Variational physical information neural network solving method based on functional constraint of self-adjoint operator

Abstract

This invention discloses a variational physics information neural network solution method based on self-adjoint operator functional constraints, involving the interdisciplinary field of neural networks and scientific computing. The method includes: S1, constructing an energy functional that forms a directly correlated energy with the constraint terms of the differential equation based on predetermined boundary and initial conditions and leveraging the symmetric conservation properties of the self-adjoint operator; wherein each component of the energy functional is mapped one-to-one with the corresponding physical field quantities in the differential equation, including diffusion coefficient and convection velocity; S2, verifying the correctness of the energy functional constructed in S1; S3, substituting the verified energy functional into the variational physics information neural network for iterative training. This invention, by introducing functional constraints into the variational physics information neural network, can more precisely control the neural network's synergistic satisfaction of multiple constraints (equation-boundary-functional), thereby improving the overall accuracy and spatial stability of the solution.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of neural networks and scientific computing. More specifically, this invention relates to a method for solving variational physical information neural networks based on self-adjoint operator functional constraints. Background Technology

[0002] Physical system modeling is a core technology in fields such as engineering simulation and scientific computing (e.g., fluid evolution, heat conduction analysis, and biomechanics). Its core task is to characterize the evolutionary laws of physical phenomena by solving partial differential equations (PDEs), providing quantitative support for scenarios such as industrial design, disaster prediction, and medical simulation. In this type of modeling process, the accuracy and efficiency of PDE solutions directly determine the reliability of the final analysis results. Low reliability affects the rationality of engineering solutions, while high reliability restricts in-depth exploration of complex physical mechanisms. Therefore, exploring efficient and stable PDE solution methods has always been a core requirement in this field.

[0003] Traditional numerical methods (such as the finite element method and the finite difference method) are classic approaches for solving PDEs, achieving high-precision approximations in low-dimensional and simple scenarios and supporting the solution of various fundamental physical problems for a long time. However, significant bottlenecks arise in high-dimensional and complex boundary scenarios. In complex boundary scenarios, poor mesh adaptability can lead to fluctuations in computational accuracy, and if physical parameters are adjusted after mesh generation, re-discretization is required, adding extra preprocessing time. In multi-physics coupled scenarios, the dual complexity of mesh and equation discretization further amplifies efficiency issues, and even leads to insufficient mesh compatibility across different fields. In high dimensions, the computational load increases exponentially with dimensionality, and existing hardware computing power struggles to support large-scale discretization operations, making it difficult to handle high-dimensional nonlinear parabolic PDEs and severely restricting modeling efficiency.

[0004] To overcome the grid dependency problem of traditional numerical methods, Physical Information Neural Networks (PINN) emerged. As an innovative framework integrating data-driven approaches and physical laws, PINN directly embeds PDE constraints, initial and boundary conditions into the neural network loss function. Leveraging the powerful function approximation capabilities of neural networks, it characterizes the PDE solution space and utilizes automatic differentiation to calculate higher-order derivatives in the PDE. Even in extreme physical scenarios where experimental data is scarce, it can maintain model reliability through physical constraints, effectively breaking down the contradiction between "data dependency" and "physical fidelity." Since its initial proposal in 2019, this framework has rapidly demonstrated its effectiveness in fields such as fluid mechanics, quantum mechanics, and biomedicine. However, PINN still faces significant bottlenecks in high-dimensional scenarios: on the one hand, calculating PDE losses through automatic differentiation to optimize network parameters in high-dimensional environments is not only costly to train but also prone to affecting model convergence stability due to gradient accumulation errors; on the other hand, PINN requires sampling points to strictly satisfy PDE constraints, and the sampling density in high-dimensional space needs to increase exponentially with dimensionality, directly leading to a sharp increase in memory consumption and further limiting the solution scale of high-dimensional problems.

[0005] In the aerospace field, wind tunnel testing is a core means of obtaining aerodynamic characteristics and optimizing design schemes for aircraft. Its core requirement is to accurately characterize the distribution of physical fields such as temperature and concentration fields across the entire flow field by solving the convection-diffusion equations in the flow field, providing quantitative support for the optimization of aerodynamic shape, thermal protection design, and improvement of flow stability of aircraft. For wind tunnel tests on irregular boundaries such as the leading edge of the wing and the curved surface of the fuselage, it is necessary to solve convection-diffusion equations with complex boundary conditions to simulate the heat exchange and diffusion process between the airflow and the aircraft surface. However, traditional Physical Information Neural Networks (PINN) do not explicitly incorporate functional constraints, which is insufficient in the synergistic satisfaction of "equation constraints-boundary conditions". Error oscillations are prone to occur near the boundary, making it difficult to meet the accuracy requirements of aerodynamic thermal protection and diffusion characteristic analysis of aircraft. Summary of the Invention

[0006] One object of the present invention is to solve at least the above-mentioned problems and / or defects, and to provide at least the advantages described below.

