Method for rapidly and accurately solving nonlinear / heterogeneous transient heat conduction problem of complex structure

Abstract

The invention provides a method for rapidly and accurately solving a nonlinear / heterogeneous transient heat conduction problem of a complex structure, and belongs to the field of heat transfer theory numerical solution. The method is realized based on a pre-training module and a physical information correction network module, the pre-training module mainly adopts a Fourier neural network, and a temperature field result of a prediction model is rapidly output according to input model grid information and parameter information; and the physical information correction network module is used for correcting the temperature field result generated in the pre-training module by utilizing physical information. According to the method, solution of a complex nonlinear heat conduction problem is further accelerated through transfer learning under the condition that too much precision is not lost, and rapid calculation of different parameters of the same model is achieved. According to the method, the nonlinear / heterogeneous transient heat conduction problem of a complex structure can be rapidly and accurately solved, and the method is beneficial to the scientific and engineering fields in which the complex nonlinear heat conduction problem needs to be solved on a large scale.

Description

Technical Field

[0001] The present invention belongs to the field of numerical solution of heat transfer, and relates to a method for quickly and accurately solving nonlinear / inhomogeneous transient heat conduction problems of complex structures, and particularly relates to a method for quickly and accurately solving nonlinear / inhomogeneous transient heat conduction problems of complex structures based on Fourier neural operators and physics-informed neural networks. Background Art

[0002] The Fourier neural operator is a generalization of neural networks. The learning of classical neural networks mainly focuses on establishing a mapping between inputs and outputs in Euclidean space, while the Fourier neural operator learns the mapping between function spaces. Previous studies have shown that once the training process is completed, the generalization ability of the Fourier neural operator is better than that of neural networks.

[0003] Physics-Informed Neural Networks (PINN), a new machine learning architecture proposed in recent years, has received extensive attention from scholars. It solves forward and inverse physical problems by constructing a loss function based on the residuals of the governing partial differential equations (PDEs) rather than labeled data. During training, by minimizing the deviation between the output and physical laws, PINN can learn the solution of PDEs in a high-dimensional parametric space, avoiding the per-case calculation in traditional numerical methods. In the current related research on solving heat conduction problems using physics-informed neural networks, there is generally a problem that when facing nonlinear / inhomogeneous heat conduction problems, the error in solving the temperature field is too large or even unsolvable; in addition, existing research only focuses on steady-state heat conduction problems and rarely studies transient problems. The reason why nonlinear / inhomogeneous transient heat conduction problems are difficult to solve is that the introduction of nonlinear / inhomogeneous thermal physical parameters and time terms causes great difficulties in solving partial differential equations. In view of the above existing problems, the present invention provides a method for quickly and accurately solving nonlinear / inhomogeneous transient heat conduction problems of complex structures.

[0004] Currently, enhancing the learning ability of physics-informed neural networks for physical laws is mainly achieved through the following two methods: (1) network architecture design and (2) loss function design. The method based on network architecture design usually constructs a special neural network structure before training to meet some physical constraints, such as modifying the architecture of convolutional neural networks to have rotational invariance, and recurrent neural networks for processing time series problems; or the method of designing the loss function incorporates a regularization term into the loss function to strengthen the prediction results that satisfy certain physical laws. The present invention constructs a Fourier operator neural network as a preprocessing network to improve the learning ability of physics-informed neural networks for nonlinear / inhomogeneous transient heat conduction equations.

[0005] The present invention has the following advantages in solving complex non-linear / heterogeneous transient heat conduction problems: (1) The addition of the Fourier operator neural operator greatly enhances the network's learning ability for complex non-linear / heterogeneous transient heat conduction problems; (2) It is sensitive to the local regions of concern. For the current methods of solving heat conduction related problems using physics-informed neural networks, the importance of all points is equal, while the grid-based input strategy of this method places more prediction points in the regions of concern (such as common interfaces / structures or interfaces with drastic changes in thermal conductivity, etc.); (3) There is no interpolation error in the prediction results. Since the output and input of this method are in one-to-one correspondence, the Fourier operator has no interpolation error. Summary of the Invention

