A method and system for solving partial differential equations based on physical information neural networks

Abstract

This invention belongs to the interdisciplinary field of artificial intelligence and computational mathematics, and relates to a method for solving partial differential equations based on a physical information neural network. The method includes: performing coordinate separation sampling on the partial differential equation to obtain a set of sampling points; training the neural network from the previous stage based on the set of sampling points to obtain an approximate solution and a preliminary neural network for the current stage; calculating the residual field and spatial gradient after substituting the predicted solution from the previous stage into the partial differential equation, and constructing a gating function based on the residual field and spatial gradient; updating the solution momentum and solution variance from the previous stage based on the approximate solution for the current stage to obtain the corrected solution momentum and solution variance for the current stage; calculating the predicted solution for the current stage based on the predicted solution from the previous stage and the gating function, solution momentum, and solution variance for the current stage; and calculating the loss function value to update the preliminary neural network for the current stage until the final predicted solution is obtained. The spatiotemporal dual-dimensional adaptive multi-stage training strategy of this invention significantly improves the solution accuracy.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of artificial intelligence and computational mathematics, and in particular relates to a method and system for solving partial differential equations based on physical information neural networks. Background Technology

[0002] Partial differential equations are a core mathematical tool for describing physical phenomena in natural sciences and engineering. Their analytical solutions are often difficult to obtain, so numerical solutions have become the mainstream. Traditional numerical solutions (such as the finite difference method, the finite element method, and the spectral method) require discretizing the continuous computational domain into a grid. As the dimensionality of the problem increases, the number of grids grows exponentially, leading to the "curse of dimensionality," which results in a surge in computational resource consumption and a significant decrease in computational efficiency.

[0003] Physical Information Neural Networks (PINNs), as a deep learning framework that integrates physical laws, embed the governing equations, initial conditions, and boundary conditions of partial differential equations into a loss function. The neural network then fits the solutions to the partial differential equations without requiring grid discretization, demonstrating significant advantages in solving high-dimensional partial differential equations. However, traditional PINNs still face key technical bottlenecks:

[0004] 1. Difficulty in balancing accuracy and efficiency: To improve the accuracy of the solution, it is often necessary to design complex network structures or increase a large number of training sampling points, which leads to a sharp increase in training costs. Moreover, due to the ill-conditioned nature of the loss function, it is easy to get trapped in local optima and fail to obtain a solution that meets the actual needs.

[0005] 2. Insufficient high-dimensional feature extraction capability: Traditional PINN processes high-dimensional coordinates as a whole, making it difficult to fully extract feature information from each dimension. This can easily lead to model instability and insufficient solution accuracy due to information loss.

[0006] 3. Poor adaptability of activation functions: Traditional PINN uses fixed activation functions (such as ReLU and Tanh), which cannot be dynamically adjusted according to the linear or nonlinear characteristics of partial differential equations, and has limited ability to capture high-frequency, multi-scale features;

[0007] 4. Training is prone to stagnation: During the training of a single neural network, the loss function is prone to plateau after a certain number of iterations, and simply increasing the number of iterations or the network size cannot effectively reduce the error of the solution.

[0008] To address the aforementioned issues, existing technologies have proposed coordinate-separated physical information neural networks, which reduce computational load and improve training speed by processing coordinates axis by axis.

[0009] However, the coordinate-separated physical information neural network has not optimized its training strategy and activation function. In high-precision scenarios, the solution accuracy still needs to be improved, and it cannot solve the error stagnation problem of single-stage training. Therefore, there is an urgent need for a physical information neural network optimization method that can significantly improve the solution accuracy of partial differential equations while maintaining computational efficiency. Summary of the Invention

[0010] To address the aforementioned problems in the prior art, one aspect of the present invention proposes a method for solving partial differential equations based on a physical information neural network, comprising:

[0011] S1: Obtain the partial differential equation to be solved, construct d parallel feature extraction neural networks, and initialize the solution momentum. Sum of variances ;