[0007] To achieve these objectives and other advantages of the present invention, a method for solving variational physical information neural networks based on self-adjoint operator functional constraints is provided, comprising:

[0008] S1. Based on predetermined boundary and initial conditions, and with the help of the symmetric conservation properties of the self-adjoint operator, construct an energy functional J that is directly related to the constraint terms of the differential equation.

[0009] In this context, each component of the energy functional J is mapped one-to-one with the corresponding physical field quantities in the differential equation, and the physical field quantities include: diffusion coefficient, convection velocity, and source term;

[0010] S2. Verify the correctness of the energy functional J constructed in S1;

[0011] S3. Substitute the verified energy functional J into the variational physics information neural network for iterative training, and construct the following loss function during the training process. L Iteratively update the parameters of the variational physics information neural network until convergence:

[0012]

[0013] In the above formula, For boundary constraint loss terms, The initial state loss term, For functional constraint loss term, , , These are the corresponding weight coefficients;

[0014] S4. The trained variational physical information neural network is applied to the wind tunnel experiment of the flight performance of an irregularly shaped aircraft so as to solve the approximate solution of the target differential equation in the whole domain through the solution function corresponding to the output parameters of the variational physical information neural network.

[0015] Preferably, in S1, the initial condition is a time-dimensional constraint, and the constraint satisfies... u ( x , y , t 0)= u 0, where, t 0 represents the initial time. u 0 represents the known value of the physical quantity at the initial moment. x , y Spatial coordinates;

[0016] When the boundary conditions are homogeneous boundary conditions and satisfy When applicable, this method is suitable for the fixed constraint of the temperature field T at the boundary in the convection-diffusion equation. Representation function u The value is 0 at the boundary of the integration region;

[0017] The energy functional J is characterized by the following equation:

[0018]

[0019] In the above formula, w For self-adjoint operators, Indicates the integration region. d It represents calculus.

[0020] Preferably, in S1, for the function u The energy functional J is characterized as And set the function Then the function u Corresponding energy functional It can be characterized as:

[0021]

[0022] In the above formula, , h Indicates a constant coefficient. x Represents spatial coordinates, This indicates the partial derivative.

[0023] Preferably, in S2, the energy functional is... The correctness verification method is as follows:

[0024] S210. When the function u is subjected to a small perturbation When, then the perturbed function Corresponding energy functional After using the linear expansion of the partial derivatives and the squared terms, it can be characterized by the following formula:

[0025]

[0026] In the above formula, , The infinitesimal parameter represents the perturbation of the function;

[0027] S211. According to the definition of variation, extract... Substituting the first-order term and The expression, ignoring higher-order minor quantities After that, we can finally get:

[0028]

[0029] S212, to After integration by parts, the derivative with respect to w yields the following equation:

[0030]

[0031] In the above formula, d represents differentiation;

[0032] S213. Substituting the formula in S212 into S211, we get:

[0033]

[0034] S214, Order ,because For any, then we have Equivalent to the original equation: .

[0035] Preferably, in the two-dimensional convection-diffusion equation, when the boundary conditions... When, function The corresponding energy functional J is characterized by the following equation:

[0036]

[0037] In the above formula, For the Laplace operator, Let T represent the thermal diffusivity, T represent the temperature field, s represent the source term, and s represent the self-adjoint operator. , .

[0038] This invention offers at least the following advantages: By introducing functional constraints into the variational physical information neural network (PINN), it can more precisely control the synergistic satisfaction of multiple constraints (equation-boundary-functional) within the neural network, thereby improving the overall accuracy and spatial stability of the solution. Specifically, this invention effectively overcomes the limitations of traditional numerical methods and PINNs in high-dimensional complex scenarios. This method achieves deep embedding of physical constraints by constructing energy functionals. Leveraging the symmetric conservation properties of self-adjoint operators, it enables a direct mapping between functional components and physical field quantities, ensuring clear physical meaning. Boundary conditions can be naturally integrated during the variational process without relying on basis function space adaptation, avoiding subjectivity in construction and the curse of high-dimensionality. Simultaneously, relying on the spectral theory of self-adjoint operators, it guarantees the existence and uniqueness of the solution and the mandatory nature of the energy functional, improving the stability of neural network optimization and providing an efficient path for high-dimensional physical modeling.