[0006] The present invention mainly aims to improve the deficiencies of the traditional physics-informed neural network in solving heat conduction problems, and proposes a method for quickly and accurately solving complex structure non-linear / heterogeneous transient heat conduction problems based on the Fourier operator and physics-informed neural network. The present invention can quickly and accurately solve complex structure non-linear / heterogeneous transient heat conduction problems, which is helpful for scientific and engineering fields that require large-scale solution of complex non-linear heat conduction problems.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] A method for quickly and accurately solving complex structure non-linear / heterogeneous transient heat conduction problems, specifically a method for quickly and accurately solving complex structure non-linear / heterogeneous transient heat conduction problems based on the Fourier operator and physics-informed neural network. The method is implemented based on a pre-training module and a physics information correction network module. The pre-training module mainly uses a Fourier neural network to quickly output the temperature field results of the prediction model according to the input model grid information and parameter information; the physics information correction network module uses physics information to correct the temperature field results generated in the pre-training module. Specifically, it includes the following steps:

[0009] Step S1: Obtain a data set, extract the required triple in the form of model grid, temperature field results, and model parameters, and integrate it into a data set input to the pre-training module. Preprocess the data set, and the preprocessing includes data cleaning, noise processing, and initialization. The preprocessed data set is divided into a training set and a test set according to a ratio of 9:1;

[0010] Step S2: Build the Fourier operator network used in the pre-training module. The Fourier operator network consists of three parts: neural network P, Fourier layer, and neural network Q. Then, take the model grid and model parameters in the training set obtained in Step S1 as inputs and input them into the Fourier operator network. After being processed by the three-layer network, the output of the Fourier operator network is obtained, which is the temperature field result in the training set. Train the Fourier operator network. After training is completed, use the test set obtained in Step S1 to complete the test and obtain the built pre-training module. Specifically:

[0011] Step S21: Complete the construction of the pre-training module. Build the Fourier operator network required for the pre-training module, namely neural network P, Fourier layer, and neural network Q. Neural network P consists of a fully connected layer, a Relu layer, and a fully connected layer; the Fourier layer is composed of an upsampling layer, a BN layer, a pooling layer, and a Relu layer stacked in sequence 2 times; neural network Q consists of two fully connected layers;

[0012] Step S22: Denote the model grid and model parameters in the training set obtained in Step S1 as input a(x), and input them into neural network P built in Step S21. Neural network P is mainly responsible for lifting the input function a(x) to a higher-dimensional space to obtain v(x);

[0013] Step S23: Input v(x) obtained in Step S22 into the Fourier layer. The Fourier layer is the core of the pre-training module. It first performs a fast Fourier transform (FFT) on v(x) to obtain a series of frequency-domain patterns. Through feature selection, filter out the high-frequency patterns of the input data and retain the low-frequency patterns to obtain ν t (x). Then, perform an inverse fast Fourier transform (IFFT) on ν t (x) to obtain the output ν T (x) of the Fourier layer;

[0014] Step S24: Input ν T (x) obtained in Step S23 into neural network Q. Neural network Q reduces the dimension of ν T (x) to the same dimension as the input a(x) in Step S22 to obtain the output of neural network Q, denoted as u(x). u(x) is the predicted model temperature field result and also the output of the pre-training module;

[0015] Step S25: Train the Fourier operator network. After training is completed, use the test set obtained in Step S1 to complete the test and verification of the Fourier operator network to obtain the built pre-training module;

[0016] Step S3: After the pre-training module is built, input any model grid and model parameters into the pre-training module, and a temperature field prediction result containing grid information and parameter information can be obtained as the preliminary prediction result of the temperature field;

[0017] Step S4: Input the temperature field prediction result containing grid information and parameter information obtained in Step S3 into the physical information correction network module, and use the boundary conditions, initial conditions, control equations, and corresponding constrained physical information to constrain the temperature field prediction result containing grid information and parameter information obtained in Step S3 to obtain the temperature result of the corrected non-linear or inhomogeneous transient heat conduction problem. Specifically:

[0018] Step S41: Build the physical information correction network module. The physical information correction network module is mainly composed of a physical information neural network, including two parts: the construction of the physical information neural network and the construction of the loss function. The network part of the physical information neural network consists of three stacked fully connected layers and Relu layers; the loss function of the physical information neural network is denoted as Loss, which mainly includes: the control equation loss denoted as MSE g and the initial condition loss denoted as MSE I and the boundary loss denoted as MSE b and the heat flux continuity condition loss denoted as MSE lp and the constraint loss of some known temperature points denoted as MSE s ;

[0019] Step S42: Collect the model grid, model parameters, and the temperature field obtained from the pre-training module in Step S3 as the input of the physical information correction network module, and obtain the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model through the model parameters and the grid;

[0020] Step S43: Convert the model grid information collected in Step S42 into domain points, and use the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model obtained in Step S42 to construct the loss function Loss, completing the construction of the physical information correction network module. Specifically as follows:

[0021] Define the control equation function g(x,t) for the non-linear and inhomogeneous transient heat conduction equation:

[0022]

[0023] where λ is the thermal conductivity of the model material; ρ(T) is the density of the model material; c(T) is the specific heat of the model material; T refers to the temperature result to be predicted; x is the spatial coordinate position; t is the calculation time; is the gradient operator.