[0012] S2: Perform coordinate separation sampling on the solution domain of the partial differential equation to obtain a set of sampling points for d coordinate axes. ; Set of sampling points for d coordinate axes Each corresponds one-to-one with d parallel feature extraction neural networks;

[0013] S3: Obtain the d feature extraction neural networks from the previous stage t-1, based on the sampling point set of the d coordinate axes. The neural networks for extracting d features from the previous stage t-1 are initially trained to obtain an approximate solution for the current stage t. And the d preliminary feature extraction neural networks at the current stage t;

[0014] S4: Obtain the predicted solution from the previous stage t-1. Calculate the predicted solution Residual field after substituting into the partial differential equation and its spatial gradient According to the residual field and its spatial gradient Constructing a spatial confidence gating function ;

[0015] S5: Obtain the solution momentum of the previous stage t-1. Sum of variances According to the approximate solution Solution momentum Sum of variances Update the solution to obtain the momentum at the current stage t. Sum of variances and its corrected solution momentum Sum of variances ;

[0016] S6: Based on the predicted solution Spatial confidence gating function Solution momentum Solution of variance Calculate the predicted solution for the current stage t. ;

[0017] S7: Calculate the loss function value, update the parameters of the d preliminary feature extraction neural networks in the current stage t based on the loss function value, and obtain the d feature extraction neural networks in the current stage t; determine whether the training meets the convergence condition. If it does, stop the training and obtain the trained d parallel feature extraction neural networks; otherwise, return to step S2 to resample and enter the training of the next stage t+1.

[0018] S8. Solve the partial differential equations using d trained parallel feature extraction neural networks to obtain numerical solutions.

[0019] Another aspect of the present invention provides a partial differential equation solving system based on a physical information neural network, the system comprising a memory and a processor; the memory is used to store an application program; the processor is used to run the application program stored in the memory and execute the aforementioned partial differential equation solving method based on a physical information neural network.

[0020] Beneficial effects:

[0021] 1. This invention designs a spatiotemporal dual-dimensional adaptive multi-stage training strategy that allows the network to learn the residual distribution of previous stages layer by layer. In the spatial dimension, a spatial confidence gating function guided by residual topology is used to automatically identify and focus on correcting discontinuous regions, abruptly changing regions, and high-gradient regions of the solution, avoiding the blindness of spatial correction caused by traditional global unified weights. In the temporal dimension, through exponential moving average estimation and bias correction of solution momentum and variance across stages, correction trend information of multiple stages is accumulated. The solution momentum is normalized and scaled using solution variance to automatically constrain the fusion step size in regions of drastic solution change to prevent overcorrection. At the same time, the global fusion learning rate is dynamically adjusted according to the rate of change of the residual field to automatically decelerate when the residual stagnates or rebounds, and to automatically slow down when the residual decreases rapidly. 1. Maintaining fusion efficiency effectively alleviates the problem of traditional physical information neural network training easily getting trapped in local optima and loss function stagnation, significantly improving solution accuracy without increasing training time; 2. This invention introduces the ACON adaptive activation function, which dynamically adjusts linear and nonlinear transition characteristics through learnable parameters, and can adaptively capture high-frequency, multi-scale features according to the linear and nonlinear characteristics of partial differential equations, improving the adaptability and accuracy of feature extraction and enhancing the network's expressive power; 3. The loss function of this invention integrates physical constraints, preceding output regularization terms, and solution momentum consistency constraints, which not only improves the stability of model training and the coherence of cross-stage optimization, reducing the risk of solution non-convergence caused by information loss, but also ensures the physical rationality and update stability of the solution. Attached Figure Description

[0022] Figure 1 A flowchart illustrating the partial differential equation solving method based on physical information neural network provided in an embodiment of the present invention;

[0023] Figure 2 This is a schematic diagram of a partial differential equation solving model based on a physical information neural network provided in an embodiment of the present invention. Detailed Implementation

[0024] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0025] like Figure 1 , Figure 2 As shown, this embodiment of the invention employs a method for solving partial differential equations based on a physical information neural network, comprising the following steps:

[0026] S1: Obtain the partial differential equation to be solved, construct d parallel feature extraction neural networks, and initialize the solution momentum. Sum of variances ;

[0027] When obtaining the partial differential equation to be solved, determine the solution domain, governing equations, initial conditions, and boundary conditions of the partial differential equation to be solved.

