Multi-fidelity physical information neural network modeling method for sharing parallel features

Abstract

The invention discloses a multi-fidelity physical information neural network modeling method for sharing parallel features, and belongs to the field of multi-fidelity agent model modeling. The method comprises the following steps: carrying out simulation calculation through CFD software (COMSOL) to obtain a large amount of low-fidelity data; constructing a physical definition of the solving range through mathematical reasoning; a neural network model for solving high-precision data is established by constructing a neural network shared by high-precision and low-precision feature parts. According to the method, under the condition that a large amount of low-precision data exists but high-precision data is lacked, the problem of low modeling precision caused by missing of high-fidelity data information is solved through the known physical control equation.

Description

Technical Field

[0001] The present invention belongs to the field of multi-fidelity proxy model modeling, and specifically relates to a multi-fidelity physical information neural network modeling method in which high-fidelity data is missing and a large amount of low-fidelity data and physical control equations are used. Background Art

[0002] With the rapid advancement of computing power, optimization algorithms, and automated computational capabilities, physically-informed neural networks (PINNs) have become a powerful tool for solving forward and inverse problems of partial differential equations (PDEs). PINN methods attempt to constrain the output of neural networks using PDEs and high-fidelity data. Using physics alone, PINN methods have proven effective for problems with reasonable stationarity and simple boundary conditions. However, they can struggle with problems exhibiting sharp spatial features or short timescales. To mitigate these issues, various enhancements to PINN methods have been proposed, such as Fourier feature embedding, implementation of adaptive sampling, and development of adaptive weighting techniques. Despite significant progress, PINNs still face significant challenges in solving problems of increasing complexity. In such cases, data may be required to further constrain the optimization of neural network parameters. High-fidelity datasets obtained from measurements or numerical simulations can be prohibitively expensive, while low-fidelity data can be abundant. Despite their lower accuracy, low-fidelity data can provide useful information to guide PINNs toward better solutions. Therefore, by combining low-fidelity and physics-based information, multi-fidelity machine learning can significantly enhance our ability to solve PDEs.

[0003] In multi-fidelity machine learning, it is crucial to uncover the correlation between low-fidelity data and high-fidelity solutions. Related research in the literature can be roughly divided into three categories:

[0004] The first type of method uses neural networks to explicitly approximate the correlation between high-fidelity and low-fidelity solutions. For example, the paper Meng X, Karniadakis G EA composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems[J]. Journal of Computational Physics, 2020, 401: 109020. proposed a composite neural network architecture, which uses three neural network modules (low-fidelity network NN L ), linear correlation network (NN H1 ) and nonlinear correlation networks (NN H2), respectively processing low-fidelity data and linear and nonlinear correlations between high and low fidelity, and further extending it to physical information neural networks (MPINNs), which adaptively learn complex relationships by decomposing correlations into linear and nonlinear components. This approach has proven effective in function approximation and inverse PDE problems, particularly outperforming Gaussian process regression in handling high-dimensional problems and adversarial data.

[0005] The second type of method is based on transfer learning, which enables cross-fidelity information transfer by sharing neural network parameters. The paper Aliakbari M, Mahmoudi M, Vadasz P, et al. Predicting high-fidelity multiphysics data from low-fidelity fluid flow and transport solvers using physics-informed neural networks [J]. International Journal of Heat and Fluid Flow, 2022, 96:109002 proposes using low-fidelity CFD data to pre-train a neural network, and then using transfer learning to initialize a physics-informed neural network (PINN), thereby improving prediction accuracy without the need for high-fidelity training data. This method significantly improves the training efficiency and accuracy of traditional PINNs in multi-physics problems such as fluid mechanics, heat transfer, and porous media transport by guiding PINN convergence with low-fidelity data. This is particularly effective when dealing with boundary layers and complex flow characteristics.

[0006] The third type of method uses the same network to predict high- and low-fidelity solutions, suitable for scenarios where the two are close in solution space. Chen W, Stinis P. Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations [J]. Journal of Computational Physics, 2024, 498:112683. They propose a "feature-adjacent multi-fidelity physics-informed machine learning" framework. This framework constructs a shared feature space through an encoder-decoder structure, constraining the representations of high- and low-fidelity solutions to adjacent regions. Specifically, the encoder maps the input to the feature space, while the decoder implements the mapping from features to the solution space, characterizing the fidelity difference through feature offset vectors. This method leverages the structural information of low-fidelity data to aid in the inference of high-fidelity solutions for both steady-state and non-steady-state PDE problems and inverse problems, effectively reducing reliance on high-fidelity data.