[0024] Then the control equation loss:

[0025]

[0026] Among them, MSE g represents the control equation loss; N g represents the number of sampled domain points inside the model; g(x, t) represents the control equation function;

[0027] For the initial condition loss:

[0028]

[0029] Among them, is the known initial condition; MSE I represents the control equation loss; N I represents the number of sampled domain points in the initial condition part; T(x) t＝0 represents the predicted temperature value at t = 0.

[0030] For the boundary loss:

[0031]

[0032] MSE b = MSE bDirichlet + MSE bNeumann + MSE bRobin (7)

[0033] Among them, MSE bDirichlet represents the Dirichlet boundary loss; N bDirichlet represents the number of sampled domain points in the Dirichlet boundary part; T(x, τ) represents the predicted temperature value of the Dirichlet boundary; represents the true temperature value of the Dirichlet boundary; τ represents any moment; MSE bNeumann represents the Neumann boundary loss; N bNeumann represents the number of sampled domain points in the Neumann boundary part; λ represents the thermal conductivity; n i represents the normal vector of the boundary; x i represents the spatial coordinate of the corresponding point; q(x, τ) represents the surface heat flux; MSE bRobin represents the Robin boundary loss; N bRobin represents the number of sampled domain points in the Robin boundary part; h(T w - T f ) represents the heat exchange between the external environment and the internal temperature; T w represents the environmental temperature; T f represents the external temperature; MSE b represents the boundary loss; j represents the current computational domain point.

[0034] For the residuals of the prediction results on the common interface or structure that should satisfy the heat flux continuity condition, denoted as MSE lp,p+1 :

[0035]

[0036] where, T p is the temperature of the current region, T p+1 is the temperature of the adjacent region, λ p is the temperature of the current region, λ p+1 is the temperature of the adjacent region, n p,p+1 is the normal vector of the common interface or structure; N lp,p+1 represents the number of sampled domain points of the common interface or structure; j represents the current computational domain point; n p,p+1 represents the normal vector of the common interface or structure.

[0037] At the same time, satisfy the residuals of the prediction result MSE lp,p+1 of the common interface or structure and the average temperature of the two adjacent regions

[0038]

[0039] where, T p is the temperature of the current region, T p+1 is the temperature of the adjacent region; represents the average temperature of the two interfaces; represents the residuals of the average temperature of the two adjacent regions; N lp,p+1 represents the number of sampled domain points of the common interface or structure.

[0040] For the sparse point hard constraint condition:

[0041]

[0042] where, T s (x,t) is the temperature data of the sparse point, the above N g 、N I 、N b 、N lp and Ns respectively represent the number of sampled domain points inside the model, the number of sampled domain points in the initial condition part, the number of sampled domain points in the boundary condition part, the number of sampled domain points in the heat flux continuity condition part, and the number of sampled domain points in the sparse point hard constraint condition.

[0043] Then the total loss is:

[0044] Loss = W g MSE g + WI MSE I +WbMSEb+Wl p MSEl p +W s MSE s (13)

[0045] Among them, W g 、W l 、W b 、W lp 、W s are the adaptive weights of each loss respectively.

[0046] Step S44: Use optimization algorithms such as Stochastic Gradient Descent (SGD) and Adaptive Moment Estimation (Adam) to train and solve the physical information correction network module built in step S43, update its network parameters, minimize its loss function, complete the training, and obtain the temperature field result output by the physical information correction network module. Compare the temperature field calculation result with the reference solution, and calculate the error index to evaluate the solution accuracy. The reference solution is the high-precision numerical simulation result or experimental data, and the error index includes RMSE, MAE, etc. According to the solution accuracy, adjust the physical information neural network structure in the physical information correction network module (such as changing the activation function type, optimizing the number of network layers), and optimize the algorithm parameters (such as adjusting the learning rate decay strategy, increasing the number of iterations) to improve the accuracy and stability of solving the problem, so that the loss is less than the rated residual or exceeds the rated number of iteration steps, and finally obtain the optimal physical information correction network module.

[0047] Step S45: Input the temperature field prediction result containing grid information and parameter information obtained in step S3 into the optimal physical information correction network module obtained in step S44 to obtain the temperature field result of the complex structure non-linear / inhomogeneous transient heat conduction problem after physical information correction;

[0048] Step S5: Save the temperature result obtained in step S45 in txt format and output it.

