Integrated method, medium and product directed to training neuro-operators based on variational principles and solving simultaneous partial differential equations

Abstract

An integrated method, medium and product directed to training a neuro operator based on variational principle and solving partial differential simultaneous equations, comprising the steps of: sampling from the parameter space of partial differential simultaneous equations to form a data set including only discrete parameter fields, dividing it into an offset set, a test set and an untagged set, dividing the untagged set into multiple batches, and forming a mask tensor of boundary conditions; predicting the node solution of the discrete parameter field samples in the untagged set using a neuro operator module, obtaining a discretized functional as an estimation of the system functional; calculating the gradient of the functional estimation with respect to the node solution, taking its norm as the minimization goal, obtaining the update step size of the current node solution, and updating the weight of the module. The present invention integrates and completes the two tasks of solving partial differential simultaneous equations operator and training a neuro operator in one frame, and introduces variational operations to construct an untagged optimization goal to replace the training error term of the data driver, thereby saving the time and computational power required to generate a large amount of tags required to construct the data driver error term.

Description

[Technical field]

[0001] The present invention relates to an integrated method, medium and product that belongs to the intersection of physical simulation and machine learning, and in particular is directed to training neuro-operators based on variational principles and solving simultaneous partial differential equations. [Background technology]

[0002] Neuro operators are deep neural networks that map operators between parameter space and solution space that can effectively learn partial differential simultaneous equations. Compared to traditional surrogate models such as Gaussian process models and radial basis function models, they can perform highly accurate instantaneous inference for the entire domain of solving partial differential simultaneous equations, and have attracted widespread attention in many physical simulation fields such as fluids, solids, electromagnetics, and heat conduction. Neuro operators have many advantages such as potential super-resolution, high flexibility, and excellent generalization performance, and have a huge birth space in industries such as industrial equipment digital twins, real-time and large-scale simulation calculations, animation game modeling rendering, and metaverse, and are likely to become the core technology of next-generation physical simulation solvers.

[0003] However, the data driver-based training method for traditional neuro-operators requires large tag data from traditional solvers, which causes heavy use of traditional solvers, thereby resulting in huge computational burdens. In addition, the training process itself also brings about certain computational costs. Summary of the Invention [Problem to be solved by the invention]

[0004] Regarding the shortcomings of the traditional data driver-based training neuro-operator method, the present invention provides an integrated method, medium and product directed to training neuro-operator based on variation principle and solving partial differential simultaneous equations, estimates discrete functional by predicting node solution of neuro-operator, minimizes gradient norm for node solution of discrete functional obtained by using automatic differentiation technique by iterative method based on simultaneous equations, and constructs tag-free optimization objective to train neuro-operator, thereby avoiding using traditional solver extensively to obtain tag and saving computational cost.Moreover, the present method utilizes generalization ability of neuro-operator to provide good initial solution for each step of iteration, and essentially combines training and solving into an integrated frame. [Means for solving the problem]

[0005] To achieve the above objectives, the technical solutions adopted in the present invention are as follows:

[0006] In a first aspect, an integrated method for training a neuro-operator based on a variational principle and solving a system of partial differential equations includes the following steps 100, 200, and 300: In step 100, the steps of sampling from a parameter space of the partial differential simultaneous equations to form a data set D including only untagged discrete parameter fields, partitioning the data set D into an offset set L, a test set T and an untagged set U, splitting the untagged set U into multiple batches, encoding the boundary conditions, and forming the mask tensor of the boundary conditions include: In step 101, a form of a parameter space of a system of partial differential equations and a sampling policy are selected; In step 102, a domain for solving the partial differential simultaneous equations is meshed, a parameter space of the partial differential simultaneous equations is sampled according to the sampling policy selected in step 101, and the parameter field obtained by the sampling is discretized at Gaussian points of the mesh to form a discrete parameter field data set D including only untagged discrete parameter fields, specifically, the number of sampling discrete parameter fields and the shape of the discrete parameter field are determined; In step 103, randomly sample (N1+N2) discrete parameter fields from the discrete parameter field data set D, and obtain node solutions of the partial differential simultaneous equations corresponding to the (N1+N2) discrete parameter fields, among which the first N1 discrete parameter fields and their corresponding node solutions (tags) are set as an offset set L, and the remaining N2 discrete parameter fields and their corresponding node solutions are set as a test set T, among which the offset set L is used to offset the range of the output, and for details, refer to step 203; In step 104, removing the (N1+N2) discrete parameter fields sampled in the previous step from the data set D to form an untagged set U as a training set; In step 105, a variable i for the number of recording iterations is set to 0, and the untagged set U is divided into several batches of discrete parameter field samples; and in step 106, encoding the boundary conditions to form a mask tensor M of the boundary conditions of the same shape as the nodal solution, setting the M elements of M corresponding to the nodal degrees of freedom whose positions are constrained to 0 and the remaining elements to 1; In step 200, the step of predicting a nodal solution of the discrete parameter field samples in the untagged set U using a neuro-operator module F and further obtaining a discretized functional based on the nodal solution as an estimate of the system functional, comprises the steps of: In step 201, extract one batch of discrete fiber angle field samples s from the untagged set U without backfilling; when all the batches are extracted, the variable i itself is incremented by 1; judge whether the training process for the neuro operator reaches the algorithm convergence condition; if convergence, output the weight θ of the neuro operator; execute step 304; if not, perturb the untagged set U, re-divide the perturbed untagged set U into several batches, and re-extract from the first batch and train; wherein the convergence condition may be that i reaches the maximum number of training iterations or the inference accuracy of the neuro operator reaches the accuracy requirement; In step 202, the sample s is input to the neuro-operator module F to perform inference, and the mask tensor M is used to perform a mask operation on the tensor F(s) output from F to obtain a node solution a, as shown in equation (1.1): In step 203, the mean value and standard deviation of all tags in the offset set L are used to perform offset processing on the node solution a. In step 204, a convolution operation is performed on the node solution a to obtain a solution at a Gaussian point and its spatial derivative; In step 205, processing the solution and its spatial derivatives at the Gaussian points using tensor operations to find values ​​at the Gaussian points of the integrand of the functional; In step 206, a discretized functional estimate II is obtained based on the values ​​of the integrand of the functional at the Gaussian points using a Gaussian integration method; In step 300, a variation operation is performed on the discretized functional to construct an optimization goal. Specifically, the gradient R of the node solution a of the functional estimation II is calculated, and its norm is set as the minimization goal. An iterative method based on simultaneous equations is adopted to obtain the update step size Δa of the current node solution, and Δa is used to update the weight θ of the neuro-operator module F. In step 301, an automatic differentiation method is used to perform backpropagation on the functional estimation II, a gradient R for the node solution a of the functional estimation II is recorded, and the norm of the gradient R is set as a minimization target; In step 302, the gradient R is input to an iterative method based on simultaneous equations to obtain an update step size Δa for the current node solution; In step 303, the weight θ of the neuro-operator is updated using Δa, and the process returns to step 201; In step 304, the neuro-operator module F is inferred on the test set T and the test set indices are calculated.

[0007] Furthermore, in step 101, the parameter space of the partial differential simultaneous equations may be a space formed by the coefficients or free terms of the partial differential simultaneous equations, the parameter space form of the partial differential simultaneous equations may be 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 argument, and the sampling policy may be a sampling method such as a simple random sampling method and a Latin parcube sampling method.

[0008] Furthermore, in step 102, the mesh division settings can be adjusted automatically based on the specific problem being investigated. Sampling the parameter space of the partial differential simultaneous equations can mean directly sampling the parameters of the partial differential simultaneous equations, or sampling the arguments that control the parameters of the partial differential simultaneous equations. For example, for the heat conduction partial differential equations that control heat conduction, the heat source terms can be directly sampled; for the elastic mechanics partial differential simultaneous equations that control a fiber-laminated elastic plate, the principal direction coordinate axis angles of the material that controls the elastic coefficient in this structural simultaneous equation can be sampled, that is, the fiber angle field is sampled.

[0009] Furthermore, in step 103, the analysis method used to solve the partial differential simultaneous equations may be finite element analysis, boundary element analysis, geometric analysis and similar analysis methods, such as meshless methods.