[0007] Based on the third type of method, the present invention proposes a multi-fidelity physical information neural network modeling method with shared parallel features. By introducing a linear and nonlinear dual-module architecture, a shared parallel feature layer and a high- and low-fidelity feature mapping layer are constructed in the nonlinear module to decouple the trend information of low-fidelity data from the detailed features of high-fidelity data. Specifically, the linear module captures the direct correlation between high and low fidelity, and the nonlinear module learns complex mappings through a multi-layer perceptron. The two realize information fusion through a shared feature space. This method uses rich samples of low-fidelity data to guide the learning process of the high-fidelity model, and by adaptively adjusting the mapping relationship in the feature space, it significantly improves the prediction accuracy of the model under a small amount of high-fidelity data, providing a new approach for efficient modeling of multi-physics field problems.

[0008] The core concept of the method proposed in this paper is to treat low-precision data and high-precision information as known information and construct a multi-fidelity physical information neural network to solve the high- and low-fidelity data simultaneously. This not only utilizes the physical field information contained in the low-fidelity data, but also constrains the entire physical field through the partial differential governing equations of the physical field, achieving a close approximation to the real high-fidelity data, thereby improving the convergence speed and accuracy of the neural network. Summary of the Invention

[0009] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is as follows:

[0010] A multi-fidelity physical information neural network modeling method with shared parallel features includes the following steps:

[0011] A large amount of low-precision data of the physical field (x l ,y l ,u l ,v l ,p l );

[0012] Based on the description of the problem to be solved, combined with mathematical and physical principles, the entire physical field control partial differential equations and boundary conditions are constructed;

[0013] Build a neural network with shared parallel features to load low-precision data and high-precision physical information;

[0014] A neural network model is trained using low-precision data and physical information (PDE), and the model is used to make predictions on high-fidelity data.

[0015] The present invention provides a multi-fidelity physical information neural network modeling method with shared parallel features. It proposes a multi-fidelity physical information neural network that integrates physical information when high-fidelity data is missing but a large amount of low-fidelity data is available. This solves the problem of incomplete data or the inability to directly solve the problem using a physical information neural network (PINN), thereby improving the high-fidelity prediction accuracy of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings used in the embodiments.

[0017] Figure 1 is a flowchart of the steps of the multi-fidelity physical information neural network modeling method with shared parallel features described in the embodiment;

[0018] Figure 2 The diagram is a high-precision comparison diagram, a low-precision data diagram, a prediction result diagram, and a relative error diagram of the X-direction velocity of the unsteady cover-driven flow problem in the embodiment;

[0019] Figure 3 The diagram is a high-precision comparison diagram, a low-precision data diagram, a prediction result diagram, and a relative error diagram of the Y-direction velocity of the unsteady cover-driven flow problem in the embodiment;

[0020] Figure 4 The high-precision comparison diagram, low-precision data diagram, prediction result diagram, and relative error diagram of the pressure P of the unsteady cover drive flow problem in the embodiment;

[0021] Figure 5 These are a high-precision comparison diagram, a low-precision data diagram, a prediction result diagram, and a relative error diagram of the total directional velocity of the unsteady cover-driven flow problem in the embodiment.

[0022] Figure 6 These are the high-precision comparison graph, low-precision data graph, prediction result graph, and relative error graph of the Burger problem speed in the embodiment. DETAILED DESCRIPTION

[0023] To make the technical solution of the present invention clearer, the present invention will be clearly and completely explained below with reference to the accompanying drawings of the embodiments of the invention. The embodiments described only use a part of the embodiments, not all of the embodiments. The present invention is specifically implemented according to the following steps:

[0024] (1) A large amount of low-precision data of the physical field (x) is obtained through rapid simulation calculation using CFD software (COMSOL) l ,y l ,u l ,v l ,p l );

[0025] (2) Based on the description of the problem to be solved, combining mathematical and physical principles, construct the partial differential equations and boundary conditions governing the entire physical field;

[0026] (3) Build a neural network that shares parallel features and loads low-precision data and high-precision physical information.

