WO2024198599A1 - 基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品 - Google Patents

基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品 Download PDF

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WO2024198599A1
WO2024198599A1 PCT/CN2024/070340 CN2024070340W WO2024198599A1 WO 2024198599 A1 WO2024198599 A1 WO 2024198599A1 CN 2024070340 W CN2024070340 W CN 2024070340W WO 2024198599 A1 WO2024198599 A1 WO 2024198599A1
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neural
operator
partial differential
node
solution
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徐腾飞
郝鹏
刘大川
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Dalian University of Technology
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    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12—Simultaneous equations, e.g. systems of linear equations
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13—Differential equations
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/15—Correlation function computation including computation of convolution operations
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    • G06F30/00—Computer-aided design [CAD]
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
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    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/0464—Convolutional networks [CNN, ConvNet]
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    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/048—Activation functions
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/08—Learning methods
    • G06N3/084—Backpropagation, e.g. using gradient descent
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    • G06—COMPUTING OR CALCULATING; COUNTING
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    • G06N5/04—Inference or reasoning models
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    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C60/00—Computational materials science, i.e. ICT specially adapted for investigating the physical or chemical properties of materials or phenomena associated with their design, synthesis, processing, characterisation or utilisation
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2113/00—Details relating to the application field
    • G06F2113/24—Sheet material

Definitions

  • the present invention belongs to the intersection of physical simulation and machine learning, and in particular relates to a unified method, medium and product for neural operator training and partial differential equation solution based on variational principle.
  • neural operators can perform high-precision instant reasoning on the entire solution domain of the group of partial differential equations compared to traditional proxy models such as Gaussian process models and radial basis function models. They have received extensive attention in many physical simulation fields such as fluids, solids, electromagnetics, and heat transfer. Neural operators have many advantages such as potential super-resolution, high flexibility, and excellent generalization performance. They have huge landing space in industries such as industrial equipment digital twins, real-time and large-scale simulation solutions, animation and game modeling and rendering, and the metaverse, and are very likely to become the core technology of the next generation of physical simulation solvers.
  • the present invention proposes a unified method, medium and product for neural operator training and partial differential equation solution based on the variational principle.
  • the discrete functional is estimated by the predicted node solution of the neural operator, and the gradient norm of the discrete functional obtained by the automatic differentiation technique to the node solution is minimized by an iterative method based on the system of equations.
  • the label-free optimization objective is constructed to train the neural operator, thereby avoiding the large-scale use of traditional solvers to obtain labels and saving computational costs.
  • the method uses the generalization ability of the neural operator to provide a good initial solution for each iteration, essentially incorporating training and solution into a unified framework.
  • a unified method for neural operator training and partial differential equation solution based on variational principle comprises the following steps:
  • Step 100 Sample from the parameter space of the partial differential equation system to form an unlabeled dataset D containing only discrete parameter fields, divide the dataset D into an offset set L, a test set T and an unlabeled set U, divide the unlabeled set U into multiple batches, encode the boundary conditions, and form a mask tensor of the boundary conditions, including the following sub-steps:
  • Step 101 Select the form and sampling strategy of the parameter space of the partial differential equation system
  • Step 102 Grid the solution domain of the partial differential equation system, and sample the parameter space of the partial differential equation system according to the sampling strategy selected in step 101, discretize the sampled parameter field at the Gaussian points of the grid, and form an unlabeled discrete parameter field data set D containing only discrete parameter fields. Specifically: determine the number of sampled discrete parameter fields and the discrete parameter field shape;
  • Step 103 randomly sample (N 1 +N 2 ) discrete parameter fields from the discrete parameter field data set D, solve the node solutions of the partial differential equations corresponding to the (N 1 +N 2 ) discrete parameter fields, and use the first N 1 discrete parameter fields and their corresponding node solutions (labels) as the offset set L, and the remaining N 2 discrete parameter fields and their corresponding node solutions as the test set T, wherein the offset set L is used to offset the output range, see step 203 for details;
  • Step 104 exclude the (N 1 +N 2 ) discrete parameter fields sampled in the previous step from the data set D to form an unlabeled set U as a training set;
  • Step 105 Set a variable i that records the number of iterations and set its value to 0, dividing the unlabeled set U into several batches of discrete parameter field samples;
  • Step 106 Encode the boundary conditions to form a mask tensor M of the boundary conditions with the same shape as the node solution, in which the M elements corresponding to the constrained node degrees of freedom are set to 0, and the remaining elements are set to 1;
  • Step 200 Use the neural operator module F to predict the node solutions of the discrete parameter field samples in the unlabeled set U, and further obtain the discretized functional as an estimate of the system functional based on the node solutions, including the following sub-steps:
  • Step 201 extract a batch of discrete fiber angle field samples s from the unlabeled set U without replacement. If all batches are extracted, the variable i is incremented by 1 to determine whether the training process of the neural operator has reached the algorithm convergence condition. If converged, the weight ⁇ of the neural operator is output and step 304 is executed. Otherwise, the unlabeled set U is shuffled and the shuffled unlabeled set U is re-divided into several batches, and re-extracted and trained from the first batch; wherein the convergence condition may be that i reaches the maximum number of iterations of training or the inference accuracy of the neural operator has reached the accuracy requirement;
  • ⁇ represents the element-wise product between tensors, the same below;
  • Step 204 performing a convolution operation on the node solution a to obtain a solution at the Gaussian point and its spatial derivative;
  • Step 205 Process the solution at the Gaussian point and its spatial derivative using tensor operations to obtain the value of the functional integrand at the Gaussian point;
  • Step 206 using the Gaussian integral method, according to the value of the functional integrand at the Gaussian point, obtain the discretized functional estimate ⁇ ;
  • Step 300 Apply variational operations to the discretized functional to construct an optimization target. Specifically, calculate the gradient R of the functional estimate ⁇ with respect to the node solution a, and use its norm as the minimization target, use an iterative method based on a system of equations to obtain the update step length ⁇ a of the current node solution, and use ⁇ a to update the weight ⁇ of the neural operator module F, including the following sub-steps:
  • Step 301 using the automatic differentiation method, back-propagating the functional estimate ⁇ , recording the gradient R of the functional estimate ⁇ to the node solution a, and taking its norm as the minimization target;
  • Step 302 input the gradient R into the iterative method based on the equation system to obtain the update step length ⁇ a of the current node solution;
  • Step 303 Use ⁇ a to update the weight ⁇ of the neural operator, and return to execute step 201;
  • Step 304 Perform inference on the neural operator module F on the test set T and calculate the test set indicators.
  • the parameter space of the partial differential equations may be a space formed by the coefficients or free terms of the partial differential equations
  • the parameter space of the partial differential equations may be in the form of a continuous parameter field controlled by parameters such as a B-spline surface and a Gaussian random field, or a function with such a parameter field as an independent variable
  • the sampling strategy may be a sampling method such as a simple random sampling method and a Latin hypercube sampling method.