[0049] The present invention has the following beneficial effects compared with the prior art:

[0050] (1) Compared with the traditional physics-informed neural network that cannot capture the influence of non-linear / inhomogeneous thermal physical parameters on heat conduction, the present invention enhances the learning of non-linear / inhomogeneous transient heat conduction problems through the introduction of Fourier neural operators and the improvement of the loss function;

[0051] (2) Compared with the traditional physics-informed neural network that has interpolation errors in solving the temperature field, since the output and input of this method are one-to-one corresponding, the Fourier operator has no interpolation errors;

[0052] (3) Compared with the traditional physics-informed neural network, which has excessive errors or even fails to solve the nonlinear / heterogeneous transient heat conduction problem, the present invention can solve the nonlinear / heterogeneous transient heat conduction problem with high precision. Description of the Drawings

[0053] Figure 1 It is the overall structure designed by the present invention. Figure 2 It is the physical model diagram of the comparative verification example of Comparative Example 1.

[0054] Figure 3 It is the comparison of the loss function curves between the method designed in Embodiment 1 of the present invention and the case where only the physics neural network is used to solve the two-dimensional example in Comparative Example 1.

[0055] Figure 4 It is the comparison of the average errors at different times between the method designed in Embodiment 1 of the present invention and the case where only the physics neural network is used to solve the two-dimensional example in Comparative Example 1.

[0056] Figure 5 It is the comparison of the temperature in the y direction between the method designed in Embodiment 1 of the present invention and the cases where only the physics neural network and the finite element method are used to solve the two-dimensional example in Comparative Example 1.

[0057] Figure 6 The comparison of the temperature in the x direction between the method designed in Embodiment 1 of the present invention and the cases where only the physics neural network and the finite element method are used to solve the two-dimensional example in Comparative Example 1. Detailed Embodiments

[0058] Next, the technical solutions of the present invention will be further described in conjunction with specific embodiments and drawings.

[0059] Comparative Example 1

[0060] In order to further verify and compare the advantages of the present invention compared with the traditional physics-informed neural network, the following examples are selected for comparative analysis:

[0061] Consider the transient heat conduction problem of a two-dimensional flat plate with a hole, as Figure 2 shown. The side length is 1*1 m, the density of the material is 1 kg·m -3 , the specific heat capacity is 1 J·kg -1 ·K -1 . It is divided into two regions on the left and right sides. The thermal conductivity of the material on the left side is heterogeneous, λ1 = x + y W·m -1 ·K -1 The material on the right side is nonlinear, λ2 = 0.1T W·m -1 ·K -1 , the radius of the hole is 0.2 m, the position of the center of the circle is (0.3, 0.6), the boundary of the hole is adiabatic, and the calculation time is 1 s.

[0062] Embodiment 1

[0063] A method for quickly and accurately solving complex structure non - linear / inhomogeneous transient heat conduction problems based on Fourier operators and physics - informed neural networks, the steps of which include:

[0064] Step S1: Prepare the dataset. Extract the required triple in the form of model grids, temperature field results, and model parameters, and integrate them into a dataset for input to the pre - training module. Pre - process the dataset, and the pre - processing includes data cleaning, noise processing, and initialization. After pre - processing, the dataset is divided into a training set and a test set according to a ratio of 9:1;

[0065] Step S2: Build the Fourier operator network to be used in the pre - training module. The Fourier operator network consists of three parts: neural network P, Fourier layer, and neural network Q. Then, use the model grids and model parameters in the training set obtained in step S1 as inputs and input them into the Fourier operator network. After being processed by three - layer network, the output of the Fourier operator network is obtained, which is the temperature field result in the training set. Train the Fourier operator network, and after training is completed, use the test set obtained in step S1 to complete the test to obtain the built pre - training module. Specifically:

[0066] Step S21: Complete the construction of the pre - training module. Build the Fourier operator network required for the pre - training module, that is, neural network P, Fourier layer, and neural network Q. Neural network P consists of a fully - connected layer, a Relu layer, and a fully - connected layer; the Fourier layer is composed of an up - sampling layer, a BN layer, a pooling layer, and a Relu layer stacked in sequence 2 times; neural network Q consists of two fully - connected layers;

[0067] Step S22: Denote the model grids and model parameters in the training set obtained in step S1 as input a(x), and input them into neural network P built in step S21. Neural network P is mainly responsible for lifting the input function a(x) to a higher - dimensional space to obtain v(x);