[0010] Moreover, in step 105, the batch size and number can be adjusted independently based on the specific considerations.

[0011] Furthermore, in step 202, 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 structures and super parameters of the network structures of all the neural networks of the neural operator module F can be adjusted by themselves according to the specific problem being studied.

[0012] Furthermore, in step 205, the specific form of obtaining the tensor operation of the integrand of the functional 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 simultaneous equation to be solved.

[0013] Furthermore, in step 302, the iterative method based on simultaneous equations includes, but is not limited to, the steepest descent method and the conjugate gradient method, the number of iteration steps of the iterative method based on simultaneous equations can be adjusted automatically based on the specific problem being studied, and the maximum number of training iterations can be adjusted automatically based on the specific problem being studied.

[0014] Furthermore, in step 303, various superparameters required in the updating process, such as learning rate, momentum, etc., can be adjusted by themselves according to the specific problem being studied.

[0015] Furthermore, in step 304, the test set index can be determined autonomously based on the specific study problem.

[0016] In a second aspect, a computer readable storage medium comprising instructions that, when executed on a computer, cause a computer to carry out the method of any one of the first aspects.

[0017] In a third aspect, there is provided a computer program product comprising instructions which, when executed on a computer, cause the computer to carry out a method according to any one of the first aspects. Effect of the Invention

[0018] The beneficial effects of the present invention are as follows:

[0019] The present invention provides an integrated method, medium and product for training a neuro operator based on the variation principle and solving a system of partial differential equations, and integrates the solving and training processes into one integrated frame. The solving and training processes of a pure data-driven method are performed separately, and require a large amount of tags to construct an optimization goal, but the present invention estimates the output functional of a neuro operator and introduces variational operations to construct an optimization goal without tags, thereby reducing the tags required for the training process and further reducing the use of traditional solvers. In fields such as optimization design and uncertainty quantization, it is necessary to solve a set of parameters of a system of partial differential equations rather than a single parameter, and traditional solvers, such as solvers based on finite elements, only independently solve each parameter instance in the parameter set, and then ergodically solve the entire parameter set, but the present invention solves the operator from one parameter space to the solution space while learning the weights of a neuro operator with one generalization ability, which ensures that the present invention obtains accuracy improvement in other parameters while solving a specific parameter. The present invention is expected to replace conventional solvers in fields that require solving large numbers of parameters of simultaneous partial differential equations, such as optimization design and uncertainty quantization. [Brief description of the drawings]

[0020] [Figure 1]1 is a flowchart showing an implementation of an integrated method, medium and product directed to training neuro-operators based on variational principles and solving simultaneous partial differential equations provided in an embodiment of the present invention. [Diagram 2] FIG. 2 is a diagram showing boundary conditions in the first embodiment. [Diagram 3] FIG. 2 is a diagram showing the structure of a neuro operator network in the first embodiment. [Figure 4] This is a display diagram of randomly sampling five samples of the test set in Example 1. [Diagram 5] FIG. 13 is a diagram showing boundary conditions of a rectangular region P including a variable heat source in the second embodiment. [Figure 6] FIG. 13 is a diagram showing random sampling of five samples of the test set in Example 2. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0021] In order to provide a more detailed description of the technical problems to be solved, the technical solutions adopted, and the technical effects achieved by the present invention, the present invention will be described in more detail below with reference to the drawings and examples. It should be understood that the specific embodiments described herein are for the purpose of interpreting the present invention, and do not limit the present invention. The described embodiments are only some of the embodiments of the present invention, and are not all of the embodiments. Any other embodiments obtained based on the embodiments of the present invention without the need for creative efforts by those skilled in the art are within the scope of protection of the present invention. For ease of explanation, the drawings show only parts related to the present invention, rather than the entire contents. EXAMPLES

