipes. In order to optimize the structure of the 2D battery pack based on the Eq. (20) , the governing PDEs need to be solved to obtain the physical quantity, that is, the temperature T.

According to aspects of the disclosure, the PDEs of Eq. (21a) to Eq. (21e) are reformulated by adding extra fields according to the HCNN model. The introduced extra fields is p (x, t) shown in Eq. (22b) , and the reformulated PDEs are:

k (n (x) ·p (x, t) ) =h (T _a-T (x, t) ) , x∈γ _ou, t∈ (0, 1] , (22c)

k (n (x) ·p (x, t) ) =h (T _c-T (x, t) ) , x∈γ _c, i, t∈ (0, 1] , i=1, …, n _c, (22d)

k (n (x) ·p (x, t) ) =h (T _w-T (x, t) ) , x∈γ _p, i, t∈ (0, 1] , i=1, …, n _w, (22e)

T (x, 0) =T ₀, x∈Ω, (22f)

Since the solution of the PDEs or reformulated PDEs is a scalar function, that is, the temperature T (x, t) ) , the solution is denoted by u (x, t) =T (x, t) ) . Let

denote (u, p) , where p is the extra field shown in Eq. (22b) . The general solutions at boundaries γ _ou, γ _c, i, γ _p, i are derived respectively.

For any point at the outer boundary, that is, for x∈γ _ou, the coefficients of the BCs shown in Eq. (9) may be obtained from BC shown in Eq. (22c) , that is, a (x) =h, b (x) =k, g (x) =hT _a. According to Eq. (10) , the general solution

at the outer boundary is given by

where B (x) is given by Eq. (10a) or Eq. (10c) , with

For any point at the cell boundary, that is, for x∈γ _c, i, the coefficients of the BCs shown in Eq. (9) may be obtained from BC shown in Eq. (22d) , that is, a (x) =h, b (x) =k, g (x) =hT _c. According to Eq. (10) , the general solution

at the cell boundary is given by

For any point at the cooling pipe boundary, that is, for x∈γ _p, i, the coefficients of the BCs shown in Eq. (9) may be obtained from BC shown in Eq. (22e) , that is, a (x) =h, b (x) =k, g (x) =hT _w. According to Eq. (10) , the general solution

at the cooling pipe boundary is given by

The Extended Distance Function

in Eq. (11) may be implemented in various ways according to aspects of the disclosure. In an embodiment, for the cell boundary γ _c, i and the cooling pipe boundary γ _p, i, since they are 2D circles, the extended distance functions

and

can be chosen as the distance between x and the center minus the radius. For the rectangular outer boundary γ _ou, supposing that it is given by [a ₁, a ₂] × [b ₁, b ₂] , the extended distance function

can be constructed as follows

where x= (x ₁, x ₂) , SoftMin is a differentiable version of min function. In an implementation, the SoftMin is implemented by LogSumExp in PyTorch, i.e., SoftMin (y) = LogSumExp (-βy) / (-β) , β = 4. And the extended function

is computed by taking the SoftMin of the distances to all the boundaries

Fig. 5 illustrates an architecture of HCNN 500 for predicting PDE solutions of the 2D battery pack according to aspects of the disclosure.

The HCNN 500 includes a main-NN denoted as M-NN 510, and three sub-NNs denoted as S-NNs 520-1, 520-2 and 520-3. In an implementation, the main-NN is a multilayer perceptron (MLP) of size [3] + 4 × [50] + [3] , which means 3 inputs, 4 hidden layers of width 50, and 3 outputs. The three inputs are spatial coordinates x= (x ₁, x ₂) and time t, and the three outputs are temperature T and its two derivatives

The sub-NNs 520-1, 520-2 and 520-3 are MLPs of size [3] + 3 × [20] + [3] , which means 3 inputs, 3 hidden layers of width 20, and 3 outputs. The three inputs are spatial coordinates x= (x ₁, x ₂) and time t, and the three outputs are temperature T and its two derivatives

The main-NN 510 is responsible for the prediction of the PDE solutions in the out-of-boundary region of the battery pack, and the sub-NNs 520-1, 520-2 and 520-3 are respectively responsible for the prediction of the PDE solutions at the BCs corresponding to Eq. (22c) , Eq. (22d) , and Eq. (22e) . In an implementation, the hyper-parameters of “hardness” are β _s = 5 and β _t = 10.