  • the grid division settings can be adjusted according to the specific research problems.
  • Sampling the parameter space of the partial differential equation group can refer to direct sampling of the parameters of the partial differential equation group, or can refer to sampling of the independent variables that control the parameters of the partial differential equation group.
  • the heat source term can be directly sampled
  • the elastic mechanics partial differential equation group that controls the fiber-laminated elastic plate the material principal direction coordinate axis angles that control the elastic coefficients in the constitutive equation group can be sampled, that is, the fiber angle field can be sampled.
  • the analysis method used to solve the partial differential equations may be a finite element analysis method, a boundary element analysis method, an isogeometric analysis method, a meshless method, or the like.
  • step 105 the size and number of batches can be adjusted according to specific research questions.
  • the neural operator module F includes but is not limited to a single neural operator and a combination of multiple neural operators, and the network structure and hyperparameters of all neural networks of the neural operator module F can be adjusted according to specific research problems.
  • step 205 the specific form of the tensor operation of obtaining the functional integrand at the Gaussian point from the solution at the Gaussian point and its spatial derivative can be adjusted according to the form of the partial differential equation group to be solved.
  • the iterative method based on the system of equations includes but is not limited to the steepest descent method and the conjugate gradient method.
  • the number of iterations of the iterative method based on the system of equations can be adjusted according to the specific research problem, and the maximum number of iterations of training can be adjusted according to the specific research problem.
  • step 303 various hyperparameters required for the update process, such as learning rate, momentum, etc., can be adjusted according to specific research problems.
  • test set index can be determined according to the specific research question.
  • a computer-readable storage medium comprises instructions, which, when executed on a computer, cause the computer to execute any one of the methods of the first aspect.
  • a computer program product comprising instructions, when it is run on a computer, causes the computer to execute the On the one hand, any method.
  • the present invention proposes a unified method, medium and product for neural operator training and partial differential equation solution based on variational principle, which incorporates the solution and training process into a unified framework.
  • the solution and training process of pure data-driven method are carried out separately, and a large number of labels are required to construct optimization targets.
  • the present invention estimates the output functional of the neural operator and introduces variational operation to construct unlabeled optimization targets, thereby reducing the labels required in the training process and reducing the use of traditional solvers.
  • Fields such as optimization design and uncertainty quantification require solving the parameter set of partial differential equations rather than a single parameter.
  • Traditional solvers such as finite element-based solvers, can only independently solve each parameter instance in the parameter set, and then traversely solve the entire parameter set.
  • the present invention solves an operator from parameter space to solution space and simultaneously learns the weight of a neural operator with generalization ability, which ensures that while solving a specific parameter, the present invention can obtain accuracy improvement on other parameters.
  • the present invention is expected to replace traditional solvers in fields such as optimization design and uncertainty quantification, which require solving a large number of parameters of partial differential equations.
  • FIG1 is a flowchart of a unified method, medium, and product for neural operator training and partial differential equation solution based on the variational principle provided by an embodiment of the present invention
  • FIG2 is a schematic diagram of boundary conditions of Example 1;
  • FIG3 is a schematic diagram of the neural operator network structure of Example 1;
  • FIG4 is a diagram showing 5 samples randomly sampled from the test set of Example 1;
  • FIG5 is a schematic diagram of boundary conditions of a rectangular region P containing a variable heat source in Example 2;
  • FIG6 is a diagram showing 5 samples randomly sampled from the test set of Example 2.
  • the embodiment of the present invention provides a unified method for neural operator training and partial differential equation solution based on variational principle, comprising the following steps:
  • Step 100 Study the variable stiffness fiber laminate elastic plate P.
  • the size, material properties and boundary conditions of the plate P are shown in step 103.
  • the fiber angle field of the plate P is sampled to form an unlabeled dataset D containing only discrete fiber angle fields.
  • the dataset D is further divided into an offset set L, a test set T and an unlabeled set U.
  • the unlabeled set U is divided into multiple batches, and the boundary conditions are encoded to form a mask tensor of the boundary conditions, including the following sub-steps:
  • Step 101 The fiber angle field is selected in the form of the z coordinate component of a second-order B-spline surface controlled by a 4 ⁇ 4 control point grid.
  • the x and y components of the control points are fixed.
  • Z( xp , yp ) is determined by formula (1.1):
  • Ni ,2 and Nj,2 are the basis functions of the (i+1)th and (j+1)th control points in the x and y directions respectively, i and j are the control point numbers in the x and y directions respectively, and the sampling strategy is Latin hypercube sampling;
  • Step 102 For the fiber-laminated elastic plate P under study, the unit is selected as a bilinear reduced integration unit with four plane nodes.
  • the grid setting adopts uniform grid division, the grid division density is 32 ⁇ 32 units, and there are 33 ⁇ 33 nodes in total.
  • the z coordinates of the 4 ⁇ 4 control points of the fiber angle field are sampled by Latin hypercube, and the range of all z coordinate sampling is In this way, the fiber angle field is sampled.
  • the number of sampled fiber angle fields is 100,000. 14,005 fiber angle fields are randomly selected and discretized to form an unlabeled discrete fiber angle field dataset D containing only discrete fiber angle fields.
  • the fiber angle field is discretized at the Gaussian points of the grid, and the shape of the discrete fiber angle field is 32 ⁇ 32 ⁇ 1.
  • Step 103 Randomly sample 2005 discrete fiber angle fields from the data set D, solve the node displacement solutions of the partial differential equations corresponding to the 2005 discrete fiber angle fields, and use the first 5 discrete fiber angle fields and their corresponding node displacement solutions as the offset set L, and the remaining 2000 discrete fiber angle fields and their corresponding node displacement solutions as the test set T.
  • the node displacement solutions in this example are obtained by solving the ABAQUS commercial software.
  • the S4R unit in the ABAQUS commercial software unit library is used in the calculation.
  • the grid division in the ABAQUS software adopts uniform division, and the density of the grid is 32 ⁇ 32 units, with a total of 33 ⁇ 33 nodes.
  • the boundary conditions of the variable stiffness fiber laminate elastic plate P studied in this example are shown in Figure 2.
  • test set randomly samples 5 samples and the display diagram is shown in FIG4 , wherein the 5 samples are displayed in five columns, each column is a sample, wherein the first row is the fiber path display of the 5 samples, the second row is the fiber angle field (unit: rad) of the 5 samples, and the third row is the x-component prediction (unit: mm) of the displacement node solution of the 5 samples by the neural operator.
  • the fourth row is the label of the x-component of the displacement node solutions of the 5 samples (unit: mm)
  • the fifth row is the absolute value error of the neural operator's prediction of the x-component of the displacement node solutions of the 5 samples (unit: mm)
  • the sixth row is the neural operator's prediction of the y-component of the displacement node solutions of the 5 samples (unit: mm)
  • the seventh row is the label of the y-component of the displacement node solutions of the 5 samples (unit: mm)
  • the eighth row is the absolute value error of the neural operator's prediction of the y-component of the displacement node solutions of the 5 samples (unit: mm).