[0028] The construction of the feature extraction neural network includes: building d parallel feature extraction neural networks with the same number of dimensions as the coordinate axes. All networks adopt a unified lightweight fully connected architecture, including an input layer, several hidden layers, a tensor operation layer, and an output layer. The input layer is a 1D model adapted to 1D coordinate sampling points. The number of neurons in the hidden layers is set according to the required solution accuracy. The tensor operation layer is used to generate a high-dimensional feature matrix with a fixed rank. The output layer is a 1D model to complete the feature mapping. An ACON-C adaptive activation function is integrated into each hidden layer to initialize learnable parameters. , With hyperparameters ,in, , Control the asymptotic values ​​of the gradient in the positive and negative regions respectively. To control the steepness of the transition region between linear and nonlinear modes, set a high value during the initial training phase. Value enhancement nonlinearity to quickly fit the low-frequency components of the equation, setting low values ​​in the later stages of training. Value enhancement linearity is used to finely adjust high-frequency errors, enabling adaptive capture of the linear and nonlinear characteristics of partial differential equations.

[0029] The mathematical expression for the ACON-C adaptive activation function is:

[0030]

[0031] in, The sigmoid function is used to achieve a smooth transition between linear and nonlinear states. , Optimize in real time using the backpropagation algorithm.

[0032] S2. Perform coordinate separation sampling on the solution domain of the partial differential equation to obtain a set of sampling points for d coordinate axes. ; Set of sampling points for d coordinate axes Each corresponds one-to-one with d parallel feature extraction neural networks;

[0033] Coordinate separation sampling of the solution domain of partial differential equations includes: decomposing the d-dimensional solution domain according to the coordinate axis dimensions, and independently performing one-dimensional sampling on each coordinate axis using uniform or random sampling methods to obtain a set of sampling points. The set of sampling points for each coordinate axis i n is the number of sampling points for each coordinate axis, and the total number of sampling points is determined by the traditional d-dimensional space. One reduced to This significantly reduces the amount of sampling data and computational costs.

[0034] S3: Obtain the d feature extraction neural networks from the previous stage t-1, based on the sampling point set of the d coordinate axes. The neural networks for extracting d features from the previous stage t-1 are initially trained to obtain an approximate solution for the current stage t. And the d preliminary feature extraction neural networks at the current stage t;

[0035] The initial training of the d feature extraction neural networks from the previous stage t-1 includes:

[0036] S31. Set the sampling points of d coordinate axes Input the corresponding feature extraction neural network from the previous stage t-1 into each of the d coordinate axes to obtain the feature matrices. ;

[0037] Each feature extraction neural network independently performs feature extraction and mapping for the corresponding one-dimensional sampling points, applying a non-linear transformation to the hidden layer features using the ACON-C adaptive activation function. After feature extraction, each neural network generates an r-dimensional high-dimensional feature matrix of the corresponding dimension. ,in, Let r be the trainable parameters such as weights and biases of the network in the previous stage t, and r be the feature enhancement dimension, which can be flexibly adjusted according to the complexity of the equation.

[0038] S32, Characteristic matrix of d coordinate axes Perform outer product summation and fusion to obtain the fused result. ;

[0039] The characteristic matrix of d coordinate axes The outer product summation and fusion process involves performing an outer product summation and fusion operation on the r-dimensional high-dimensional feature matrices generated from each dimension. The fusion formula is as follows:

[0040]

[0041] in, This is the outer product operation of the eigenvectors corresponding to each coordinate axis. The eigenvector of the j-th row of the eigenmatrix of the i-th coordinate axis;

[0042] S33. Calculate the physical constraint loss function value based on the fusion result. Use the gradient descent algorithm to fine-tune the parameters of the d feature extraction neural networks in the previous stage t-1 based on the physical constraint loss function value. When the stopping condition of the initial training is met, the final fusion result and the d preliminary feature extraction neural networks in the current stage t are obtained.