[0068] ν(x) = P(a(x)) (1)

[0069] Step S23: Input v(x) obtained in step S22 into the Fourier layer. The Fourier layer is the core of the pre - training module. It first performs a fast Fourier transform (FFT) on v(x) to obtain a series of frequency - domain patterns. Through feature selection, filter out the high - frequency patterns of the input data and retain the low - frequency patterns to obtain ν t (x). Then, perform an inverse fast Fourier transform (IFFT) on ν t (x) to obtain the output ν T (x) of the Fourier layer;

[0070] ν t (x) = ∫ Dν(x)e -2iπ(x,k) dx (2)

[0071] ν T (x) = ∫ D νt(x)e 2iπ(x,k) dk (3)

[0072] Step S24: Input the ν T (x) obtained in Step S23 into the neural network Q. The neural network Q reduces the dimension of ν T (x) to the same dimension as the input a(x) in Step S22, and the output of the neural network Q is denoted as u(x). u(x) is the predicted result of the model temperature field and also the output of the pre-training module;

[0073] u(x) = Q(ν T (x)) (4)

[0074] Step S25: Train the Fourier operator network. After the training is completed, use the test set obtained in Step S1 to test and verify the Fourier operator network to obtain the built pre-training module;

[0075] Step S3: After the pre-training module is built, input any model grid and model parameters into the pre-training module to obtain the preliminary predicted result of the model temperature field;

[0076] Step S4: Input the temperature field prediction result containing grid information and parameter information obtained in Step S3 into the physical information correction network module, and use the boundary conditions, initial conditions, control equations, and corresponding constraint physical information to constrain the temperature field with grid information obtained in Step S3 to obtain the temperature result of the corrected model for the nonlinear or inhomogeneous transient heat conduction problem. Specifically:

[0077] Step S41: Build the physical information correction network module. The physical information correction network module mainly consists of a physical information neural network. It mainly includes two parts: the construction of the physical information neural network and the construction of the loss function. The network part of the physical information neural network is composed of a fully connected layer and a Relu layer stacked 3 times; the loss function of the physical information neural network is denoted as Loss, which mainly includes: the control equation loss denoted as MSE g 、the initial condition loss denoted as MSE I 、the boundary loss denoted as MSE b 、the heat flux continuity condition loss denoted as MSE lp and the constraint loss of some known temperature points denoted as MSE s ;

[0078] Step S42: Collect the model grid, model parameters, and the temperature field obtained by the pre-training module in Step S3 as the input of the physical information correction network module, and obtain the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model through the model parameters and the grid;

[0079] Step S43: Convert the model grid information collected in Step S42 into domain points, and use the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model obtained in Step S42 to construct the loss function Loss, completing the construction of the physical information correction network module. Specifically as follows:

[0080] Define the governing equation function g(x,t) for the non-linear and inhomogeneous transient heat conduction equation:

[0081]

[0082] where λ is the thermal conductivity of the model material; ρ(T) is the density of the model material; c(T) is the specific heat of the model material; T refers to the temperature result to be predicted; x is the spatial coordinate position; t is the calculation time; is the gradient operator.

[0083] Then the governing equation loss:

[0084]

[0085] where MSE g represents the governing equation loss; N g represents the number of domain points sampled inside the model; g(x,t) represents the governing equation function;

[0086] For the initial condition loss:

[0087]

[0088] where, is the known initial condition; MSE I represents the governing equation loss; N I represents the number of domain points sampled in the initial condition part; T(x) t＝0 represents the predicted temperature value at t = 0.

[0089] For the boundary loss:

[0090]

[0091] MSE b = MSE bDirichlet + MSE bNeumann + MSE bRobin (11)

[0092] where MSEbDirichlet Denotes the Dirichlet boundary loss; N bDirichlet Denotes the number of sampled domain points in the Dirichlet boundary part; T(x,τ) denotes the predicted temperature value of the Dirichlet boundary; Denotes the true temperature value of the Dirichlet boundary; τ denotes any moment; MSE bNeumann Denotes the Neumann boundary loss; N bNeumann Denotes the number of sampled domain points in the Neumann boundary part; λ denotes the thermal conductivity; n i Denotes the normal vector of the boundary; x i Denotes the spatial coordinates of the corresponding point; q(x,τ) denotes the surface heat flux; MSE bRobin Denotes the Robin boundary loss; Nb R obin denotes the number of sampled domain points in the Robin boundary part; h(T w -T f ) denotes the heat exchange between the external environment and the internal temperature; T w Denotes the environmental temperature; T f Denotes the external temperature; MSE b Denotes the boundary loss; j denotes the current computational domain point.