  • Step 104 exclude the 2005 discrete fiber angle fields sampled in the previous step from the data set D, and obtain an unlabeled set U containing 12000 discrete fiber angle fields as a training set;
  • Step 105 set a variable i that records the number of iterations and set its value to 0, divide the unlabeled set U into 12,000 batches of discrete fiber angle field samples, and the size of each batch is 1;
  • Step 106 Encode the boundary conditions to form a boundary condition mask tensor M of shape (1, 33, 33, 2), in which the M elements corresponding to the constrained node degrees of freedom are set to 0, and the remaining elements are set to 1;
  • Step 200 Using a neural operator module F, F includes two implicit Fourier neural operators (IFNO) with the same settings, each of which is responsible for predicting a displacement component.
  • the settings of each Fourier neural operator are shown in FIG3 .
  • the node displacements of the shape of 32 ⁇ 32 ⁇ 2 corresponding to the discrete fiber angle field samples in the unlabeled set U are predicted. Further, the discretized functional is obtained according to the node displacement as an estimate of the system functional, including the following sub-steps:
  • Step 201 extract a batch of discrete fiber angle field samples s from the unlabeled set U without replacement. If all batches are extracted, the variable i is incremented by 1 to determine whether the training process of the neural operator reaches the algorithm convergence condition. If converged, the weight ⁇ of the neural operator is output and step 304 is executed. Otherwise, the unlabeled set U is shuffled and the shuffled unlabeled set U is re-divided into 12,000 batches, and re-extracted and trained from the first batch; wherein the convergence condition is that i reaches the maximum number of iterations of training, and the maximum number of iterations of training in this embodiment is 500;
  • Step 204 performing a convolution operation on the node displacement a to obtain a solution at the Gaussian point and its spatial derivative
  • Step 205 using tensor operations to process the solution and spatial derivative at the Gaussian point to obtain the value of the functional integrand at the Gaussian point;
  • Step 206 Using the Gaussian integral method, according to the value of the integrand of the functional at the Gaussian point, the discretized functional ⁇ is obtained, as shown in formula (1.4);
  • ae represents the nodal displacement of each element
  • the subscript “l” is taken at the lth integration point in the element
  • N represents the shape function matrix
  • B represents the strain-displacement matrix
  • D represents the material property matrix
  • Je represents the Jacobian matrix of the element
  • f represents the body force.
  • H l and I ml represent the weight of the lth integration point in the unit and the weight of the lth integration point on the boundary of the mth unit, respectively.
  • Step 300 Discretize the functional Apply variational operations, specifically: calculate functional estimates Regarding the gradient R of the node displacement a, and taking its norm as the minimization target, an iterative method based on the equation group is used to obtain the update step length ⁇ a of the current node displacement, and ⁇ a is used to update the weight ⁇ of the neural operator module F, including the following sub-steps:
  • Step 301 Use automatic differentiation technology to estimate the functional Perform back propagation and record functional estimates
  • the gradient R of the node displacement a is shown in formula (1.5), and the norm of R is taken as the minimization target;
  • Step 302 input the gradient R into the conjugate gradient method, and the conjugate gradient method runs two iterations to obtain the update step length ⁇ a of the current node displacement;
  • Step 303 Based on the stochastic gradient descent method, the weight ⁇ of the neural operator is updated using ⁇ a, and the learning rate is set to 1, and then the process returns to step 201;
  • Step 304 Perform inference on the neural operator module F on the test set T, and select the test set indicator as the average relative 2-norm error, as shown in formula (1.6);
  • a computer-readable storage medium comprises instructions, which, when executed on a computer, cause the computer to execute any one of the methods of the first aspect.
  • a computer program product comprising instructions, when the computer program product is run on a computer, causes the computer to execute any one of the methods in the first aspect.
  • the present invention proposes a unified method, medium and product for neural operator training and partial differential equation solution based on variational principle.
  • Composite materials are widely used in aerospace, sports equipment, automobile manufacturing and other industries.
  • Variable stiffness composite materials are favored in lightweight design because they can maximize mechanical bearing performance through fiber path optimization design.
  • Variable angle fiber ply is a basic component of variable stiffness composite materials. Realizing batch real-time analysis of the mechanical response of variable angle fiber ply is of great significance to accelerating the optimization design of variable stiffness composite materials.
  • anisotropic variable angle fiber ply simulation is achieved.
  • the present invention provides a unified method for neural operator training and partial differential equation solution based on variational principle, comprising the following steps:
  • Step 100 Study the heat transfer problem of a rectangular region P containing a variable heat source.
  • the region size and boundary conditions are shown in step 103.
  • the unlabeled set U is divided into multiple batches. Encode the boundary conditions to form a mask tensor of the boundary conditions, including the following sub-steps:
  • Step 101 Select the form of the heat source field as Gaussian random field:
  • Step 102 For the rectangular region P containing the variable heat source under study, the unit is selected as a bilinear unit with four plane nodes, the grid setting adopts uniform grid division, the grid division density is 30 ⁇ 30 units, and there are 31 ⁇ 31 nodes in total.
  • the heat source field is simply randomly sampled and discretized, and the number of samples is 12010, thereby forming an unlabeled data set D containing only discrete heat source fields, wherein the heat source field is discretized at the Gaussian points of the grid, and the shape of the discrete heat source field is 30 ⁇ 30 ⁇ 1;
  • Step 103 randomly sample 2010 discrete heat source fields from the data set D, solve the node temperature solutions of the partial differential equations corresponding to the 2010 discrete heat source fields, and use the first 10 discrete heat source fields and their corresponding node temperature solution labels as the offset set L, and the remaining 2000 discrete heat source fields and their corresponding node temperature solution labels as the test set T.
  • the node temperature solution labels in this example are obtained by solving the COMSOL commercial software.
  • the planar 4-node bilinear unit in the COMSOL commercial software unit library is used in the calculation.
  • the mesh division in the COMSOL software adopts uniform division, and the mesh division density is 30 ⁇ 30 units, with a total of 31 ⁇ 31 nodes.
  • the boundary conditions of the rectangular area P containing a variable heat source studied in this example are shown in FIG5.
  • the display diagram of 5 samples randomly sampled from the test set is shown in FIG6 , where the 5 samples are displayed in five columns, each column is a sample, where the first row is the heat source field of the 5 samples (unit: W/m 2 ), the second row is the temperature node solution of the 5 samples by the neural operator (unit: °C), the third row is the temperature node solution label of the 5 samples (unit: °C), and the fourth row is the predicted absolute value error of the temperature node solution of the 5 samples by the neural operator (unit: °C);
  • Step 104 Exclude the 2010 discrete heat source fields sampled in the previous step from the data set D, and obtain a data set containing 10,000 discrete heat source fields.