[0043] The physical constraint loss function value is calculated based on the fusion results, including:

[0044] Combining forward mode with fusion results Automatic differentiation yields partial derivatives of various orders for each coordinate. The formula for calculating partial derivatives of any order is:

[0045]

[0046] in, The eigenvector pair of the j-th row of the characteristic matrix of the i-th coordinate axis The first-order partial derivatives. The role of each derivative is to construct the physical constraint loss and characterize the degree to which the current fusion result satisfies the partial differential equation.

[0047] Substituting the partial derivatives of each coordinate into the differential operators of the partial differential equation to be solved yields the fused result. The corresponding partial differential equation residuals;

[0048] The residuals of the partial differential equations, together with the deviations corresponding to the boundary conditions and initial conditions, constitute the physical constraint loss function in the initial training process, i.e. , , , These represent the residual loss, boundary condition loss, and initial condition loss for the partial differential equation, respectively.

[0049] Based on the physical constraint loss function value, the gradient descent algorithm is used to fine-tune the parameters of the d feature extraction neural network in the previous stage t-1, including: the parameters of each feature extraction neural network with respect to the physical constraint loss function value. Calculate the gradient and iteratively update each parameter according to the gradient descent algorithm, so that the updated fusion result gradually approaches the solution that satisfies the governing equation, boundary conditions and initial conditions.

[0050] The initial training is stopped when the fusion result satisfies the physical constraints of the partial differential equation (i.e., the governing equation, initial conditions, and boundary conditions of the partial differential equation).

[0051] S34. The final fusion result is used as an approximate solution to the partial differential equation of the current training phase t. .

[0052] S4: Obtain the predicted solution from the previous stage t-1. Calculate the predicted solution Residual field after substituting into the partial differential equation and its spatial gradient According to the residual field and its spatial gradient Constructing a spatial confidence gating function ;

[0053] Calculate the predicted solution Residual field after substituting into the partial differential equation and its spatial gradient This includes: the predicted solution from the previous stage Substitute the differential operators of the partial differential equation to calculate the PDE residual field. Then, the spatial gradient of the residual field is calculated by automatic differentiation. This gradient reflects the degree of drastic local variation of the residual in space;

[0054] If the partial differential equation is written as Then there is ,in, For differential operators of partial differential equations, For the right-hand side of the equation, the required derivatives are obtained through automatic differential calculation.

[0055] Constructing a spatial confidence gating function based on the absolute value of the residual field and its spatial gradient include:

[0056]

[0057] in, This represents the global mean over the solution domain. The basic fusion rate has a range of values. In this embodiment, the value is taken as ; This is a gradient sensitivity factor, and its value range is... In this embodiment, the value is taken as ; This is the numerical stability constant, and its range is: In this embodiment, the value is taken as The physical meaning of this gating function is as follows: regions with large absolute residual values ​​indicate that the predicted solution from the previous stage deviates significantly from the true solution in that region, requiring greater correction; regions with large residual spatial gradients correspond to discontinuities or abrupt changes in the solution, representing bottlenecks in solution accuracy, and requiring further amplification of the correction weights. The function constrains the gate value within a reasonable range to prevent overcorrection caused by abnormally large residuals.

[0058] S5: Obtain the solution momentum of the previous stage t-1. Sum of variances According to the approximate solution Solution momentum Sum of variances Update the solution to obtain the momentum at the current stage t. Sum of variances and its corrected solution momentum Sum of variances ;

[0059] Time Dimension—Cross-Stage Solution Momentum Memory and Adaptive Learning Rate: An exponential moving average estimate of cross-stage solution momentum and variance is updated based on the current stage's approximate solution.