[0093] The residual of the predicted result on the common interface or structure that should satisfy the heat flux continuity condition is denoted as MSE lp,p+1 :

[0094]

[0095] Among them, T p is the temperature of the current region, T p+1 is the temperature of the adjacent region, λ p is the temperature of the current region, λ p+1 is the temperature of the adjacent region, n p,p+1 is the normal vector of the common interface or structure; N lp,p+1 Denotes the number of sampled domain points of the common interface or structure; j denotes the current computational domain point; n p,p+1 Denotes the normal vector of the common interface or structure.

[0096] At the same time, the predicted result MSE of the common interface or structure is satisfied lp,p+1 The residual with the average temperature of the two adjacent regions

[0097]

[0098]

[0099] Among them, T p is the temperature of the current region, Tp+1 is the temperature of adjacent regions; represents the average temperature of two interfaces; represents the residual of the average temperature of two adjacent regions; N lp,p+1 represents the number of points sampled in the common interface or structure sampling domain.

[0100] For the sparse point hard constraint condition:

[0101]

[0102] where, T s (x, t) is the temperature data of sparse points, and the above N g 、N I 、N b 、N lp and Ns represent the number of points sampled in the model internal sampling domain, the number of points sampled in the initial condition part, the number of points sampled in the boundary condition part, the number of points sampled in the heat flux continuity condition part, and the number of points sampled in the sparse point hard constraint condition, respectively.

[0103] Then the total loss is:

[0104] Loss = W g MSE g + W I MSE I + WbMSEb + Wl p MSEl p + W s MSE s (17)

[0105] where, W g 、W l 、W b 、W lp 、W s are the adaptive weights of each loss item, respectively.

[0106] Step S44: Use optimization algorithms such as Stochastic Gradient Descent (SGD) and Adaptive Moment Estimation (Adam) to train and solve the physical information correction network module built in Step S43, update its network parameters to minimize its loss function, complete the training, and obtain the temperature field result output by the physical information correction network module. Compare the temperature field calculation result with the reference solution, and calculate error metrics to evaluate the solution accuracy. The reference solution is the high-precision numerical simulation result or experimental data, and the error metrics include RMSE, MAE, etc. According to the solution accuracy, adjust the physical information neural network structure in the physical information correction network module (such as changing the activation function type, optimizing the number of network layers), and optimize the algorithm parameters (such as adjusting the learning rate decay strategy, increasing the number of iterations) to improve the accuracy and stability of solving the problem, so that the loss is less than the rated residual or exceeds the rated number of iteration steps, and finally obtain the optimal physical information correction network module.

[0107] Step S45: Input the temperature field prediction result containing grid information and parameter information obtained in Step S3 into the optimal physical information correction network module obtained in Step S44 to obtain the temperature field result of the complex structure nonlinear / heterogeneous transient heat conduction problem after physical information correction;

[0108] Step S5: Save the temperature result obtained in Step S45 in txt format and output it.

[0109] It can be seen that Figure 3 for the traditional physics-informed neural network, the Loss cannot reach the rated value and cannot converge when solving the nonlinear / heterogeneous transient heat conduction problem, while for the present invention, the Loss can converge quickly when solving the nonlinear / heterogeneous transient heat conduction problem;

[0110] It can be seen that Figure 4 the error of the traditional physics-informed neural network is too large when solving the nonlinear / heterogeneous transient heat conduction problem, while the present invention can solve the nonlinear / heterogeneous transient heat conduction problem with high precision;

[0111] It can be seen that Figure 5 the traditional physics-informed neural network cannot capture the influence of the nonlinear / heterogeneous thermal physical parameters on heat conduction in the y direction and there are a large number of interpolation errors; on the contrary, the present invention greatly enhances the learning of the nonlinear / heterogeneous transient heat conduction problem in the y direction through the introduction of the Fourier neural operator and the improvement of the loss function;

[0112] Figure 6 The traditional physics-informed neural network cannot capture the influence of the nonlinear / heterogeneous thermal physical parameters on heat conduction in the x direction and there are a large number of interpolation errors; on the contrary, the present invention greatly enhances the learning of the nonlinear / heterogeneous transient heat conduction problem in the x direction through the introduction of the Fourier neural operator and the improvement of the loss function.

[0113] The above-described embodiments merely represent the implementation modes of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that for those skilled in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all fall within the protection scope of the present invention.