  • the unlabeled set U of scattered heat source fields is used as the training set;
  • Step 105 Set the variable i that records the number of iterations to 0, and divide the unlabeled set U into 157 batches of discrete heat source field samples, of which 156 batches have a size of 64 and the last batch has a size of 16;
  • Step 106 Encode the boundary conditions to form a boundary condition mask tensor M of shape (1, 31, 31, 1), in which the M elements corresponding to the constrained node degrees of freedom are set to 0, and the remaining elements are set to 1;
  • Step 200 Use the neural operator module F, which includes an implicit Fourier neural operator (IFNO) responsible for temperature prediction, predicting the node temperature of the shape of 31 ⁇ 31 ⁇ 1 corresponding to the discrete heat source field samples in the unlabeled set U, and further obtain the discretized functional as the estimation of the system functional according to the node temperature, including the following sub-steps:
  • IFNO implicit Fourier neural operator
  • Step 201 extract a batch of discrete fiber angle field samples s from the unlabeled set U without replacement. If all batches are extracted, the variable i is incremented by 1 to determine whether the training process of the neural operator reaches the algorithm convergence condition. If converged, the weight ⁇ of the neural operator is output and step 304 is executed. Otherwise, the unlabeled set U is shuffled and the shuffled unlabeled set U is re-divided into 157 batches, of which 156 batches have a size of 64 and the last batch has a size of 16. The first batch is re-extracted and trained; wherein the convergence condition is that i reaches the maximum number of iterations of training, and the maximum number of iterations of training in this embodiment is 5000;
  • Step 204 performing a convolution operation on the node temperature a to obtain a solution at the Gaussian point and its spatial derivative
  • Step 205 using tensor operations to process the solution and spatial derivative at the Gaussian point to obtain the value of the functional integrand at the Gaussian point;
  • Step 206 Using the Gaussian integral method, the discretized temperature functional is obtained based on the value of the integrand of the functional at the Gaussian point.
  • Step 300 Discretize the temperature functional Apply variational operations, specifically: calculate functional estimates Regarding the gradient R of the node temperature a, and taking its norm as the minimization target, an iterative method based on a system of equations is used to obtain the update step length ⁇ a of the current node temperature, and ⁇ a is used to update the weight ⁇ of the neural operator module F, including the following sub-steps:
  • Step 301 Use automatic differentiation technology to estimate the functional Perform back propagation and record functional estimates
  • the gradient R of the node temperature a is as shown in equation (1.5), and the norm of R is taken as the minimization target;
  • Step 302 input the gradient R into the conjugate gradient method, and the conjugate gradient method runs two iteration steps to obtain the update step length ⁇ a of the current node temperature;
  • Step 303 Based on the stochastic gradient descent method, the weight ⁇ of the neural operator is updated using ⁇ a, and the learning rate is set to 1e-5, and then the process returns to step 201;
  • Step 304 Perform inference on the neural operator module F on the test set T, and select the test set indicator as the average relative 2-norm error, as shown in formula (1.6);
  • a computer-readable storage medium comprises instructions, which, when executed on a computer, cause the computer to execute any one of the methods of the first aspect.