The HCNN 500 further includes an assemble unit denoted as AS 530. The assemble unit 530 includes a main assemble unit denoted as M-AS 530-4 and three sub-assemble units denoted as S-ASs 530-1, 530-2 and 530-3. The S-ASs 530-1, 530-2 and 530-3 respectively take the outputs of sub-NNs 520-1, 520-2 and 520-3 as input and obtain the general solutions at the respective boundaries of the outer boundary, the cell boundary and the cooling pipe boundary. In an implementation, the S-ASs 530-1, 530-2 and 530-3 respectively output the general solutions

at the three boundaries according to Eq. (23) , Eq. (24) , and Eq. (25) . The main assemble unit 530-4 takes the outputs of M-NN 510, S-ASs 530-1, 530-2 and 530-3 as input and obtain the solution of the PDEs according to Eq. (11) . The solution of PDEs of HCNN 500 is the temperature T and its two derivatives

It is appreciated that the output of the HCNN 500 can be only the temperature T.

In order to train the HCNN 500, the loss may be obtained based on Eq. (13) , the parameters or weights of the NNs of the HCNN 500 can be updated based on the loss. In an implementation, the NN models are trained for e.g., 5000 Adam iterations, followed by a L-BFGS optimization until convergence. In an implementation, the mean squared error (MSE) is used for the loss function and tanh is used for the activation function.

After the training of the HCNN 500, the output, i.e., the temperature T, of the trained HCNN 500 is used to optimize the structure of the 2D battery packet. For example, the structure of the 2D battery packet may be optimized based on Eq. (20) . In an implementation, in each iteration of the structural optimization of the object such as the 2D battery packet, the HCNN 500 is trained based on the current structural quantities such as the positions and radius of the cells and the cooling pipes. Therefore the improvement of the efficiency and accuracy of HCNN significantly enhance the performance of the structure design process for the object such as the 2D battery packet and so on.

Fig. 6 is a schematic diagram illustrating the structural optimization of an airfoil according to aspects of the disclosure.

As shown in Fig. 6, the airfoil boundary γ _af located in domain Ω is the structure to be optimized. The goal of the structural optimization is to reach the desired pressure distribution p _ref on the airfoil surface by changing the structural parameter θ.

The structural optimization problem for the airfoil shown in Fig. 6 may be represented by

where p is the pressure, p _ref is the reference pressure.

The governing PDEs describing the physical system related to the airfoil may be formulated according to physical laws, for example, the PDEs may be NS equations and may be given by:

u (x) =u ₀ (x) , x∈γ _il ∪γ _tp ∪γ _bt, (29c)

p (x) =1, x∈γ _ol , (29d)

n (x) ·u (x) =0, x∈γ _af , (29e)

where x= (x ₁, x ₂) is the spatial coordinate. u (x) = (u ₁ (x) , u ₂ (x) ) is the velocity of the fluid, p (x) is the pressure of the fluid, v is viscosity of the fluid and is set to 1/50 in this example. u ₀ (x) = (1, 0) .

is a Jacobian matrix,

In Fig. 6, γ _il stands for the inlet boundary of the domain, γ _tp stands for the top boundary, γ _bt stands for the bottom boundary, γ _ol stands for the outlet boundary, γ _af stands for the airfoil boundary.

The structural quantities include the shape of the airfoil which is represented by the boundary γ _af. In order to optimize the structure of the airfoil based on the Eq. (28) , the governing PDEs need to be solved to obtain the physical quantity, that is, the velocity u and pressure p.

According to aspects of the disclosure, the PDEs of Eq. (29a) to Eq. (29e) are reformulated by adding extra fields according to the HCNN model. The introduced extra fields are

shown in Eq. (30c) and Eq. (30d) , and the reformulated PDEs are:

u (x) =u ₀ (x) , x∈γ _il ∪γ _tp ∪γ _bt, (30e)

p (x) =1, x∈γ _ol , (30f)

n (x) ·u (x) =0, x∈γ _af , (30g)

The solution of the PDEs or reformulated PDEs is the velocity and pressure (u (x) , p (x) ) . For the pressure p (x) , the general solution in the outlet boundary γ _ol is exactly

The ansatz for the pressure p is given by

where [3] means taking the third elements of the output of NN _main (x) .

For the velocity u (x) , the general solution in the inlet, top and bottom boundaries γ _*=γ _il ∪γ _tp ∪γ _bt is exactly

The general solution at the airfoil boundary γ _af is given by

where

according to Eq. (10b) and the output of

is a scalar.