[0022] An embodiment of the present invention provides an integrated method for training a neuro-operator based on a variational principle and solving a system of partial differential equations, which includes the following steps 100, 200, and 300: In step 100, a variable stiffness fiber laminated elastic plate P is considered, and the size, material attributes and boundary conditions of the plate P are described in detail in step 103. The steps of sampling the fiber angle field of the plate P to form a dataset D including only the discrete untagged fiber angle field, further dividing the dataset D into an offset set L, a test set T and an untagged set U, dividing the untagged set U into multiple batches, encoding the boundary conditions, and forming a mask tensor of the boundary conditions are as follows: JPEG2025514582000005.jpg18170JPEG2025514582000006.jpg14170JPEG2025514582000007.jpg6170 are basis functions of the (i+1)th and (j+1)th control points in x and y directions, where i and j are the control point symbols in x and y directions, respectively, and a sampling policy is selected as Latin percube sampling; Sampling is realized for the fiber angle field by JPEG2025514582000008.jpg33170. The number of sampled fiber angle fields is 100000, of which 14005 fiber angle fields are randomly extracted and discretized to form a discrete fiber angle field dataset D containing only discrete fiber angle fields without tags, in which the fiber angle field is discretized at Gaussian points of the mesh, and the shape of the discrete fiber angle field is 32×32×1; In step 103, 2005 discrete fiber angle fields are randomly sampled from the data set D, and node displacement solutions of the partial differential simultaneous equations corresponding to the 2005 discrete fiber angle fields are obtained, among which the first 5 discrete fiber angle fields and their corresponding node displacement solutions are set as the offset set L, and the remaining 2000 discrete fiber angle fields and their corresponding node displacement solutions are set as the test set T. The node displacement solutions in this example are obtained by ABAQUS commercial software, and the S4R unit in the ABAQUS commercial software unit library is adopted in the calculation. The mesh division in the ABAQUS software adopts uniform division, the mesh density is 32×32 units, and a total of 33×33 nodes. The boundary conditions of the variable stiffness fiber laminated elastic plate P considered in this example are shown in FIG. 2, and the size of the variable stiffness fiber laminated elastic plate P considered in this example is as follows: Plate P is a square plate, the length and width are both 100 mm, i.e., OA = OB = 100 mm, the thickness of the plate is 0.125 mm, and the material attributes of plate P are as follows: the elastic modulus in the fiber direction E1 = 126000 MPa, the elastic modulus perpendicular to the fiber direction E2 = 11000 MPa, and the Poisson's ratio in the 1-2 plane V 12 = 0.28, and the shear elasticity in the 1-2 plane G 12 = 6600MPa, and the boundary conditions of the plate P are as follows: The left side OA and the bottom side OB of the plate P are fixed, an upward wire load F1 and a rightward wire load F3 are applied to the top side BC, and a rightward wire load F2 and an upward wire load F4 are applied to the right side AC, where F1 = 5 N / mm, F3 = 5x N / mm, F2 = 5 N / mm, and F4 = 5y N / mm. In this embodiment, the display diagram of the five samples of the test set randomly sampled is shown in FIG. 4, in which the five samples are displayed in five columns, each column is one sample, in which the first row displays the fiber paths of the five samples, the second row displays the fiber angle fields of the five samples (unit: rad), the third row displays the x-component predictions (unit: mm) of the neuro operator for the displacement node solutions of the five samples, and the fourth row displays the x-component predictions (unit: mm) of the displacement node solutions of the five samples. The fifth line is the tag of the x component (unit: mm), the fifth line is the prediction of the absolute value error for the x component of the displacement node solution of the five samples of the neuro operator (unit: mm), the sixth line is the prediction of the y component (unit: mm) for the displacement node solution of the five samples of the neuro operator, the seventh line is the tag of the y component (unit: mm) for the displacement node solution of the five samples of the neuro operator, and the eighth line is the prediction of the absolute value error for the y component of the displacement node solution of the five samples of the neuro operator (unit: mm); In step 104, removing the 2005 discrete fiber angle fields sampled in the previous step from the data set D to obtain an untagged set U containing 12000 discrete fiber angle fields as a training set; In step 105, a variable i for the number of recording iterations is set to 0, and the untagged set U is divided into 12000 batches of discrete fiber angle field samples, each batch having a size of 1; In step 106, encoding the boundary conditions to form a boundary condition mask tensor M of shape (1, 33, 33, 2) by setting the M elements of M corresponding to the position-constrained nodal degrees of freedom to 0 and the remaining elements to 1; In step 200, a neural operator module F is used, where F includes two identically placed hidden Fourier neural operators (IFNO), each of which is responsible for predicting one displacement component, and the placement of each Fourier neural operator is shown in FIG. 3 , to predict node displacements of shape 32×32×2 corresponding to the discrete fiber angle field samples in the untagged set U, and further to obtain a discretized functional based on the node displacements as an estimate of the system functional, In step 201, extract one batch of discrete fiber angle field samples s from the untagged set U without replacing them; when all the batches are extracted, the variable i itself is incremented by 1; judge whether the training process for the neuro operator reaches the algorithm convergence condition; if it converges, output the weight θ of the neuro operator; execute step 304; if not, scramble the untagged set U; re-divide the scrambled untagged set U into 12000 batches, and re-extract from the first batch and train; where the convergence condition is that i reaches the maximum number of training iterations, and the maximum number of training iterations in this embodiment is 500; In step 202, the discrete fiber angle field samples s are input to a neuro-operator module F to perform inference, and a mask tensor M is used to perform a mask operation on the tensor F(s) output from F to obtain a node displacement solution a, as shown in equation (1.2): In step 203, the node displacement a is offset using the mean tensor mean and standard deviation tensor std of all tags in the offset set L. In step 204, a convolution operation is performed on the node displacement a to obtain a solution at a Gaussian point and its spatial derivative; In step 205, processing the solution and its spatial derivatives at the Gaussian points using tensor operations to find values ​​at the Gaussian points of the integrand of the functional; In step 206, a discretized functional II is obtained based on the values ​​of the integrand of the functional at the Gaussian points using the Gaussian integration method, as shown in equation (1.4): JPEG2025514582000011.jpg13170(1.4) JPEG2025514582000012.jpg31170JPEG2025514582000013.jpg28170JPEG2025514582000014.jpg17170JPEG2025514582000015.jpg13170In step 302, the gradient R is input to the conjugate gradient method, which performs two iteration steps to obtain an update step size △a of the current node displacement; In step 303, update the weight θ of the neuro operator using Δa based on the random gradient descent method, and set the learning rate to 1, and then return to step 201 and execute; In step 304, a neuro-operator module F is inferred on the test set T, and the test set index is selected as the average relative 2-norm error, as shown in equation (1.6): JPEG2025514582000016.jpg20170