[0060]

[0061]

[0062] in, The momentum decay coefficient controls the accumulation of solution information in historical stages, and its value ranges from [value missing]. In this embodiment, the value is taken as ; The variance attenuation coefficient controls the tracking speed of the amplitude fluctuations in the solution, and its value range is... In this embodiment, the value is taken as .because , Zero initialization can lead to systematically underestimation of the estimated values ​​in the early stages of training, necessitating bias correction:

[0063]

[0064] in, for The power of t, the solution to momentum The update pertains to the current approximate solution itself, which undergoes changes in sign and direction, accumulating correction direction information from multiple stages. This effectively smooths out random noise in single-stage predictions; solution variance... The update is based on the square of the approximate solution. It has no positive or negative sign, only numerical magnitude, which reflects the fluctuation range of the solution at each spatial location between multiple stages. It plays a normalization role in the fusion formula—the fusion step size is automatically reduced in regions where the solution changes drastically to prevent overcorrection, while a larger fusion step size is maintained in regions where the solution changes steadily to accelerate convergence.

[0065] S6: Based on the predicted solution Spatial confidence gating function Solution momentum Solution of variance Calculate the predicted solution for the current stage t. ;

[0066] The global fusion learning rate is dynamically adjusted based on the changing trend of PDE residuals in adjacent stages:

[0067]

[0068] in, The initial fusion learning rate has a range of values. In this embodiment, the value is taken as , This is the attenuation factor, and its value range is... In this embodiment, the value is taken as ; , Representing the residual fields respectively , The L2 norm at the current sampling point, , The predicted solutions for stage t-1 are respectively Predicted solution for stage t-2 The residual field after substituting into the partial differential equation. When the residual continues to decrease... The value inside is zero. Maintain maximum fusion efficiency; when the residual increases instead of decreasing (optimization backs down), exponentially decay the fusion step size and automatically slow down to prevent divergence.

[0069] Spatiotemporal fusion: By integrating confidence gating in the spatial dimension and angular momentum normalization in the temporal dimension, an adaptive fusion update formula for spatiotemporal dual-dimensional fusion is obtained:

[0070]

[0071] In the formula Provides differentiated correction weights for spatial dimensions. It provides momentum acceleration and variance normalization over time. It provides comprehensive security. The synergistic effect of these three elements enables the fusion process to automatically identify and correct key areas spatially, accumulate correction trends across stages over time, and adaptively adjust the fusion intensity.

[0072] The solution variance after bias correction in this formula Placed in the denominator In the middle, for the solution momentum Normalization scaling is performed. Since this normalization process is conducted independently for different spatial locations, the update intensity can be adaptively adjusted based on the historical fluctuations of the solutions at each location; when a certain region... When the denominator is large, the update amount in that region automatically decreases, thereby reducing the fusion step size and suppressing over-correction; when When the value is small, the update amount is relatively increased to accelerate convergence.

[0073] In the first stage, the predicted solution for the first stage ;in, This is an approximate solution for the first stage.

[0074] S7: Calculate the loss function value, update the parameters of the d preliminary feature extraction neural networks in the current stage t based on the loss function value, and obtain the d feature extraction neural networks in the current stage t; determine whether the training meets the convergence condition. If it does, stop the training and obtain the trained d parallel feature extraction neural networks; otherwise, return to step S2 to resample and enter the training of the next stage t+1.

[0075] Training at each stage is performed using a loss function. Optimize network parameters loss function The expression combines the residual loss, boundary condition loss, initial condition loss, preceding output regularization term, and solution momentum consistency constraint term of the partial differential equation, and is as follows:

[0076]

[0077] in, , , These represent the residual loss, boundary condition loss, and initial condition loss for partial differential equations, respectively. This is the solution momentum after correction in the previous stage t-1. To address the momentum consistency constraint term, this term constrains the deviation between the current approximate solution and the cumulative momentum direction from historical stages, improving the coherence of cross-stage optimization. This ensures that the solution satisfies physical laws while also achieving smooth solution updates. , , , , The weight hyperparameter for the loss term. Let be the network parameters for stage t.