Assembling

and

according to Eq. (11) , the ansatz for velocity u can be obtained by

where [1 : 2] means taking the first two elements of the output of NN _main (x) and

as well as

are similarly defined as in Eq. (12) .

The Extended Distance Function

in Eq. (11) may be implemented in various ways according to aspects of the disclosure. In an embodiment, For the airfoil boundary, a direct way of obtaining

is to calculate the distance between x and the airfoil γ _af . However, it may be time consuming since the airfoil boundary γ _af is complicated. In another implementation, the true distance may be approximated with an MLP. For example, a NN with 3 hidden layers of width 30 may be trained with 1024 × 6 points sampled in the domain Ω shown in Fig. 6. A part of the points are sampled in the bounding box of the airfoil, and the rest are sampled in Ω along with their truth distances, which may be computed by using the formula of the distance to a polygon.

For the outlet boundary γ _ol , it is a vertical line, for example, the outlet boundary is x ₁=a. The extended distance function can be computed as

where x= (x ₁, x ₂) .

For the collection of the inlet, top and bottom boundaries γ _*=γ _il ∪γ _tp ∪γ _bt, the boundary γ _* is an open rectangle, so the extended distance function

can be constructed similarly to the case of the rectangle as shown in Eq. (26) while ignoring the right side. Besides,

is computed by taking the SoftMin of the distances to all the boundaries as shown in Eq. (27) .

Fig. 7 illustrates an architecture of HCNN 700 for predicting PDE solutions of the airfoil according to aspects of the disclosure.

The HCNN 700 includes a main-NN denoted as M-NN 710, and a sub-NN denoted as S-NN 720. In an implementation, the main-NN is a multilayer perceptron (MLP) of size [2] + 6 × [50] + [7] , which means 2 inputs, 6 hidden layers of width 50, and 7 outputs. The two inputs are spatial coordinates x= (x ₁, x ₂) , and the seven outputs are velocity u= (u ₁, u ₂) , pressure p, and two derivatives of u ₁ and two derivatives of u ₂. The sub-NN 720 is a MLP of size [2] + 4 × [40] + [1] , which means 2 inputs, 4 hidden layers of width 40, and 1 output. As illustrated in Eq. (30e) , Eq. (30f) and Eq. (30g) , the BCs do not include the extra fields p ₁ (x) and p ₂ (x) , therefore the sub-NN 720 does not need to output the corresponding terms. The BCs of Eq. (30e) and Eq. (30f) provide constant velocity and pressure at the corresponding boundaries, therefore the sub-NN 720 does not need to provide output for these BCs. As for the BC of Eq. (30g) , by using the

as the basis of solution, the sub-NN 720 only needs to provide a scalar as illustrated in Eq. (32) . The main-NN 710 is responsible for the prediction of the PDE solutions in the out-of-boundary region of the airfoil, and the sub-NN 720 is responsible for the prediction of the PDE solutions at the BC corresponding to Eq. (30g) . And the PDE solutions at the BCs corresponding to Eq. (30e) and Eq. (30f) can be analytically obtained. In an implementation, the hyper-parameters of “hardness” are β _s = 5.

The HCNN 700 further includes an assemble unit denoted as AS 730. The assemble unit 730 includes a main assemble unit denoted as M-AS 730-2 and a sub-assemble units denoted as S-AS 730-1. The S-AS 730-1 takes the output of sub-NN 720 as input and obtains the general solution at the corresponding airfoil boundary γ _af, the S-AS 730-1 outputs the general solutions

at the boundary according to Eq. (30g) or Eq. (32) . The main assemble unit 730-2 takes the outputs of M-NN 710 and S-AS 730-1 as input and obtains the solution of the PDEs according to Eq. (11) or according to Eq. (31) and Eq. (33) . The solution of PDEs of HCNN 700 is the velocity u= (u ₁, u ₂) , pressure p, and derivatives

of velocity. It is appreciated that the output of the HCNN 700 can be the velocity u= (u ₁, u ₂) and pressure p only.

In order to train the HCNN 700, the loss may be obtained based on Eq. (13) , the parameters or weights of the NNs of the HCNN 700 can be updated based on the loss. In an implementation, the NN models are trained for e.g., 5000 Adam iterations, followed by a L-BFGS optimization until convergence. In an implementation, the mean squared error (MSE) is used for the loss function and tanh is used for the activation function.