  • a computer program product comprising instructions, when the computer program product is run on a computer, causes the computer to execute any one of the methods in the first aspect.
  • the present invention proposes a unified method, medium and product for neural operator training and partial differential equation solution based on variational principle.
  • Heat transfer problems are widely present in various engineering and technical fields, such as energy and power, metallurgy, chemical industry, transportation, construction and building materials, machinery, traditional industries such as food, light industry, textile, and medicine, as well as many high-tech fields such as aerospace, nuclear energy, microelectronics, materials, biomedical engineering, environmental engineering, new energy and agricultural engineering. Accurate and real-time batch simulation of heat transfer phenomena can help us better understand and master heat transfer phenomena, so as to better apply them to actual production and life.
  • Example 2 of the present invention achieved an average relative 2-norm error of only 2.20% on a test set with a capacity of 2000, when only 10 labels were used in the offset set L and the training set was completely unlabeled, reaching the level of pure data-driven training error, thereby proving that the present invention can be applied to accurate real-time batch simulation of heat transfer phenomena.
  • the present invention can save time and computing power for generating a large number of labels required to construct data-driven error terms, and unify the two tasks of solving partial differential equations and training neural operators in one framework, providing strong support for downstream applications such as optimization design and inverse problems.

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Abstract

一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品，包括：从偏微分方程组的参数空间中抽样形成仅含离散参数场的数据集，划分为偏移集、测试集、无标签集，将无标签集分为多个批次，并形成边界条件的掩码张量；使用神经算子模块预测无标签集中离散参数场样本的节点解，得到离散化的泛函作为系统泛函的估计；计算泛函估计关于节点解的梯度，将其范数作为最小化目标，获取当前节点解的更新步长，更新模块的权重。本发明将求解偏微分方程组算子和训练神经算子两个任务统一在一个框架内完成，引入变分操作构建无标签的优化目标以替代数据驱动的训练误差项，节省生成用于构建数据驱动误差项所需的大量标签的时间与算力。

Description

基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品 技术领域
本发明属于物理仿真与机器学习的交叉领域，尤其涉及一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品。
背景技术
神经算子作为一种能够有效学习偏微分方程组的参数空间与解空间之间映射算子的深度神经网络，相对于高斯过程模型、径向基函数模型等传统的代理模型，能够对整个偏微分方程组求解域进行较高精度的即时推理，在流体、固体、电磁、传热等诸多物理仿真领域受到广泛关注。神经算子具有潜在超分辨性、高度的灵活性与优良的泛化性能等诸多优点，在工业装备数字孪生、实时与大规模仿真解算、动漫游戏建模渲染、元宇宙等行业具有巨大的落地空间，非常可能成为下一代物理仿真求解器的核心技术。
然而，现有针对神经算子的基于数据驱动的训练方法需要来自传统求解器的海量标签数据，导致对传统求解器的大量使用，从而带来巨大的计算负担。此外，训练过程本身也会带来一定的计算成本。
发明内容
针对现有基于数据驱动的训练神经算子方法的不足，本发明提出一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品，通过神经算子的预测节点解估计离散的泛函，并通过基于方程组的迭代法最小化利用自动微分技术得到的离散泛函对节点解的梯度范数，构建无标签的优化目标来训练神经算子，从而避免大量地使用传统求解器获得标签，节约了计算成本。同时，本方法利用神经算子的泛化能力为每步迭代提供良好的初始解，本质上将训练与求解纳入了一个统一的框架。
为了达到上述目的，本发明采用的技术方案如下：
第一方面，一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，包括以下步骤：
步骤100：从偏微分方程组的参数空间中抽样形成无标签的仅含离散参数场的数据集D，将数据集D划分为偏移集L、测试集T和无标签集U，将无标签集U分为多个批次，对边界条件进行编码，形成边界条件的掩码张量(the mask tensor)，包括以下子步骤：
步骤101：选定偏微分方程组的参数空间的形式和抽样策略；
步骤102：对偏微分方程组的求解域进行网格划分，并按照步骤101选定的抽样策略对偏微分方程组的参数空间进行抽样，对抽样得到的参数场在网格的高斯点处离散化，形成无标签的仅含离散参数场的离散参数场数据集D，具体的：确定抽样离散参数场数目和离散参数场的 形状；
步骤103：从离散参数场数据集D随机抽样出(N1+N2)个离散参数场，求解这(N1+N2)个离散参数场对应的偏微分方程组的节点解，并将其中前N1个离散参数场及其对应的节点解(标签)作为偏移集L，剩余的N2个离散参数场及其对应的节点解作为测试集T，其中，偏移集L用于对输出的范围进行偏移，详见步骤203；
步骤104：将上一步骤抽样的(N1+N2)个离散参数场排除出数据集D，形成无标签集U作为训练集；
步骤105：设定记录迭代次数的变量i，并设置其值为0，将无标签集U分为若干批次的离散参数场样本；
步骤106：对边界条件进行编码，形成形状和节点解相同的边界条件的掩码张量M，在M当中，位置对应于被约束的节点自由度的M元素被设为0，其余元素被设为1；
步骤200：使用神经算子模块F预测无标签集U中离散参数场样本的节点解，进一步根据节点解得到离散化的泛函作为系统泛函的估计，包括以下子步骤：