[0039] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0040] Figure 1 This is a two-dimensional heatmap of the predicted solution from the variational physics information neural network (FC-VPINN) in Example 1;

[0041] Figure 2 This is a two-dimensional heatmap of the exact solution of the variational physics information neural network (FC-VPINN) in Example 1;

[0042] Figure 3 This is a prediction thermodynamic error diagram of the variational physics information neural network (FC-VPINN) in Example 1;

[0043] Figure 4This is a two-dimensional heatmap of the predicted solution from the conventional Physical Information Neural Network (PINN) in Example 1;

[0044] Figure 5 This is a two-dimensional heatmap of the exact solution of the conventional Physical Information Neural Network (PINN) in Example 1;

[0045] Figure 6 This is the prediction thermodynamic error diagram of the conventional physical information neural network (PINN) in Example 1;

[0046] Figure 7 The L2 error spatial distribution diagram is shown for the Variational Physics Information Neural Network (FC-VPINN) in solving the two-dimensional convection-diffusion equation;

[0047] Figure 8 The L2 error spatial distribution diagram of the traditional physical information neural network (PINN) in solving the two-dimensional convection-diffusion equation;

[0048] Figure 9 The training loss curves of the variational physical information neural network (FC-VPINN) and the traditional physical information neural network (PINN) in solving the two-dimensional convection-diffusion equation are shown.

[0049] Figure 10 for Figure 9 A partially enlarged schematic diagram. Detailed Implementation

[0050] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0051] This invention proposes a variational physical information neural network based on self-adjoint operator functional constraints for efficiently solving PDE problems in physical systems. This method achieves deep embedding of physical constraints by constructing an energy functional. Leveraging the symmetric conservation properties of the self-adjoint operator, it establishes a direct correlation between the functional and the constraint terms of the differential equation. This means that each component of the energy functional can be mapped one-to-one with physical field quantities such as diffusion coefficients and convection velocities, effectively ensuring the clarity and intuitiveness of the functional's physical meaning and avoiding model bias caused by ambiguity in physical interpretation. Simultaneously, it uses the spectral theory of the self-adjoint operator as a rigorous mathematical foundation, relying on its guarantee of the existence and uniqueness of solutions to functional extremum problems and the mandatory properties of the energy functional to theoretically improve the stability of the model solution. Furthermore, this method directly transforms the strong form of the differential equation using variational principles, without relying on the weighted integral weak form of the test basis functions. This not only avoids the subjectivity problem of basis function space construction but also fundamentally solves the curse of dimensionality in high-dimensional scenarios, providing a more efficient and reliable new path for modeling high-dimensional complex physical systems. Specifically, the specific content of this invention includes:

[0052] Step 1: Construction of Self-Adjoint Operators. Constructing self-adjoint operators in Hilbert space (H) is to address problems such as fuzzy physical constraints, boundary handling errors, and insufficient convergence in traditional variational methods, achieving compatibility between physical constraints and numerical optimization. Self-adjoint operators are linear operators that satisfy (and are adjoint operators), with the core characteristic being symmetry conservation (…).<u,v> =<u,v> The relationship between symmetry conservation and domain consistency is crucial. Symmetry conservation establishes an explicit mapping between the energy functional and physical field quantities; domain consistency, combined with boundary condition adaptation, eliminates boundary term errors; and spectral theory ensures model convergence. The model construction must be based on domain density, first satisfying symmetry and then upgrading to self-adjointness. Boundary constraints (such as function value or derivative value restrictions) eliminate boundary terms in integration by parts, ultimately achieving cooperative constraints.

[0053] Step 2: Functional Construction. In the variational method, we aim to construct a functional J(u) such that the variation of the functional is... Can correspond to the original equation

[0054] According to the variational principle: if right If it is established, then .

[0055] The variation of a functional is the linear response of the functional to small perturbations of the function, defined as:

[0056]

[0057] in, It is a variation of the function u (i.e., a small perturbation that satisfies) Equal boundary conditions, where, Representation function u (It takes the value 0 at the boundary of the integration region). It is an infinitesimal parameter.