[0078] The residual loss, boundary condition loss, and initial condition loss of partial differential equations are existing losses, which can be specifically expressed as:

[0079]

[0080]

[0081]

[0082] in, , , These represent the number of sampling points within the solution domain, the number of sampling points at the boundary, and the number of sampling points at the initial time step, respectively. , , These represent sampling points within the solution domain, boundary sampling points, and initial sampling points, respectively. For differential operators of partial differential equations, Let be the right-hand term of the equation. For boundary condition functions, This is the initial condition function.

[0083] The convergence condition is set as follows: the rate of change of the error of the predicted solution in adjacent stages is less than a preset threshold or the maximum number of training stages T is reached. The rate of change of the error of the predicted solution in adjacent stages is defined as follows:

[0084]

[0085] in, Let be the rate of change of the predicted solution error at stage t relative to stage t-1. This represents the L2 norm at the current sampling point. To prevent the numerical stability constant from having a denominator of zero, we can take... The preset threshold value range is: In this embodiment, the value is taken as .

[0086] S8. Solve the partial differential equations using d trained parallel feature extraction neural networks to obtain numerical solutions.

[0087] Solving partial differential equations using d trained parallel feature extraction neural networks involves: inputting the values ​​of the d coordinate axes of any coordinate within the solution domain of the partial differential equation into the d trained parallel feature extraction neural networks to obtain feature matrices of the d coordinate axes; performing an outer product of the feature matrices of the d coordinate axes and summing them to obtain the fused result, i.e., the numerical solution.

[0088] The method of this invention has good adaptability to partial differential equations of different dimensions such as 2D, 3D, and 4D, linear and nonlinear types, and regular and irregular geometric domains. For 2D Poisson equations, 3D Helmholtz equations, and 3D diffusion equations, only the core parameters such as coordinate axis dimension d, number of sampling points n, feature enhancement dimension r, and number of training stages T need to be adjusted according to the characteristics of the equation, while the rest of the implementation steps remain the same. It can achieve efficient and high-precision solutions, solving the problem that traditional physical information neural networks cannot balance accuracy and efficiency when solving high-dimensional and complex partial differential equations.

[0089] In one embodiment, taking the solution of a 3D Klein-Gordon nonlinear hyperbolic partial differential equation as an example, the equation is expressed as follows:

[0090]

[0091]

[0092]

[0093] The true solution is The coordinate axis dimension is set to d = 3, the number of sampling points per axis is n = 64, the feature augmentation dimension is r = 128, the number of training stages is T = 5, the Adam optimizer learning rate is 0.001, and the weight hyperparameters for each loss term are set. Solving the momentum consistency constraint weights The spatiotemporal dual-dimensional adaptive fusion parameters are set as follows: basic fusion rate. gradient sensitive factor Momentum decay coefficient variance attenuation coefficient Initial fusion learning rate Attenuation factor Numerical stability constant The convergence condition is that the rate of change of the solution error in adjacent stages is less than 1%. The method of this invention requires only 3 × 64 = 192 sampling points. The forward-mode automatic differentiation improves the efficiency of derivative calculation by over 80%. The spatiotemporal dual-dimensional adaptive fusion mechanism achieves significant accuracy improvement in the third stage, specifically in the high-gradient region of the solution. The error change rate in the fifth stage reaches [a certain percentage]. The convergence condition is met, and the final output approximate solution converges at twice the speed of the traditional SPINN method. The error is reduced by 44.16% compared to SPINN and by 98.20% compared to the traditional PINN. This significantly reduces the computational cost while improving the solution accuracy.

[0094] In one embodiment, the method of the present invention can be used for transient thermal management analysis of power battery modules in new energy vehicles, and the specific steps include:

[0095] S1: The internal temperature field of the battery module As the object to be solved, a three-dimensional transient heat conduction partial differential equation is constructed, and four parallel feature extraction neural networks are built to initialize the solution momentum. Sum of variances ;

[0096] The three-dimensional transient heat conduction partial differential equation (governing equation) is as follows:

[0097]

[0098] The solution domain is ;

[0099] The boundary conditions are:

[0100]

[0101] The initial conditions are:

[0102]

[0103] in, For battery material density, For specific heat capacity, Thermal conductivity, The divergence of heat flux density represents the heat conduction term. For temperature distribution, This refers to the internal heat generation of the battery cell at different operating times t. For the geometric region to be solved, To solve for the time interval, For boundary cooling conditions, The convective heat transfer coefficient is... To cool the ambient temperature, This represents the initial temperature distribution.