步骤201：不放回地抽取无标签集U中一批离散纤维角度场样本s，如果所有的批次都被抽出，变量i自增1，判断对神经算子的训练过程是否达到算法收敛条件，如果收敛，则输出神经算子的权重θ，并执行步骤304，否则，打乱无标签集U，并将打乱后的无标签集U重新分为若干个批次，从第一个批次重新抽取、训练；其中，所述的收敛条件可以是i达到训练的最大迭代次数或者神经算子的推理精度已达到精度要求；
步骤202：将样本s输入神经算子模块F进行推理，利用掩码张量M对F输出的张量F(s)进行掩码操作，得到节点解a，如式(1.1)所示；
a＝F(s)⊙M        (1.1)
式(1.1)中，⊙代表张量之间进行元素积，下同；
步骤203：使用偏移集L内所有标签的均值mean和标准差std，对节点解a进行偏移处理：
a＝a⊙std+mean          (1.2)
步骤204：对节点解a进行卷积操作，得到在高斯点处的解及其空间导数；
步骤205：利用张量操作处理高斯点处的解及其空间导数，求得泛函被积函数在高斯点处的值；
步骤206：使用高斯积分法，根据泛函被积函数在高斯点处的值，求得离散化的泛函估计Π；
步骤300：对离散化的泛函施加变分操作构建优化目标，具体的：计算泛函估计Π关于节点解a的梯度R，并将其范数作为最小化目标，采用基于方程组的迭代法获取当前节点解的更新步长△a，并利用△a更新神经算子模块F的权重θ，包括以下子步骤：
步骤301：利用自动微分方法，对泛函估计Π进行反向传播，记录泛函估计Π对节点解a的梯度R，并将其范数作为最小化目标；
步骤302：将梯度R输入基于方程组的迭代法，得到当前的节点解的更新步长△a；
步骤303：使用△a对神经算子的权重θ进行更新，并返回执行步骤201；
步骤304：将神经算子模块F在测试集T上进行推理，并统计测试集指标。
进一步，所述的步骤101中，偏微分方程组的参数空间可以是偏微分方程组的系数或自由项形成的空间，偏微分方程组的参数空间形式可以是B样条曲面和高斯随机场等由参数控制的连续的参数场或由此类参数场作为自变量的函数，抽样策略可以是简单随机抽样法和拉丁超立方抽样法等抽样方法。
进一步，所述的步骤102中，网格划分的设置可根据具体的研究问题自行调整，对偏微分方程组的参数空间进行抽样可以指对偏微分方程组参数的直接抽样，也可以指对控制偏微分方程组参数的自变量的抽样，例如，对控制热传导的热传导偏微分方程，可以对热源项进行直接抽样，而对于控制纤维铺层弹性板的弹性力学偏微分方程组，可以对控制本构方程组中的弹性系数的材料主方向坐标轴角度进行抽样，即对纤维角度场进行抽样。
进一步，所述的步骤103中，求解偏微分方程组所用到的分析方法可以是有限元分析法、边界元分析法、等几何分析法和无网格法等类似分析方法。
进一步，所述的步骤105中，批次的大小与数目可以根据具体的研究问题自行调整。
进一步，所述的步骤202中，神经算子模块F包括但不限于单个神经算子和多个神经算子的组合，神经算子模块F所有的神经网络的网络结构与网络结构的超参数可根据具体的研究问题自行调整。
进一步，所属步骤205中，从高斯点处的解及其空间导数得到高斯点处泛函被积函数的张量操作的具体形式可根据解决的偏微分方程组形式自行调整。
进一步，所述的步骤302中，基于方程组的迭代法包括但不限于最速下降法和共轭梯度法，基于方程组的迭代法的迭代步数可以根据具体的研究问题自行调整，训练的最大迭代次数可以根据具体的研究问题自行调整。
进一步，所述的步骤303中，更新过程所需的各种超参数，如学习率、动量等可以根据具体的研究问题自行调整。
进一步，所述的步骤304中，测试集指标可以根据具体的研究问题自行确定。
第二方面，一种计算机可读存储介质，包括指令，当指令其在计算机上运行时，使得计算机执行如第一方面任意一项的方法。
第三方面，一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行第 一方面任意一项的方法。
本发明的有益效果是：
本发明提出基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品，将求解与训练过程纳入一个统一的框架。纯粹的数据驱动方式的求解与训练过程是分离进行的，需要大量的标签构建优化目标，而本发明通过估计神经算子的输出泛函，并引入变分操作构建无标签的优化目标，从而减少了训练过程中所需的标签，进而减少了传统求解器的使用。优化设计与不确定性量化等领域需要对偏微分方程组的参数集而非单个参数进行求解，传统的求解器，例如基于有限元的求解器，只能针对参数集当中的每一个参数实例进行独立求解，继而遍历地求解整个参数集，而本发明则在求解一个从参数空间到解空间的算子，并同时地学习一个具有泛化能力的神经算子的权重，这保证了在具体地求解某一个参数的同时，本发明能够在其他的参数上获得精度提升。本发明有望在优化设计与不确定性量化等需要对偏微分方程组的大量参数求解的领域替代传统的求解器。
附图说明
图1为本发明实施例提供的基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品的实现流程图；
图2为实施例1边界条件示意图；
图3为实施例1神经算子网络结构示意图；
图4为实施例1的测试集随机抽样5个样本展示图；
图5为实施例2的含变化热源的矩形区域P边界条件示意图；
图6为实施例2的测试集随机抽样5个样本展示图。
具体实施方式
为使本发明解决的技术问题、采用的技术方案和达到的技术效果更为详尽，下面结合附图和实施例对本发明作进一步的详细说明。可以理解的是，此处所描述的具体实施例仅用于解释本发明，而非对本发明的限定。所描述的实施例仅仅是本发明的一部分实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有付出创造性劳动前提下所获得的其他实施例，都属于本发明保护的范围。另外还需要说明的是，为了便于描述，附图中仅示出了本发明相关的部分而非全部内容。
实施例1
本发明实施例提供基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，包括以下步骤：
步骤100：研究变刚度纤维铺层弹性板P，板P的尺寸，材料属性和边界条件见步骤103， 对板P的纤维角度场抽样形成无标签的仅含离散纤维角度场的数据集D，进一步将数据集D划分为偏移集L，测试集T和无标签集U，将无标签集U分为多个批次，对边界条件进行编码，形成边界条件的掩码张量，包括以下子步骤：
步骤101：选定纤维角度场的形式为由4×4控制点网格控制的2阶B样条曲面的z坐标分量，固定控制点的x和y分量，则对于点(xp,yp)，其纤维角Z(xp,yp)由式(1.1)确定：
其中，(up,vp)是点(xp,yp)的参数坐标，Ni,2和Nj,2分别是x和y方向的第(i+1)和第(j+1)个控制点的基函数，i和j分别为x和y方向的控制点标号，抽样策略选为拉丁超立方抽样；
步骤102：对所研究的纤维铺层弹性板P，选择单元为平面4节点的双线性缩减积分单元，网格设置采用均匀的网格划分，网格划分密度为32×32个单元，共33×33个节点，并对纤维角度场的4×4控制点的z坐标进行拉丁超立方抽样，对所有z坐标抽样的范围均为从而实现对纤维角度场的抽样，抽样的纤维角度场数目为100000个，随机抽取并离散其中14005个纤维角度场，形成无标签的仅含离散纤维角度场的离散纤维角度场数据集D，其中，纤维角度场在网格的高斯点处进行离散，离散纤维角度场形状为32×32×1；
步骤103：从数据集D随机抽样2005个离散纤维角度场，求解这2005个离散纤维角度场对应的偏微分方程组的节点位移解，并将其中前5个离散纤维角度场及其对应的节点位移解作为偏移集L，剩余的2000个离散纤维角度场及其对应的节点位移解作为测试集T，本例中的节点位移解由ABAQUS商用软件求解获得，计算时采用ABAQUS商用软件单元库中的S4R单元，ABAQUS软件中的网格划分采用均匀划分，网格的密度为32×32个单元，共33×33个节点，本例中研究的变刚度纤维铺层弹性板P的边界条件如图2所示，本例中研究的变刚度纤维铺层弹性板P的尺寸为：板P为正方形板，长和宽均为100mm，即OA＝OB＝100mm，板厚度为0.125mm，板P的材料属性为：纤维方向的弹性模量E1＝126000MPa，垂直于纤维方向的弹性模量为E2＝11000MPa，1-2面内的泊松比ν12＝0.28；1-2面内的剪切模量为G12＝6600MPa，板P的边界条件为：固定板P的左边OA和下边OB，并对上边BC施加向上的线载荷F1和向右的线载荷F3，对右边AC施加向右的线载荷F2和向上的线载荷F4，F1＝5N/mm,F3＝5x N/mm，F2＝5N/mm，F4＝5y N/mm；