[0058] Step 3: Verify the correctness of the functional.

[0059] For example, for the function Construct the following functional:

[0060]

[0061] in, , h Indicates a constant coefficient. x Represents spatial coordinates, This indicates the partial derivative.

[0062] When the function u is subjected to a small perturbation At that time, the new function is Substituting into the functional, we get:

[0063]

[0064] Using the linearity of partial derivatives Expand the square terms:

[0065]

[0066] Therefore, the perturbed functional can be written as:

[0067]

[0068] According to the definition of variation, it is necessary to extract First-order terms (second-order and higher-order terms due to) (can be ignored), substitute and The expression, after expansion:

[0069]

[0070] neglect The term (higher-order minor) can ultimately yield:

[0071]

[0072] right Perform integration by parts.

[0073] The formula for integration by parts is as follows:

[0074]

[0075] Therefore, , but ;

[0076]

[0077] For the treatment of boundary terms, the Dirichlet condition and the Neumann condition are referenced respectively, and the boundary term is 0.

[0078] Taking the derivative with respect to w, we can calculate:

[0079]

[0080] therefore

[0081]

[0082] Substituting the above equation into the equation From this, we can obtain:

[0083]

[0084] make ,because For any, then we have Equivalent to the original equation: .

[0085] Step 4: Substitute into the variational physical information neural network learning based on functional constraints.

[0086] This step involves applying the trained variational physics information neural network to wind tunnel tests on aircraft with irregular shapes. The irregularity refers to the irregular structure of the aircraft's wing leading edge, fuselage curved surface, etc. In the application scenario where it is necessary to solve convection-diffusion equations with complex boundary conditions to simulate the heat exchange and medium diffusion process between airflow and aircraft surface, the solution function corresponding to the output parameters of the variational physics information neural network is used to solve the approximate solution of the target differential equation over the whole domain.

[0087] In practical applications, based on the predetermined boundary conditions and initial conditions of the physical problem, as well as the functional expression derived for the physical problem, boundary constraint loss terms, initial state loss terms, and functional constraint loss terms are constructed respectively. Finally, the three types of loss terms are fused by weighted summation to obtain the total loss function L required for neural network training.

[0088]

[0089] in, , , These represent the boundary constraint loss, initial state loss, and functional constraint loss, respectively. , , These are weighting coefficients, which can be adjusted according to the actual situation of the problem. By minimizing the total loss function L, the powerful learning ability of neural networks can be used to solve physical problems.

[0090] The algorithm of this invention utilizes the symmetric conservation property of the self-adjoint operator to construct an energy functional that is directly related to the constraint terms of the convection-diffusion equation. Each component of the functional is mapped one-to-one with the diffusion coefficient, convection velocity, and source term in the equation. This allows complex boundary conditions to be naturally integrated into the functional construction without relying on mesh adaptation, significantly improving the solution accuracy and spatial stability of convection-diffusion fields with complex shapes.

[0091] Example 1

[0092] In practical applications, taking the two-dimensional convection-diffusion equation in an aircraft wind tunnel test as an example, we first construct the corresponding functional, and then train it using a variational physics information neural network based on self-adjoint operator functional constraints. The mathematical expression of the two-dimensional convection-diffusion equation is as follows:

[0093]

[0094] Laplace operator:

[0095]

[0096] in, T represents the temperature field ( ); a is the thermal diffusivity , u1 and u2 are the components of the convection velocity in the x and y directions, respectively. Convection velocity Source term function

[0097] The initial and boundary conditions are: The analytical solution is: ;

[0098] Based on the fundamental principles of variational methods, we consider the following functional from the perspective of energy functionals:

[0099]

[0100] Among them, self-adjoint operator ,and ;

[0101] Figures 1-2 , Figures 4-5 Two-dimensional heatmaps of the predicted and exact solutions of the functionally constrained variational physical information neural network (FC-VPINN) and the traditional physical information neural network (PINN) are presented respectively. The heatmaps visually reflect the distribution characteristics of the physical quantity U(x,y) through color intensity, showing a high degree of consistency between the predicted and exact solutions in spatial distribution, especially in the matching of high and low value regions. Figure 3 , Figure 6 The error distribution shows that the error of the FC-VPINN of this invention is mainly concentrated near the boundary and in a few local areas, with the error range being within... arrive Between, while the PINN error range is within arrive Between. By comparing the functional constraint-based method and the PDE constraint method, it is shown that the functional constraint-based method has higher prediction accuracy overall.