After the training of the HCNN 700, the output, i.e., the temperature T, of the trained HCNN 700 is used to optimize the structure of the airfoil. For example, the shape of the airfoil may be optimized based on Eq. (28) . In an implementation, in each iteration of the structural optimization of the object such as the airfoil, the HCNN 700 is trained based on the current structural quantities such as a set of control points which consist of splines representing the shape of the airfoil. Therefore the improvement of the efficiency and accuracy of HCNN significantly enhance the performance of the structure design process for the object such as the airfoil and so on.

Fig. 8 illustrates an architecture of HCNN 800 for predicting PDE solutions of an object to be designed or manufactured according to aspects of the disclosure.

The HCNN 800 includes at least one main-NN denoted as M-NN 810-1 to M-NN 810-m, and at least one sub-NNs denoted as S-NN 820-1 to S-NN 820-n. It is appreciated that although one main-NN can be used to predict all the solutions u = (u ₁, …, u _n) of PDEs, it is feasible to predict respective parts of all the solutions with multiple main-NNs. The one or more main-NNs 810-1 to 810-m are responsible for the prediction of the PDE solutions in the out-of-boundary region of the object, and the one or more sub-NNs 820-1 to 820-n are respectively responsible for the prediction of the PDE solutions at the respective boundaries according to corresponding BCs.

The HCNN 800 further includes an assemble unit denoted as AS 830. The assemble unit 830 includes a main assemble unit denoted as M-AS 830-k and at least one sub-assemble units denoted as S-AS 830-1 to S-AS 830-n. The one or more S-ASs 830-1 to 830-n respectively take the outputs of sub-NNs 820-1 to 820-n as input and obtain the general solutions at the respective boundaries of the object. The main assemble unit 830-k takes the outputs of the one or more main-NNs 810-1 to 810-m and the one or more sub-AS 830-1 to S-AS 830-n as input and obtain the solution of the PDEs according to Eq. (11) .

In order to train the HCNN 800, the loss may be obtained based on Eq. (13) , the parameters or weights of the NNs of the HCNN 800 can be updated based on the loss. In an implementation, the NN models are trained for e.g., a defined number of Adam iterations, followed by a L-BFGS optimization until convergence.

After the training of the HCNN 800, the output, i.e., the prediction of the solution u = (u ₁, …, u _n) of PDEs, of the trained HCNN 800 is used to optimize the structure of the object. Therefore the improvement of the efficiency and accuracy of HCNN significantly enhance the performance of the structure design process for the object to be manufactured.

Fig. 9 illustrates an exemplary process for optimizing structure of an object according to aspects of the disclosure. It is appreciated that the process may be implemented with computers or processors.

At block 910, structural quantities are received by a HCNN framework, wherein the structural quantities are used to describe boundaries of the object. The HCNN comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein PDEs are formulated to characterize a physical system related to the object with physical quantities of the object.

At block 920, a first prediction of solutions of reformulated PDEs is generated by the at least one primary NN. The reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object.

At block 930, a second prediction of solutions of BCs of the reformulated PDEs is generated by at least using the at least one secondary NN. The BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object.

At block 940, a final prediction of solutions of the reformulated PDEs is obtained by assembling the first prediction of solution and the second prediction of solution by the assemble unit. The final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object.

At block 950, the structural quantities representing the boundaries of the object are updated based on the physical quantities of the final prediction.

According to an embodiment, the BCs of the boundaries of the object comprise one or more of Dirichlet BC, Neumann BC, and Robin BC.

According to an embodiment, the primary NN corresponds to out-of-boundary region of the object, and the at least one secondary NN corresponds to at least one of the boundaries of the object.

According to an embodiment, the second prediction of the solutions of the BCs of the reformulated PDEs further comprises prediction of the additional quantities representing gradients of the physical quantities of the object.

According to an embodiment, the generating a second prediction of solutions of BCs of the reformulated PDEs at block 930 further comprises: for each of at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and output of a corresponding secondary NN.

According to an embodiment, the generating a second prediction of solutions of BCs of the reformulated PDEs at block 930 further comprises: for the each of the at least one boundary corresponding to the at least one secondary NN, generating the corresponding part of the second prediction based further on a basis matrix or vector of null space for the boundary, wherein the basis matrix or vector is obtained based on a unit normal of the boundary.

According to an embodiment, the obtaining a final prediction of solutions of the reformulated PDEs at block 940 further comprises: assembling the first prediction of solution and the second prediction of solution based on a distance of a sampled point of object to the boundaries of the object.