[0023] The function description and super parameter settings of each layer of the neuro-operator network of this embodiment are shown in Table 1.

[0024] JPEG2025514582000017.jpg153170

[0025] In a second aspect, a computer readable storage medium comprising instructions that, when executed on a computer, cause a computer to carry out the method of any one of the first aspects.

[0026] In a third aspect, there is provided a computer program product comprising instructions which, when executed on a computer, cause the computer to carry out a method according to any one of the first aspects.

[0027] Regarding the deficiencies of the traditional data driver-based neuro operator training method and the traditional solver, the present invention provides an integrated method, medium and product oriented to training neuro operators based on the variation principle and solving partial differential simultaneous equations. Composite materials are widely applied in industries such as aerospace, sports equipment and automobile manufacturing, and variable stiffness composite materials are also favored in lightweight design because they can maximize mechanical load performance through the optimization design of fiber paths. Variable angle fiber lamination is a basic component of variable stiffness composite materials, and realizing batch real-time analysis of the mechanical response of variable angle fiber lamination is of great significance to accelerating the optimization design of variable stiffness composite materials. Example 1 of the present invention is aimed at anisotropic variable angle fiber lamination simulation, and when only 5 tags are used in the offset set L and there are no tags in the training set, it achieves an average relative 2-norm error of 2.93% on a test set with a capacity of 2000, which has already achieved the training error level of pure data drivers, thereby proving that the present invention can be applied to batch real-time, low-cost and high-efficiency simulation analysis of the mechanical response of variable angle fiber lamination. Because the data driver training error item is not used, the present invention can save the time and computational power required to generate a large number of tags required to construct the data driver error item, and integrates and completes the two tasks of solving partial differential simultaneous equations and training neuro-operators in one frame, providing strong support for the optimization design, inverse problems, etc. of downstream applications. EXAMPLES