[0102] Comparing the L2 error spatial distribution of the functionally constrained variational physical information neural network (FC-VPINN) and the traditional physical information neural network (PINN) in solving the two-dimensional convection-diffusion equation, we can see that: Figure 7It can be seen that the error field of the FC-VPINN used in this invention exhibits lower amplitude and more uniform spatial distribution, reflecting its strong ability to satisfy the coupling relationship of "convection-diffusion equation-boundary condition-functional constraint"; while... Figure 8 As shown, the error field of traditional PINN not only has a significantly higher overall amplitude, but also exhibits intensified error oscillations and local concentration in regions with drastic gradient changes (such as near the boundary or at abrupt curvature changes in the solution). The root cause lies in the fact that PINN only learns through PDEs and boundary condition constraints, without explicitly incorporating the variational structure of the functional, leading to an imbalance in the coupling between "equation constraint satisfaction" and "functional objective fit." In contrast, FC-VPINN, guided by functional constraints, more precisely regulates the network's synergistic satisfaction of multiple constraints across "equation-boundary-functional," thereby improving the overall accuracy and spatial stability of the solution. Figure 9 The loss convergence characteristics during training of PINN and FC-VPINN are shown. PINN (solid blue line) experiences a rapid decrease in loss during the initial training phase (Epoch < 500) followed by a period of stabilization, but exhibits early oscillations during the overall convergence process. In contrast, FC-VPINN (dashed red line) achieves rapid loss convergence within the Epoch range (0-300). Figure 10 The study further demonstrates that the functional constraint mechanism accelerates the gradient descent process, resulting in a steeper and more stable loss curve. This indicates that the FC-VPINN optimization framework is more suitable for the convection-diffusion equation, significantly shortening the convergence period and avoiding the oscillations that occur in PINN during training, thereby improving training efficiency and stability.

[0103] Example 2

[0104] The three-dimensional Poisson equation has important applications in practical scenarios such as engineering numerical simulation, physical field analysis, and fluid dynamics calculation. The three-dimensional Poisson equation is a typical elliptic partial differential equation that broadly describes the steady-state physical field distribution, such as the electrostatic potential distribution, the displacement potential function in elasticity, and the pressure field in fluid dynamics. It is also a core simplified form of complex fluid dynamics models such as the three-dimensional convection-diffusion equation and the incompressible Navier-Stokes equations. Based on three-dimensional spatial variables (x, y, z) and using the unknown function u(x, y, z) as the state variable, the core of this equation is to obtain the steady-state physical field distribution that satisfies the source term f(x, y, z) constraint by solving the elliptic equation with variable coefficients under the Dirichlet boundary condition constraint in the integration region Ω. Simultaneously, it needs to accurately characterize complex physical features such as the boundary layer and large gradient regions. Its weak solution exists and is unique. Variational solutions can be achieved by constructing an energy functional using a self-adjoint operator. The core idea is to utilize the self-adjoint property of the operator to transform the original equation into a functional minimization problem, i.e., finding a function u that satisfies the boundary conditions, such that the energy functional reaches its minimum. The Euler-Lagrange equation is derived through the variational conditions of the functional, and is equivalent to the original equation. Adapting to the architecture design of a variational physical information neural network, the network can approximate the optimal solution corresponding to the functional minimum, efficiently handling complex boundary scenarios.

[0105] 1. The mathematical expression is as follows:

[0106]

[0107] in, These are spatial coordinates. It is the physical quantity to be solved. Its source term function, boundary conditions, and corresponding analytical solutions are as follows:

[0108] Source term function:

[0109]

[0110] Boundary conditions:

[0111]

[0112]

[0113]

[0114] Analytical solution:

[0115]

[0116] Based on the fundamental principles of variational methods, we consider the following functional from the perspective of energy functionals:

[0117]

[0118] 2. Network Training

[0119] To solve the above problem, a neural network with eight fully connected layers, each containing 80 neurons, was constructed, using tanh as the activation function. The network's input consists of spatial coordinates x, y, and z, and its output is a control variable u. The network is updated using the Adam optimizer, with a learning rate set to... .

[0120] Given the boundary conditions of the three-dimensional Poisson equation, the approximate state variables output by the neural network are embedded into the elliptic governing equation and the source term, resulting in a modified governing equation that replaces the unknown analytical solution with a neural network. Adaptive dimensionless parameters (such as dimensionless spatial coordinates, diffusion coefficient, and source term strength) are defined, and Dirichlet boundary conditions conforming to the physical scenario are selected. A variational physical information neural network based on the energy functional is constructed using self-adjoint operators, and the equation residuals and boundary constraints are integrated into the loss function for optimization.