本实施例中，测试集随机抽样5个样本展示图如图4所示，其中，5个样本以五列展示，每一列是一个样本，其中，第一行为5个样本的纤维路径展示，第二行为5个样本的纤维角度场(单位：rad)，第三行为神经算子对5个样本的位移节点解的x分量预测(单位：mm)， 第四行为5个样本位移节点解的x分量的标签(单位：mm)，第五行为神经算子对5个样本的位移节点解的x分量预测绝对值误差(单位：mm)，第六行为神经算子对5个样本的位移节点解的y分量预测(单位：mm)，第七行为5个样本位移节点解的y分量的标签(单位：mm)，第八行为神经算子对5个样本的位移节点解的y分量预测绝对值误差(单位：mm)。
步骤104：将上一步骤抽样的2005个离散纤维角度场排除出数据集D，得到包含12000个离散纤维角度场的无标签集U作为训练集；
步骤105：设定记录迭代次数的变量i，并设置其值为0，将无标签集U分为12000个批次的离散纤维角度场样本，每个批次的大小为1；
步骤106：对边界条件进行编码，形成形状为(1，33，33，2)的边界条件掩码张量M，在M当中，位置对应于被约束的节点自由度的M元素被设为0，其余元素被设为1；
步骤200：使用神经算子模块F，F包含两个相同设置的隐式傅里叶神经算子(Implicit Fourier neural operator,IFNO)，每个隐式傅里叶神经算子负责一个位移分量的预测，每个傅里叶神经算子的设置如图3所示，预测无标签集U中离散纤维角度场样本所对应形状为32×32×2的节点位移，进一步根据节点位移得到离散化的泛函作为系统泛函的估计，包括以下子步骤：
步骤201：不放回地抽取无标签集U中一批离散纤维角度场样本s，如果所有的批次都被抽出，变量i自增1，判断对神经算子的训练过程是否达到算法收敛条件，如果收敛，则输出神经算子的权重θ，并执行步骤304，否则，打乱无标签集U，并将打乱后的无标签集U重新分为12000个批次，从第一个批次重新抽取、训练；其中，所述的收敛条件是i达到训练的最大迭代次数，本实施例中训练的最大迭代次数为500；
步骤202：将离散纤维角度场样本s输入神经算子模块F进行推理，利用掩码张量M对F输出的张量F(s)进行掩码操作，得到节点位移解a，如式(1.2)所示；
a＝F(s)⊙M        (1.2)
步骤203：使用偏移集L内所有标签的均值张量mean和标准差张量std，对节点位移a进行偏移处理；
a＝a⊙std+mean         (1.3)
步骤204：对节点位移a进行卷积操作，得到在高斯点处的解及其空间导数；
步骤205：利用张量操作处理高斯点处的解和空间导数求得泛函被积函数在高斯点处的值；
步骤206：使用高斯积分法，根据泛函的被积函数在高斯点处的值，求得离散化的泛函Π，如式(1.4)所示；
其中，ae代表每个单元的节点位移，下标“l”在单元内第l个积分点处取值，N代表形函数矩阵，B代表应变-位移矩阵，D代表材料属性阵，Je代表单元的雅克比矩阵，f代表体力，代表面力，Hl，Iml分别代表在单元内第l个积分点的权重和第m个单元边界的第l个积分点的权重。
步骤300：对离散化的泛函施加变分操作，具体的：计算泛函估计关于节点位移a的梯度R，并将其范数作为最小化目标，采用基于方程组的迭代法获取当前节点位移的更新步长△a，并利用△a更新神经算子模块F的权重θ，包括以下子步骤：
步骤301：利用自动微分技术，对泛函估计进行反向传播，记录泛函估计对节点位移a的梯度R，如式(1.5)所示，并将R的范数作为最小化目标；
步骤302：将梯度R输入共轭梯度法，共轭梯度法运行两个迭代步，得到当前的节点位移的更新步长△a；
步骤303：基于随机梯度下降法，利用△a对神经算子的权重θ进行更新，学习率设为1，之后返回执行步骤201；
步骤304：将神经算子模块F在测试集T上进行推理，并选取测试集指标为平均相对2-范数误差，如式(1.6)所示；
本实施例神经算子网络各个层功能描述与超参数设置如表1所示：
表1
第二方面，一种计算机可读存储介质，包括指令，当指令其在计算机上运行时，使得计算机执行如第一方面任意一项的方法。
第三方面，一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行第一方面任意一项的方法。
针对现有基于数据驱动的训练神经算子方法与传统求解器的不足，本发明提出了基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品。复合材料在航空航天、体育装备、汽车制造等行业应用广泛，变刚度复合材料更是因其能通过纤维路径优化设计实现力学承载性能的最大化，而在轻量化设计中受到青睐。变角度纤维铺层是变刚度复合材料的基本组成部分，实现变角度纤维铺层的力学响应的批量实时分析对加速变刚度复合材料的优化设计具有重要意义。本发明实施例1面向各向异性的变角度纤维铺层仿真，在偏移集L仅使用5个标签、训练集完全无标签的情况下，在容量为2000的测试集上达到了2.93％的平均相对2-范数误差，已达到纯粹数据驱动的训练误差水平，从而证明了本发明能够应用于变角度纤维铺层的力学响应的批量实时、低成本高效能的仿真分析。由于没有使用数据驱动的训练误差项，本发明可以节省生成用于构建数据驱动误差项所需的大量标签的时间与算力，并且将求解偏微分方程组和训练神经算子两个任务统一在一个框架内完成，为下游应用如优化设计、反问题等 提供有力支持。
实施例2
第一方面，本发明提供基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，包括以下步骤：
步骤100：研究含变化热源的矩形区域P的传热问题，区域尺寸和边界条件见步骤103，对热源抽样形成无标签的仅含离散的热源场数据集D，进一步将数据集D划分为偏移集L，测试集T和无标签集U，将无标签集U分为多个批次，对边界条件进行编码，形成边界条件的掩码张量，包括以下子步骤：
步骤101：选定热源场的形式为高斯随机场：
其中，抽样策略选为简单随机抽样；
步骤102：对所研究含变化热源的矩形区域P，选择单元为平面4节点的双线性单元，网格设置采用均匀的网格划分，网格划分密度为30×30个单元，共31×31个节点，并对热源场进行简单随机抽样和离散化，抽样的数目为12010个，从而形成无标签的仅含离散热源场的数据集D，其中，热源场在网格的高斯点处进行离散，离散的热源场形状为30×30×1；
步骤103：从数据集D随机抽样2010个离散的热源场，求解这2010个离散的热源场对应的偏微分方程组的节点温度解，并将其中前10个离散的热源场及其对应的节点温度解标签作为偏移集L，剩余的2000个离散的热源场及其对应的节点温度解标签作为测试集T，本例中的节点温度解标签由COMSOL商用软件求解获得，计算时采用COMSOL商用软件单元库中的平面4节点的双线性单元，COMSOL软件中的网格划分采用均匀划分，网格划分密度为30×30个单元，共31×31个节点，本例中研究的含变化热源的矩形区域P的边界条件如图5所示，本例中，区域P长和宽均为1m，即OA＝OB＝1m，区域P的四个边界均设为温度为0℃的狄利克雷边界条件；
本实施例中，测试集随机抽样5个样本展示图如图6所示，其中，5个样本以五列展示，每一列是一个样本，其中，第一行为5个样本的热源场(单位：W/m2)，第二行为神经算子对5个样本的温度节点解(单位：℃)，第三行为5个样本温度节点解标签(单位：℃)，第四行为神经算子对5个样本的温度节点解的预测绝对值误差(单位：℃)；
步骤104：将上一步骤抽样的2010个离散的热源场排除出数据集D，得到包含10000个离 散的热源场的无标签集U作为训练集；
步骤105：设定记录迭代次数的变量i，并设置其值为0，将无标签集U分为157个批次离散的热源场样本，其中156个批次的大小为64，最后1个批次大小为16；
步骤106：对边界条件进行编码，形成形状为(1，31，31，1)的边界条件掩码张量M，在M当中，位置对应于被约束的节点自由度的M元素被设为0，其余元素被设为1；
步骤200：使用神经算子模块F，F包含一个隐式傅里叶神经算子(Implicit Fourier neural operator,IFNO)，负责温度的预测，预测无标签集U中离散的热源场样本所对应形状为31×31×1的节点温度，进一步根据节点温度得到离散化的泛函作为系统泛函的估计，包括以下子步骤：
步骤201：不放回地抽取无标签集U中一批离散纤维角度场样本s，如果所有的批次都被抽出，变量i自增1，判断对神经算子的训练过程是否达到算法收敛条件，如果收敛，则输出神经算子的权重θ，并执行步骤304，否则，打乱无标签集U，并将打乱后的无标签集U重新分为157个批次，其中156个批次的大小为64，最后1个批次大小为16，从第一个批次重新抽取、训练；其中，所述的收敛条件是i达到训练的最大迭代次数，本实施例中训练的最大迭代次数为5000；