[0028] In a first aspect, the present invention provides an integrated method for training a neuro-operator based on a variational principle and solving a system of partial differential equations, comprising the following steps 100, 200, and 300: In step 100, a heat conduction problem in a rectangular region P containing a varying heat source is considered, and the region size and boundary conditions are as shown in step 103. The steps of sampling the heat source to form a dataset D containing only discrete untagged heat source fields, further dividing the dataset D into an offset set L, a test set T and an untagged set U, dividing the untagged set U into multiple batches, encoding the boundary conditions, and forming a mask tensor of the boundary conditions are as follows: In step 101, the form of the heat source field is selected as a Gaussian random field; JPEG2025514582000018.jpg23170JPEG2025514582000019.jpg17170In step 102, for the rectangular region P containing the considered variable heat source, a bilinear unit with a planar 4-node unit is selected, the mesh installation adopts uniform mesh division, the mesh division density is 30×30 units, and a total of 31×31 nodes; simple random sampling and discretization is performed on the heat source field, and the number of samplings is 12010, thereby forming a data set D containing only untagged discrete heat source fields, in which the heat source field is discretized at Gaussian points of the mesh, and the shape of the discrete heat source field is 30×30×1; In step 103, 2010 discrete heat source fields are randomly sampled from the data set D, and the node temperature solutions of the partial differential simultaneous equations corresponding to the 2010 discrete heat source fields are obtained. The first 10 discrete heat source fields and their corresponding node temperature solution tags are set as an offset set L, and the remaining 2000 discrete heat source fields and their corresponding node temperature solution tags are set as a test set T. The node temperature solution tags in this example are obtained by the COMSOL commercial software and are used during the calculation. The planar 4-node bilinear unit in the COMSOL commercial software unit library is adopted, and the mesh division in the COMSOL software adopts uniform division, and the mesh division density is 30 × 30 units, for a total of 31 × 31 nodes. The boundary conditions of the rectangular domain P containing the variable heat source considered in this example are shown in Figure 5. In this example, the length and width of the domain P are both 1m, that is, OA = OB = 1m, and the four boundaries of the domain P are all set to Dirichlet boundary conditions with a temperature of 0. In this embodiment, the display diagram of the five samples of the test set randomly sampled is shown in FIG. 6, in which the five samples are displayed in five columns, each column is one sample, in which the first row is the heat source field (unit: W / m 2 ), the second line is the temperature node solution (unit: °C) for the five samples of the neuro operator, the third line is the five sample temperature node solution tag (unit: °C), and the fourth line is the absolute error prediction (unit: °C) for the five samples of the neuro operator temperature node solution; In step 104, the 2010 discrete heat source fields sampled in the previous step are removed from the data set D, and an untagged set U containing 10000 discrete heat source fields is obtained as a training set; In step 105, a variable i for the number of recording iterations is set to 0, and the untagged set U is divided 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; and in step 106, encoding the boundary conditions to form a boundary condition mask tensor M of shape (1, 31, 31, 1), setting the M elements of M corresponding to the position-constrained nodal degrees of freedom to 0 and the remaining elements to 1; In step 200, a neural operator module F, which includes an implicit Fourier neural operator (IFNO), is responsible for predicting the temperature and predicting the nodal temperatures of a shape 31×31×1 corresponding to the discrete heat source field samples in the untagged set U, and further obtaining a discretized functional based on the nodal temperatures as an estimate of the system functional, comprising: In step 201, extract one batch of discrete fiber angle field samples s from the untagged set U without replacing them; when all the batches are extracted, the variable i itself is incremented by 1; judge whether the training process for the neuro operator reaches the algorithm convergence condition; if it converges, output the weight θ of the neuro operator; execute step 304; if not, scramble the untagged set U; re-divide the scrambled untagged set U into 157 batches, of which the size of 156 batches is 64, and the size of the last batch is 16; re-extract from the first batch and train, where the convergence condition is that i reaches the maximum number of training iterations, and the maximum number of training iterations in this embodiment is 5000; In step 202, the discrete heat source field samples s are input to a neuro-operator module F to perform inference, and a mask tensor M is used to perform a masking operation on the tensor F(s) output from F to obtain the node temperature solution a, as shown in equation (1.2): In step 203, the node temperature a is offset using the mean value tensor mean and standard deviation tensor std of all tags in the offset set L. In step 204, a convolution operation is performed on the node temperature a to obtain a solution at a Gaussian point and its spatial derivative; In step 205, processing the solution and its spatial derivatives at the Gaussian points using tensor operations to find values ​​at the Gaussian points of the integrand of the functional; JPEG2025514582000022.jpg11170JPEG2025514582000023.jpg28170JPEG2025514582000024.jpg17170JPEG2025514582000025.jpg12170In step 302, the gradient R is input to the conjugate gradient method, which performs two iteration steps to obtain an update step size △a for the current node temperature; In step 303, update the weight θ of the neuro operator using Δa based on the random gradient descent method, and set the learning rate to 1e-5, and then return to step 201 and execute; In step 304, a neuro-operator module F is inferred on the test set T, and the test set index is selected as the average relative 2-norm error, as shown in equation (1.5): JPEG2025514582000026.jpg20170