According to an embodiment, the assembling the first prediction of solution and the second prediction of solution at block 940 further comprises: assembling respective parts of the second prediction corresponding to the boundaries of the object by ensuring that a part of the second prediction corresponding to any one boundary decays at the other boundaries to an extent of having insignificant influence to the other boundaries.

According to an embodiment, the process 900 further comprises: obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions including the physical quantities of the object and the gradients of the physical quantities; obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; obtaining a loss based on the first discrepancy and the second discrepancy; and updating the at least one primary NN and the at least one secondary NN based on the loss.

According to an embodiment, the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss and updating the at least one primary NN and the at least one secondary NN are repeated, until a training requirement is met.

According to an embodiment, the at least one primary NN and the at least one secondary NN are MLP NNs.

According to an embodiment, the object is one of a fuel cell bipolar plate, a part of a car, a part of a plane, a part of a building, pipes of a reactor, flow baffles and so on, that are to be manufactured or designed.

According to an embodiment, the structural quantities comprise at least one of position, radius, width, height, length, anchor points, and so on. The physical quantities comprise at least one of velocity, pressure, temperature, and so on.

Fig. 10 illustrates an exemplary process for predicting physical quantities of an object according to aspects of the disclosure. It is appreciated that the process may be implemented with computers or processors.

At block 1010, structural quantities are received by a HCNN framework, wherein the structural quantities are used to describe boundaries of the object. The HCNN comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein PDEs are formulated to characterize a physical system related to the object with physical quantities of the object.

At block 1020, a first prediction of solutions of reformulated PDEs is generated by the at least one primary NN. The reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object.

At block 1030, a second prediction of solutions of BCs of the reformulated PDEs is generated by at least using the at least one secondary NN. The BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object.

At block 1040, a final prediction of solutions of the reformulated PDEs is obtained by assembling the first prediction of solution and the second prediction of solution by the assemble unit. The final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object.

According to an embodiment, the generating a second prediction of solutions of BCs of the reformulated PDEs at block 1030 further comprises: for each of at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and output of a corresponding secondary NN.

According to an embodiment, the generating a second prediction of solutions of BCs of the reformulated PDEs at block 1030 further comprises: for the each of the at least one boundary corresponding to the at least one secondary NN, generating the corresponding part of the second prediction based further on a basis matrix or vector of null space for the boundary, wherein the basis matrix or vector is obtained based on a unit normal of the boundary.

According to an embodiment, the obtaining a final prediction of solutions of the reformulated PDEs at block 1040 further comprises: assembling the first prediction of solution and the second prediction of solution based on a distance of a sampled point of object to the boundaries of the object.

According to an embodiment, the assembling the first prediction of solution and the second prediction of solution at block 1040 further comprises: assembling respective parts of the second prediction corresponding to the boundaries of the object by ensuring that a part of the second prediction corresponding to any one boundary decays at the other boundaries to an extent of having insignificant influence to the other boundaries.

According to an embodiment, the process 1000 further comprises: obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions including the physical quantities of the object and gradients of the physical quantities; obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; obtaining a loss based on the first discrepancy and the second discrepancy; and updating the at least one primary NN and the at least one secondary NN based on the loss.

According to an embodiment, the at least one primary NN and the at least one secondary NN areMLP NNs.

Fig. 11 illustrates an exemplary computing system according to aspects of the disclosure. The computing system 1100 may comprise at least one processor 1110. The computing system 1100 may further comprise at least one storage device 1120. The storage device 1120 may store computer-executable instructions that, when executed, cause the processor 1110 to perform any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-10.

The embodiments of the present disclosure may be embodied in a computer-readable medium such as non-transitory computer-readable medium. The non-transitory computer-readable medium may comprise instructions that, when executed, cause one or more processors to perform any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-10.

The embodiments of the present disclosure may be embodied in a computer program product comprising computer-executable instructions that, when executed, cause one or more processors to perform any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-10.

It should be appreciated that all the operations in the methods described above are merely exemplary, and the present disclosure is not limited to any operations in the methods or sequence orders of these operations, and should cover all other equivalents under the same or similar concepts.

It should also be appreciated that all the modules in the apparatuses described above may be implemented in various approaches. These modules may be implemented as hardware, software, or a combination thereof. Moreover, any of these modules may be further functionally divided into sub-modules or combined together.

The previous description is provided to enable any person skilled in the art to practice the various aspects described herein. Various modifications to these aspects will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein. All structural and functional equivalents to the elements of the various aspects described throughout the present disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims.