步骤202：将离散的热源场样本s输入神经算子模块F进行推理，利用掩码张量M对F输出的张量F(s)进行掩码操作，得到节点温度解a，如式(1.2)所示；
a＝F(s)⊙M         (1.2)
步骤203：使用偏移集L内所有标签的均值张量mean和标准差张量std，对节点温度a进行偏移处理；
a＝a⊙std+mean       (1.3)
步骤204：对节点温度a进行卷积操作，得到在高斯点处的解及其空间导数；
步骤205：利用张量操作处理高斯点处的解和空间导数求得泛函被积函数在高斯点处的值；
步骤206：使用高斯积分法，根据泛函的被积函数在高斯点处的值，求得离散化的温度泛函
步骤300：对离散化的温度泛函施加变分操作，具体的：计算泛函估计关于节点温度a的梯度R，并将其范数作为最小化目标，采用基于方程组的迭代法获取当前节点温度的更新步长△a，并利用△a更新神经算子模块F的权重θ，包括以下子步骤：
步骤301：利用自动微分技术，对泛函估计进行反向传播，记录泛函估计对节点温度a的梯度R，如式(1.5)所示，并将R的范数作为最小化目标；
步骤302：将梯度R输入共轭梯度法，共轭梯度法运行两个迭代步，得到当前的节点温度的更新步长△a；
步骤303：基于随机梯度下降法，利用△a对神经算子的权重θ进行更新，学习率设为1e-5，之后返回执行步骤201；
步骤304：将神经算子模块F在测试集T上进行推理，并选取测试集指标为平均相对2-范数误差，如式(1.6)所示；
第二方面，一种计算机可读存储介质，包括指令，当指令其在计算机上运行时，使得计算机执行如第一方面任意一项的方法。
第三方面，一种包含指令的计算机程序产品，当其在计算机上运行时，使得计算机执行第一方面任意一项的方法。
针对现有基于数据驱动的训练神经算子方法与传统求解器的不足，本发明提出了基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品。传热问题广泛存在于各类工程技术领域，如能源动力、冶金、化工、交通、建筑建材、机械以及食品、轻工、纺织、医药等传统工业，以及航空航天、核能、微电子、材料、生物医学工程、环境工程、新能源以及农业工程等很多高新技术领域。对传热现象进行准确实时批量仿真，可以帮助我们更好地理解和掌握传热现象，从而更好地应用于实际生产和生活中。例如，可以用于模拟和优化工业生产过程中的传热现象，提高生产效率和产品质量；可以用于模拟和优化建筑物的传热现象，提高建筑物的节能性能；可以用于模拟和优化电子设备的传热现象，提高电子设备的散热性能等。本发明实施例2在研究含变化热源的矩形区域P的传热问题中，在偏移集L仅使用10个标签、训练集完全无标签的情况下，在容量为2000的测试集上实现了仅2.20％的平均相对2-范数误差，已达到纯粹数据驱动的训练误差水平，从而证明本发明能够应用于传热现象的准确实时批量仿真。由于没有使用数据驱动的训练误差项，本发明可以节省生成用于构建数据驱动误差项所需的大量标签的时间与算力，并且将求解偏微分方程组算子和训练神经算子两个任务统一在一个框架内完成，为下游应用如优化设计、反问题等提供有力支持。
最后应说明的是：以上各实施例仅用以说明本发明的技术方案，而非对其限制；尽管参照前述各实施例对本发明进行了详细的说明，本领域的普通技术人员应当理解：其对前述各实施例所记载的技术方案进行修改，或者对其中部分或者全部技术特征进行等同替换，并不使相应技术方案的本质脱离本发明各实施例技术方案的范围。

Claims (7)

  1. 一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，其特征在于，包括以下步骤：
    步骤100：从偏微分方程组的参数空间中抽样形成无标签的仅含离散参数场的数据集D，将数据集D划分为偏移集L、测试集T和无标签集U，将无标签集U分为多个批次，对边界条件进行编码，形成边界条件的掩码张量；
    步骤200：使用神经算子模块F预测无标签集U中离散参数场样本的节点解，根据节点解得到离散化的泛函作为系统泛函的估计；
    步骤300：对离散化的泛函施加变分操作构建优化目标，计算泛函估计Π关于节点解a的梯度R，并将其范数作为最小化目标，采用基于方程组的迭代法获取当前节点解的更新步长△a，并利用△a更新神经算子模块F的权重θ。
  2. 根据权利要求1所述的一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，其特征在于，所述的步骤100子步骤为：
    步骤101：选定偏微分方程组的参数空间的形式和抽样策略；
    步骤102：对偏微分方程组的求解域进行网格划分，并按照步骤101选定的抽样策略对偏微分方程组的参数空间进行抽样，对抽样得到的参数场在网格的高斯点处离散化，形成无标签的仅含离散参数场的离散参数场数据集D；
    步骤103：从离散参数场数据集D随机抽样出(N1+N2)个离散参数场，求解这(N1+N2)个离散参数场对应的偏微分方程组的节点解，并将其中前N1个离散参数场及其对应的节点解作为偏移集L，剩余的N2个离散参数场及其对应的节点解作为测试集T，其中，偏移集L用于对输出的范围进行偏移；
    步骤104：将上一步骤抽样的(N1+N2)个离散参数场排除出数据集D，形成无标签集U作为训练集；
    步骤105：设定记录迭代次数的变量i，并设置其值为0，将无标签集U分为若干批次的离散参数场样本；
    步骤106：对边界条件进行编码，形成形状和节点解相同的边界条件的掩码张量M，在M当中，位置对应于被约束的节点自由度的M元素被设为0，其余元素被设为1。
  3. 根据权利要求2所述的一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，其特征在于，所述的步骤200子步骤为：
    步骤201：不放回地抽取无标签集U中一批离散纤维角度场样本s，如果所有的批次都被抽出，变量i自增1，判断对神经算子的训练过程是否达到算法收敛条件，如果收敛，则输出神经算子的权重θ，并执行步骤304，否则，打乱无标签集U，并将打乱后的无标签集U重新分 为若干个批次，并从第一个批次开始重新进行抽取、训练；其中，所述的收敛条件可以是i达到训练的最大迭代次数或者神经算子的推理精度已达到精度要求；
    步骤202：将样本s输入神经算子模块F进行推理，利用掩码张量M对F输出的张量F(s)进行掩码操作，得到节点解a，如式(1.1)所示；
    a＝F(s)⊙M      (1.1)
    式(1.1)中，⊙代表张量之间进行元素积，下同；
    步骤203：使用偏移集L内所有标签的均值mean和标准差std，对节点解a进行偏移处理：
    a＝a⊙std+mean       (1.2)
    步骤204：对节点解a进行卷积操作，得到在高斯点处的解及其空间导数；
    步骤205：利用张量操作处理高斯点处的解及其空间导数，求得泛函被积函数在高斯点处的值；
    步骤206：使用高斯积分法，根据泛函被积函数在高斯点处的值，求得离散化的泛函估计Π。
  4. 根据权利要求3所述的一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，其特征在于，所述的步骤300子步骤为：
    步骤301：利用自动微分方法，对泛函估计Π进行反向传播，记录泛函估计Π对节点解a的梯度R，并将其范数作为最小化目标；
    步骤302：将梯度R输入基于方程组的迭代法，得到当前的节点解的更新步长△a；
    步骤303：使用△a对神经算子的权重θ进行更新，并返回执行步骤201；
    步骤304：将神经算子模块F在测试集T上进行推理，并统计测试集指标。
  5. 根据权利要求3所述的一种基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法，其特征在于，所述的步骤202中，神经算子模块F为单个神经算子或多个神经算子的组合。
  6. 一种计算机可读存储介质，其特征在于，包括指令，当指令其在计算机上运行时，使得计算机执行如权利要求1-5任意一项所述的统一方法。
  7. 一种包含指令的计算机程序产品，其特征在于，当其在计算机上运行时，使得计算机执行如权利要求1-5任意一项所述的统一方法。
PCT/CN2024/070340 2023-03-30 2024-01-03 基于变分原理的面向神经算子训练与偏微分方程组求解的统一方法、介质及产品 Ceased WO2024198599A1 (zh)

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