[0121] This method replaces traditional numerical discretization methods such as finite difference and finite element methods commonly used to solve the three-dimensional Poisson equation. It allows partial differential equation problems that previously required complex three-dimensional mesh generation, discretization scheme derivation, or iterative solutions to large algebraic equation systems to directly obtain approximate solutions through end-to-end learning of neural networks. This method not only avoids the dependence of traditional numerical methods on mesh quality, accurately capturing complex physical features such as boundary layers and high gradient regions without complex coordinate transformations or mesh stretching, but also reduces the need for specialized knowledge of high-order discretization scheme derivation, making the solution process for the three-dimensional Poisson equation more efficient and automated.

[0122] The above solution is merely an illustration of a preferred example and is not limited thereto. When implementing this invention, appropriate substitutions and / or modifications can be made according to the user's needs.

[0123] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. It can be applied to various fields suitable for the present invention. Other modifications can be readily made by those skilled in the art. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and examples shown and described herein.

Claims

1. A method for solving variational physical information neural networks based on self-adjoint operator functional constraints, characterized in that, include: S1. Based on predetermined boundary and initial conditions, and with the help of the symmetric conservation properties of the self-adjoint operator, construct an energy functional J that is directly related to the constraint terms of the differential equation. In this context, each component of the energy functional J is mapped one-to-one with the corresponding physical field quantities in the differential equation, and the physical field quantities include: diffusion coefficient, convection velocity, and source term; S2. Verify the correctness of the energy functional J constructed in S1; S3. Substitute the verified energy functional J into the variational physics information neural network for iterative training, and construct the following loss function during the training process. L Iteratively update the parameters of the variational physics information neural network until convergence: In the above formula, For boundary constraint loss terms, The initial state loss term, For functional constraint loss term, , , These are the corresponding weight coefficients; S4. The trained variational physical information neural network is applied to the wind tunnel experiment of the flight performance of an irregularly shaped aircraft so as to solve the approximate solution of the target differential equation in the whole domain through the solution function corresponding to the output parameters of the variational physical information neural network.

2. The method for solving variational physical information neural networks based on self-adjoint operator functional constraints as described in claim 1, characterized in that, In S1, the initial condition is a time-dimensional constraint, and the constraint satisfies... u ( x , y , t 0)= u 0, where, t 0 represents the initial time. u 0 represents the known value of the physical quantity at the initial moment. x , y Spatial coordinates; When the boundary conditions are homogeneous boundary conditions and satisfy When the temperature field T is fixed at the boundary in the convection-diffusion equation, it is applicable. Representation function u The value is 0 at the boundary of the integration region; The energy functional J is characterized by the following equation: In the above formula, w For self-adjoint operators, Indicates the integration region. d It represents calculus.

3. The method for solving variational physical information neural networks based on self-adjoint operator functional constraints as described in claim 2, characterized in that, In S1, for the function u The energy functional J is characterized as And set the function Then the function u Corresponding energy functional It can be characterized as: In the above formula, , h Indicates a constant coefficient. x Represents spatial coordinates, This indicates the partial derivative.

4. The method for solving variational physical information neural networks based on self-adjoint operator functional constraints as described in claim 3, characterized in that, In S2, the energy functional The correctness verification method is as follows: S210. When the function u is subjected to a small perturbation When, then the perturbed function Corresponding energy functional After using the linear expansion of the partial derivatives and the squared terms, it can be characterized by the following formula: In the above formula, , The infinitesimal parameter represents the perturbation of the function; S211. According to the definition of variation, extract... Substituting the first-order term and The expression, ignoring higher-order minor quantities After that, we can finally get: S212, to After performing integration by parts and differentiating with respect to w, we obtain the following equation: S213. Substituting the formula in S212 into S211, we get: S214, Order ,because For any, then we have Equivalent to the original equation: .

5. The method for solving variational physical information neural networks based on self-adjoint operator functional constraints as described in claim 2, characterized in that, In the two-dimensional convection-diffusion equation, when the boundary conditions When, function The corresponding energy functional J is characterized by the following equation: In the above formula, For the Laplace operator, Let T represent the thermal diffusivity, T represent the temperature field, s represent the source term, and s represent the self-adjoint operator. , .

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