Claims

A computer implemented method for optimizing structure of an object, comprising:

receiving structural quantities by a hard-constraint neural network (NN) framework, wherein the structural quantities are used to describe boundaries of the object, wherein the hard-constraint NN framework comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system related to the object with physical quantities of the object;

generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, the reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object;

generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object;

obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit, wherein the final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object; and

updating the structural quantities representing the boundaries of the object based on the physical quantities of the final prediction.
The method of claim 1, wherein the BCs of the boundaries of the object comprising one or more of Dirichlet BC, Neumann BC, and Robin BC.
The method of claim 1, wherein the primary NN corresponds to out-of-boundary region of the object, wherein the at least one secondary NN corresponds to at least one of the boundaries of the object.
The method of claim 1, wherein the second prediction of the solutions of the BCs of the reformulated PDEs further comprises prediction of the additional quantities representing gradients of the physical quantities of the object.
The method of claim 2, the generating a second prediction of solutions of BCs of the reformulated PDEs further comprising:

for each of at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and output of a corresponding secondary NN.
The method of claim 5, the generating a second prediction of solutions of BCs of the reformulated PDEs further comprising:

for the each of the at least one boundary corresponding to the at least one secondary NN, generating the corresponding part of the second prediction based further on a basis matrix or vector of null space for the boundary, wherein the basis matrix or vector is obtained based on a unit normal of the boundary.
The method of claim 2, wherein the obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit further comprising:

assembling the first prediction of solution and the second prediction of solution based on a distance of a sampled point of object to the boundaries of the object.
The method of claim 7, wherein the assembling the first prediction of solution and the second prediction of solution further comprising assembling respective parts of the second prediction corresponding to the boundaries of the object by ensuring that a part of the second prediction corresponding to any one boundary decays at the other boundaries to an extent of having insignificant influence to the other boundaries.
The method of claim 2, further comprising:

obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions including the physical quantities of the object and gradients of the physical quantities;

obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions;

obtaining a loss based on the first discrepancy and the second discrepancy; and

updating the at least one primary NN and the at least one secondary NN based on the loss.
The method of claim 9, further comprising:

repeating the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss and updating the at least one primary NN and the at least one secondary NN, until a training requirement is met.
The method of claim 1, wherein the at least one primary NN and the at least one secondary NN are multilayer perceptron (MLP) NNs.
The method of claim 1, wherein the object is one of a fuel cell bipolar plate, a part of a car, a part of a plane, a part of a building, pipes of a reactor, flow baffles.
The method of claim 1, wherein the structural quantities comprise at least one of position, radius, width, height, length, anchor points, and the physical quantities comprise at least one of velocity, pressure, temperature.
A computer implemented method for predicting physical quantities of an object, comprising:

receiving structural quantities by a hard-constraint neural network (NN) framework, wherein the structural quantities are used to describe boundaries of the object, wherein the hard-constraint NN framework comprises at least one primary NN, at least one secondary NN and an assemble unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system related to the object with physical quantities of the object;

generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, the reformulated PDEs are obtained by substituting gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object;

generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system related to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises prediction of the physical quantities of the object;

obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit, wherein the final prediction of solutions of the reformulated PDEs represents prediction of the physical quantities of the object.
The method of claim 14, wherein the BCs of the boundaries of the object comprising one or more of Dirichlet BC, Neumann BC, and Robin BC.
The method of claim 15, the generating a second prediction of solutions of BCs of the reformulated PDEs further comprising:

for each of the at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and output of a corresponding secondary NN.
The method of claim 15, wherein the obtaining a final prediction of solutions of the reformulated PDEs by assembling the first prediction of solution and the second prediction of solution by the assemble unit further comprising:

assembling the first prediction of solution and the second prediction of solution based on a distance of a sampled point of object to the boundaries of the object.
The method of claim 13, further comprising:

obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions including the physical quantities of the object and the gradients of the physical quantities;

obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions;

obtaining a loss based on the first discrepancy and the second discrepancy; and

updating the at least one primary NN and the at least one secondary NN based on the loss.
A computer system, comprising:

one or more processors; and

one or more storage devices storing computer-executable instructions that, when executed, cause the one or more processors to perform the operations of the method of one of claims 1-18.
One or more computer readable storage media storing computer-executable instructions that, when executed, cause one or more processors to perform the operations of the method of one of claims 1-18.