Claims

1. A method for quickly and accurately solving the non - linear / non - homogeneous transient heat conduction problem of complex structures, characterized in that, This method is implemented based on a pre-training module and a physical information correction network module. The pre-training module uses a Fourier neural network to quickly output the temperature field results of the prediction model according to the input model grid information and parameter information; The physical information correction network module uses physical information to correct the temperature field results generated in the pre-training module; specifically, it includes the following steps: Step S1: Obtain a data set, extract the required triple in the form of a model grid, temperature field results, and model parameters, integrate it into a data set input to the pre-training module, preprocess the data set, and divide the preprocessed data set into a training set and a test set; Step S2: Build a Fourier operator network to be used in the pre-training module. The Fourier operator network consists of three parts: neural network P, Fourier layer, and neural network Q; then, the model grid and model parameters in the training set obtained in step S1 are used as inputs and input into the Fourier operator network. After being processed by the three-layer network, the output of the Fourier operator network is obtained, which is the temperature field result in the training set; train the Fourier operator network, and after training is completed, use the test set obtained in step S1 to complete the test to obtain the built pre-training module; Step S4: After the pre-training module is built, input any model grid and model parameters into the pre-training module to obtain a temperature field prediction result containing grid information and parameter information as the preliminary prediction result of the temperature field; Step S5: Input the temperature field prediction result containing grid information and parameter information obtained in step S4 into the physical information correction network module, and use boundary conditions, initial conditions, control equations, and corresponding constraint physical information to constrain the temperature field prediction result containing grid information and parameter information obtained in step S4 to obtain the temperature result of the corrected non-linear or inhomogeneous transient heat conduction problem.

2. A method for quickly and accurately solving the nonlinear / inhomogeneous transient heat conduction problem of complex structures according to claim 1, characterized in that In step S1, the preprocessing includes data cleaning, noise processing, and initialization.

3. A method for quickly and accurately solving complex structure non-linear / non-homogeneous transient heat conduction problems according to claim 1, characterized in that Specifically, for step S2: Step S21: Complete the construction of the pre-training module; build a Fourier operator network required for the pre-training module, that is, neural network P, Fourier layer, and neural network Q. Neural network P consists of a fully connected layer, a Relu layer, and a fully connected layer; the Fourier layer is composed of an upsampling layer, a BN layer, a pooling layer, and a Relu layer stacked in sequence 2 times; neural network Q consists of two layers of fully connected layers; Step S22: Denote the model grid and model parameters in the training set obtained in step S1 as input a(x), and input it into neural network P built in step S21. Neural network P is mainly responsible for lifting the input function a(x) to a higher-dimensional space to obtain v(x); Step S23: Input v(x) obtained in step S22 into the Fourier layer; the Fourier layer is the core of the pre-training module, which first performs a fast Fourier transform (FFT) on v(x) to obtain a series of frequency-domain patterns; through feature selection, the high-frequency patterns of the input data are filtered out, and the low-frequency patterns are retained to obtain ν t (x); then, ν t (x) performs an inverse fast Fourier transform (IFFT) to obtain the output ν T (x); Step S24: Input ν T (x) obtained in step S23 into neural network Q; Neural network Q reduces the dimension of ν T (x) to the same dimension as the input a(x) in step S22, and the output of neural network Q is denoted as u(x). u(x) is the result of the predicted model's temperature field and also the output of the pre-training module; Step S25: Train the Fourier operator network, and after training is completed, use the test set obtained in step S1 to complete the test verification of the Fourier operator network to obtain the built pre-training module.

4. A method for quickly and accurately solving complex structure non-linear / non-homogeneous transient heat conduction problems according to claim 3, characterized in that Specifically, for step S4: Step S41: Build a physical information correction network module; The physical information correction network module consists of a physical information neural network, including two parts: the construction of the physical information neural network and the construction of the loss function; the network part of the physical information neural network is composed of three stacked fully connected layers and Relu layers; the loss function of the physical information neural network is denoted as Loss, mainly Including: the loss of the control equation is denoted as MSE g 、the loss of the initial condition is denoted as MSE I 、the loss of the boundary is denoted as MSE b 、the loss of the heat flux continuity condition is denoted as MSE lp and the constraint loss of some known temperature points is denoted as MSE s ; Step S42: Collect the model grid, model parameters, and the temperature field obtained from the pre-training module in Step S3 as the input of the physical information correction network module, obtain the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model through the model parameters and the grid; Step S43: Convert the model grid information collected in Step S42 into domain points, and use the boundary conditions, initial conditions, and corresponding physical constraint information of the physical model obtained in Step S42 to construct the loss function Loss, completing the construction of the physical information correction network module; Step S44: Use an optimization algorithm to train and solve the physical information correction network module built in Step S43, update its network parameters to minimize its loss function, complete the training, and obtain the temperature field result output by the physical information correction network module; compare the temperature field calculation result with the reference solution, and calculate the error index to evaluate the solution accuracy, where the reference solution is the result of high-precision numerical simulation or experimental data; according to the solution accuracy, adjust the physical information neural network structure and optimization algorithm parameters in the physical information correction network module to obtain the optimal physical information correction network module; Step S45: Input the temperature field prediction result containing grid information and parameter information obtained in Step S3 into the optimal physical information correction network module obtained in Step S44 to obtain the temperature field result of the complex structure non-linear / non-homogeneous transient heat conduction problem after physical information correction.