[0104] Battery material density Specific heat capacity convective heat transfer coefficient The physical and system parameters of the battery module can be obtained directly.

[0105] S2: The solution domain (i.e., spatial coordinates) of the three-dimensional transient heat conduction partial differential equation. , , Coordinate separation sampling is performed on t) to obtain the set of sampling points. ;in, Let i be the set of sampling points along coordinate axis i, and let y be the set of sampling points along all four coordinate axes. Each of the four parallel feature extraction neural networks corresponds one-to-one;

[0106] S3: Obtain the four feature extraction neural networks from the previous stage t-1, based on physical constraints such as initial conditions and boundary conditions, and according to the sampling point set of the four coordinate axes. The four feature extraction neural networks from the previous stage t-1 are initially trained to obtain an approximate solution for the current stage t. And the four preliminary feature extraction neural networks at the current stage t;

[0107] S4: Obtain the predicted solution from the previous stage t-1. Calculate the predicted solution The residual field after substituting into the three-dimensional transient heat conduction partial differential equation and its spatial gradient According to the residual field and its spatial gradient Constructing a spatial confidence gating function ;

[0108] S5: Obtain the solution momentum of the previous stage t-1. Sum of variances According to the approximate solution Solution momentum Sum of variances Update the solution to obtain the momentum at the current stage t. Sum of variances and its corrected solution momentum Sum of variances ;

[0109] S6: Based on the predicted solution Spatial confidence gating function Solution momentum Solution of variance Calculate the predicted solution for the current stage t. ;

[0110] S7: Calculate the loss function value based on physical constraints such as initial conditions and boundary conditions, update the parameters of the four preliminary feature extraction neural networks in the current stage t according to the loss function value, and obtain the four feature extraction neural networks in the current stage t; determine whether the training meets the convergence condition. If it does, stop the training and obtain the four trained parallel feature extraction neural networks; otherwise, return to step S2 to resample and enter the training of the next stage t+1.

[0111] S8. Solve the three-dimensional transient heat conduction partial differential equation based on the four trained parallel feature extraction neural networks to obtain the numerical solution, which is the temperature prediction result of the battery module during the charging and discharging process.

[0112] The three-dimensional transient heat conduction partial differential equation is solved by training four parallel feature extraction neural networks. The process involves inputting the values ​​of the four coordinate axes of any coordinate in the solution domain of the three-dimensional transient heat conduction partial differential equation into the four trained parallel feature extraction neural networks to obtain feature matrices of the four coordinate axes. The feature matrices of the four coordinate axes are then multiplied and fused to obtain the temperature prediction result.

[0113] Another aspect of the present invention provides a partial differential equation solving system based on a physical information neural network. The system includes a memory and a processor. The memory is used to store application programs, as well as solution domain parameters, initial and boundary conditions, network hyperparameters, sampling point data, feature matrix data, etc., related to the partial differential equation solving. The processor is used to run the application programs stored in the memory and execute the aforementioned partial differential equation solving method based on a physical information neural network. The system can be deployed on hardware devices such as personal computers, servers, and cloud computing platforms, and supports CPU / GPU acceleration to further improve training and solving efficiency.

[0114] Another aspect of the present invention provides a computer storage medium, which is a non-volatile storage medium, including a USB flash drive, a portable hard drive, a read-only memory, a random access memory, a solid-state drive, an optical disc, etc., on which a computer program is stored. When the computer program is executed by a processor, it implements the above-mentioned method for solving partial differential equations based on a physical information neural network. This program can be directly implemented in the existing JAX deep learning framework without the need for additional development of underlying algorithms, and has strong engineering feasibility.