[0029] In a second aspect, a computer readable storage medium comprising instructions that, when executed on a computer, cause a computer to carry out the method of any one of the first aspects.

[0030] In a third aspect, there is provided a computer program product comprising instructions which, when executed on a computer, cause the computer to carry out a method according to any one of the first aspects.

[0031] In view of the shortcomings of the traditional data driver-based neuro operator training method and traditional solver, the present invention provides an integrated method, medium and product for training neuro operators based on the principle of variation and for solving partial differential simultaneous equations. Heat conduction problems are widely present in various engineering and technical fields, such as energy power, metallurgy, chemical engineering, transportation, building materials, machinery and food, light industry, textiles, medicine and other traditional industries, and aerospace, nuclear power, microelectronics, materials, biomedical engineering, environmental engineering, new energy and agricultural engineering and other high-tech fields. Accurate and real-time batch simulation of heat conduction phenomenon is helpful to better understand and grasp heat conduction phenomenon, so as to be better applied to real production and life. For example, it can be used to simulate and optimize heat conduction phenomenon in industrial production process to improve production efficiency and product quality, it can be used to simulate and optimize heat conduction phenomenon of buildings to improve energy saving performance of buildings, and it can be used to simulate and optimize heat conduction phenomenon of electronic equipment to improve heat dissipation performance of electronic equipment. In the second embodiment of the present invention, the heat conduction problem of a rectangular region P containing a variable heat source is considered, and when only 10 tags are used in the offset set L and there are no tags in the training set, the average relative 2-norm error of only 2.20% is achieved in the test set with a capacity of 2000, which has already achieved the training error level of pure data driver, thereby proving that the present invention can be applied to accurate real-time batch simulation of heat conduction phenomenon. Because the data driver training error item is not used, the present invention can save the time and computational power required to generate a large number of tags required to build the data driver error item, and the two tasks of solving partial differential simultaneous equations operator and training neuro operator are integrated and completed in one frame, providing strong support for the optimization design, inverse problem, etc. of downstream applications.

[0032] Finally, it should be noted that the above embodiments are used to describe the technical solutions of the present invention, and are not intended to limit the same. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art can understand that the above embodiments may be modified to modify the technical solutions described in the above embodiments, or to equally replace some or all of the technical features therein, without causing the essence of the corresponding technical solutions to depart from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. In step 100, the method includes the steps of: sampling from a parameter space of a system of partial differential equations to form a data set D including only untagged discrete parameter fields; partitioning the data set D into an offset set L, a test set T and an untagged set U; splitting the untagged set U into multiple batches; encoding boundary conditions; and forming a mask tensor for the boundary conditions; In step 200, a neuro-operator module F is used to predict a nodal solution of the discrete parameter field samples in the untagged set U, and a discretized functional is obtained based on the nodal solution as an estimate of the system functional; In step 300, the method includes: constructing an optimization goal by performing a variation operation on the discretized functional; calculating a gradient R with respect to the node solution a of the functional estimation II; setting the norm as a minimization goal; adopting an iterative method based on simultaneous equations to obtain an update step size Δa of the current node solution; and updating a weight θ of the neuro-operator module F using Δa. An integrated method directed to training neuro-operators based on the principle of variations and solving simultaneous partial differential equations, characterized in that:

2. The sub-steps of step 100 include: In step 101, a form and a sampling policy for the parameter space of the partial differential simultaneous equations are selected; In step 102, a domain for solving the simultaneous partial differential equations is meshed, a parameter space of the simultaneous partial differential equations is sampled according to the sampling policy selected in step 101, and the parameter field obtained by the sampling is discretized at Gaussian points of the mesh to form a discrete parameter field data set D including only untagged discrete parameter fields; In step 103, randomly select (N １ +N ２ ) discrete parameter fields are sampled, and the (N １ +N ２ ) discrete parameter fields, and the top N of the nodal solutions of the partial differential simultaneous equations are obtained. １ Let L be the offset set of discrete parameter fields and their corresponding node solutions, and let the remaining N ２ the discrete parameter fields and corresponding node solutions are a test set T, in which an offset set L is used to offset the range of the outputs; In step 104, the sampled (N １ +N ２ ) discrete parameter fields from the data set D to form an untagged set U as a training set; In step 105, a variable i for the number of recording iterations is set to 0, and the untagged set U is divided into several batches of discrete parameter field samples; The integrated method for training neuro-operators based on the variational principle and solving partial differential simultaneous equations according to claim 1, characterized in that in step 106, the boundary conditions are encoded to form a mask tensor M of the boundary conditions having the same shape as the nodal solution, and the M elements of M corresponding to the degrees of freedom of the nodes whose positions are constrained are set to 0, and the remaining elements are set to 1.

3. The sub-steps of step 200 include: In step 201, extract a batch of discrete fiber angle field samples s from the untagged set U without backfilling; when all the batches are extracted, the variable i itself is incremented by 1; judge whether the training process for the neuro operator reaches the algorithm convergence condition; if it converges, output the weight θ of the neuro operator; execute step 304; if not, disturb the untagged set U, re-divide the disturbed untagged set U into several batches, and re-extract from the first batch and train; wherein the convergence condition can be that i reaches the maximum number of iterations of training or the inference accuracy of the neuro operator reaches the accuracy requirement; In step 202, the sample s is input to the neuro-operator module F to perform inference, and a mask operation is performed on the tensor F(s) output from F using the mask tensor M to obtain a node solution a, as shown in equation (1.1): In step 203, the mean value (mean) and the standard deviation (std) of all tags in the offset set L are used to perform offset processing on the node solution a; In step 204, a convolution operation is performed on the node solution a to obtain the solution at the Gaussian points and its spatial derivatives; In step 205, tensor operations are used to process the solution and its spatial derivatives at the Gaussian points to find values ​​at the Gaussian points of the integrand of the functional; The integrated method for training a neuro-operator based on the variation principle and for solving partial differential simultaneous equations according to claim 2, characterized in that in step 206, a discretized functional estimate II is obtained based on the value at the Gaussian point of the integrand of the functional using the Gaussian integral method.

4. The sub-steps of step 300 include: In step 301, the automatic differentiation method is used to perform backpropagation on the functional estimation II, and the gradient R for the node solution a of the functional estimation II is recorded, and the norm of the gradient R is set as the minimization target; In step 302, the gradient R is input to an iterative method based on simultaneous equations to obtain an update step size Δa for the current node solution; In step 303, the weight θ of the neuro operator is updated using Δa, and the process returns to step 201. The integrated method for training a neuro-operator based on the variation principle and for solving partial differential simultaneous equations according to claim 3, characterized in that in step 304, the neuro-operator module F is inferred to the test set T, and the test set index is statistically calculated.

5. The integrated method for training a neuro-operator based on the variation principle and solving a system of partial differential equations according to claim 3, characterized in that in step 202, the neuro-operator module F is a combination of a single neuro-operator and multiple neuro-operators.

6. A computer readable storage medium comprising instructions which, when executed on a computer, cause the computer to carry out the method of any one of claims 1 to 5.

7. A computer program product comprising instructions which, when executed on a computer, cause the computer to carry out the method according to any one of claims 1 to 5.

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