5. A method for quickly and accurately solving the nonlinear / inhomogeneous transient heat conduction problem of complex structures according to claim 4, characterized in that, The specific content of Step S43 is as follows: Define the control equation function g(x,t) for the non-linear and non-homogeneous transient heat conduction equation: Wherein, λ is the thermal conductivity of the model material; ρ(T) is the density of the model material; c(T) is the specific heat of the model material; T refers to the temperature result to be predicted; x is the spatial coordinate position; t is the calculation time; is the gradient operator; Then the control equation loss: Among them, MSE g represents the loss of the governing equation; N g represents the number of sampled domain points inside the model; g(x,t) represents the governing equation function; For the initial condition loss: Among them, is the known initial condition; MSE I represents the loss of the governing equation; N I represents the number of sampled domain points in the initial condition part; T(x) t＝0 represents the predicted temperature value at the moment of t = 0; For the boundary loss: MSE b = MSE bDirichlet + MSE bNeumann + MSE bRobin (7) Among them, MSE bDirichlet represents the Dirichlet boundary loss; N bDirichlet represents the number of sampled domain points in the Dirichlet boundary part; T(x,τ) represents the predicted temperature value of the Dirichlet boundary; represents the true temperature value of the Dirichlet boundary; τ represents any moment; MSE bNeumann represents the Neumann boundary loss; N bNeumann represents the number of sampled domain points in the Neumann boundary part; λ represents the thermal conductivity; n i represents the normal vector of the boundary; x i represents the spatial coordinates of the corresponding point; q(x,τ) represents the surface heat flux; MSE bRobin represents the Robin boundary loss; Nb R obin represents the number of sampled domain points in the Robin boundary part; h(T w -T f ) represents the heat exchange between the external environment and the internal temperature; T w represents the ambient temperature; T f represents the external temperature; MSE b represents the boundary loss; j represents the current computational domain point; The residual of the predicted result for the common interface or structure that should satisfy the heat flux continuity condition is denoted as MSE lp,p+1 : Among them, T p is the temperature of the current region, and T p+1 is the temperature of the adjacent region, and λ p is the temperature of the current region, and λ p+1 is the temperature of the adjacent region, and n p,p+1 is the normal vector of the common interface or structure; N lp,p+1 represents the number of sampled domain points of the common interface or structure; j represents the current computational domain point; n p,p+1 represents the normal vector of the common interface or structure; Simultaneously satisfy the MSE of the prediction result of the common interface or structure lp,p+1 The residual MSE from the average temperature of two adjacent regions avgp,p+1 : MSE lp = MSE avgp,p+1 + MSE lp,p+1 (11) Among them, T p is the temperature of the current area, and T p+1 is the temperature of the adjacent area; T avgp,p+1 represents the average temperature of the two interfaces; MSE avgp,p+1 represents the residual of the average temperature of two adjacent areas; N lp,p+1 represents the number of points sampled in the common interface or structure sampling domain. For the sparse point hard constraint condition: Among them, T s (x, t) is the temperature data of the sparse points, and the above N g , N I , N b , N lp and Ns respectively represent the number of sampled domain points inside the model, the number of sampled domain points in the initial condition part, the number of sampled domain points in the boundary condition part, the number of sampled domain points in the heat flux continuity condition part, and the number of sampled domain points in the sparse point hard constraint condition; Then the total loss is: Loss=W g MSE g +W I MSE I +WbMSEb+Wl p MSEl p +W s MSE s (13) Among them, W g , W l , W b , W lp , W s are the adaptive weights of each loss respectively.

6. A method for quickly and accurately solving the non-linear / non-homogeneous transient heat conduction problem of complex structures according to claim 4, characterized in that In Step S44, the physical information correction network module built in Step S43 is trained and solved using the stochastic gradient descent method and the adaptive estimation optimization algorithm.

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