[0115] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for solving partial differential equations based on physical information neural networks, characterized in that, Includes the following steps: S1: Obtain the partial differential equation to be solved, construct d parallel feature extraction neural networks, and initialize the solution momentum. Sum of variances ; S2: Perform coordinate separation sampling on the solution domain of the partial differential equation to obtain the set of sampling points. ;in, Let be the set of sampling points along coordinate axis i, and let be the set of sampling points along coordinate axes d. Each corresponds one-to-one with d parallel feature extraction neural networks; S3: Obtain the d feature extraction neural networks from the previous stage t-1, based on the sampling point set of the d coordinate axes. The neural networks for extracting d features from the previous stage t-1 are initially trained to obtain an approximate solution for the current stage t. And the d preliminary feature extraction neural networks at the current stage t; S4: Obtain the predicted solution from the previous stage t-1. Calculate the predicted solution Residual field after substituting into the partial differential equation and its spatial gradient According to the residual field and its spatial gradient Constructing a spatial confidence gating function ; S5: Obtain the solution momentum of the previous stage t-1. Sum of variances According to the approximate solution Solution momentum Sum of variances Update the solution to obtain the momentum at the current stage t. Sum of variances and its corrected solution momentum Sum of variances ; S6: Based on the predicted solution Spatial confidence gating function Solution momentum Solution of variance Calculate the predicted solution for the current stage t. ; S7: Calculate the loss function value, update the parameters of the d preliminary feature extraction neural networks in the current stage t based on the loss function value, and obtain the d feature extraction neural networks in the current stage t; determine whether the training meets the convergence condition. If it does, stop the training and obtain the trained d parallel feature extraction neural networks; otherwise, return to step S2 to resample and enter the training of the next stage t+1. S8. Solve the partial differential equations using d trained parallel feature extraction neural networks to obtain numerical solutions.

2. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, The initial training of the d feature extraction neural networks from the previous stage t-1 includes: S31. Set the sampling points of d coordinate axes Input the corresponding feature extraction neural network from the previous stage t-1 into each of the d coordinate axes to obtain the feature matrices. ; S32, Characteristic matrix of d coordinate axes Perform outer product summation and fusion to obtain the fused result; S33. Calculate the physical constraint loss function value based on the fusion result, and fine-tune the parameters of the d feature extraction neural networks in the previous stage t-1 based on the physical constraint loss function value. When the stopping condition of the initial training is met, the final fusion result and the d preliminary feature extraction neural networks in the current stage t are obtained. S34. The final fusion result is used as an approximate solution to the partial differential equation of the current training phase t. .

3. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, Spatial confidence gating function for: in, This represents the global mean over the solution domain of the partial differential equation. Based on the fusion rate, Gradient-sensitive factor, is the numerical stability constant.

4. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, Solution momentum Sum of variances The updates include: in, The momentum decay coefficient, This is the variance attenuation coefficient.

5. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, Predicted solution for current stage t ;in, For global fusion learning rate, is the numerical stability constant.

6. The method for solving partial differential equations based on a physical information neural network according to claim 5, characterized in that, Global fusion learning rate The calculation formula is: in, The initial fusion learning rate, As the attenuation factor, , The predicted solutions for stage t-1 are respectively Predicted solution for stage t-2 The residual field after substituting into the partial differential equation, It is an L2 norm.

7. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, loss function for: in, , , These represent the residual loss, boundary condition loss, and initial condition loss for partial differential equations, respectively. This is the solution momentum after correction in the previous stage t-1. , , , , The weight hyperparameter for the loss term. Let be the network parameters for stage t.

8. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, The first stage of predicted solutions ;in, This is an approximate solution for the first stage.

9. The method for solving partial differential equations based on a physical information neural network according to claim 1, characterized in that, A parallel feature extraction neural network is integrated with the ACON adaptive activation function.

10. A partial differential equation solving system based on a physical information neural network, characterized in that, It includes a memory and a processor; the memory is used to store an application program; the processor is used to run the application program stored in the memory and execute the partial differential equation solving method based on a physical information neural network as described in any one of claims 1 to 9.

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