[0027] When building the nonlinear part of a neural network that shares parallel features, it is necessary to construct a Fourier transform layer with an appropriate dimension based on the complexity of the physical field being solved. Note that the training results of the neural network will be affected by the dimension of the Fourier transform layer, so the dimension should not be set too high during initialization.

[0028] (4) Loss function design of shared parallel feature neural network, mainly divided into low-precision data loss PDE loss and boundary loss Input low-precision data (x l ,y l ), calculate the low-precision data loss Input random sampling data (x pde ,y pde ), calculate the PDE loss Input boundary sampling data (x b ,y b ), calculate the boundary loss Described by formulas (1) to (5):

[0029]

[0030]

[0031] (5) Use As the total loss of the neural network, the neural network training is carried out, and the model is successfully trained using automatic differentiation, back propagation and neural network weight update.

[0032] The present invention provides two embodiments, namely, the unsteady cover driven flow problem with the velocity u in the X and Y directions when the Reynolds number is 10000. x ,u y , pressure P and total velocity V, and the viscous Burgers equation. The data used in the experiment are low-precision data, which come from COMSOL and MATLAB simulation calculation results.

[0033] Example 1: A multi-fidelity physical information neural network with shared parallel features is constructed for the unsteady lid-driven flow problem to predict the velocity and pressure of the physical field. The details are as follows:

[0034] This embodiment uses the unsteady cover-driven flow data in the COMSOL case library as the data source, sets the data calculated by the coarse grid as low-precision data, and the data calculated by the very dense grid as high-precision data. It is only used as a control group and does not participate in model training. This embodiment uses a set of data with a Reynolds number of 10,000 for experiments, takes the position information (x, y) as the input of the model, and outputs the velocity and pressure on the X and Y axes. Only 540 low-fidelity data are provided, and high-fidelity data is not provided. Physical information is used, that is, partial differential control equations. The sampling method of the partial differential control equation is random sampling, and all low-fidelity data will be used for model training to prove the effectiveness and accuracy of the model. The method of the present invention is used to predict the velocity and pressure result graphs of unsteady cover-driven flow and their corresponding relative errors, as shown in the figure. Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 As shown in , the prediction results are nearly 30% better than those predicted by PINN, solving the problem of incomplete solution of PINN.

[0035] Example 2: A multi-fidelity physical information neural network with shared parallel features is constructed for the viscous Burgers equation to predict velocity. The details are as follows:

[0036] The experimental data of this embodiment are the same as those of embodiment 1, except that the input of the model becomes position and time (x, t), and the output of the model is only velocity. The velocity prediction result diagram of the viscous Burgers equation and its corresponding relative error are shown in Figure 1. Figure 6 As shown in the table, the absolute error of the prediction results is within 1e -2 the following.

[0037] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may be modified and varied in various ways.

Claims

1. A multi-fidelity physical information neural network modeling method with shared parallel features, characterized in that: The steps include: A large amount of low-precision data of the physical field (x l ,y l ,u l ,v l ,p l ); Based on the description of the problem to be solved, combined with mathematical and physical principles, the entire physical field control partial differential equations and boundary conditions are constructed; Build a neural network with shared parallel features to load low-precision data and high-precision physical information; A neural network model is trained using low-precision data and physical information, and then used to make predictions on high-fidelity data.

2. A multi-fidelity physical information neural network modeling method with shared parallel features according to claim 1, characterized in that: The low-precision data comes from the two-dimensional NS equation examples in the COMSOL software learning case library, including: Determine the dependent variable (x l ,y l ), that is, the coordinates of the physical field; The physical field data (u l ,v l ,p l ).

3. A multi-fidelity physical information neural network modeling method with shared parallel features according to claim 1, characterized in that: Construct the entire physics field governing partial differential equations and boundary conditions, including: u(x,t)=(u w (x,t),0) (x,t)∈Γ1×[0,T] u(x,t)=0 (x,t)∈Γ2×[0,T] u(x,0)=u0(x) x∈Ω The time span T is 1, the Reynolds number is 10000, the geometry is a square with a boundary of 1 and the boundary conditions are initial velocity 0 and the velocity of the moving cover is 1. Since it is a steady state, the time t is ignored.

4. A multi-fidelity physical information neural network modeling method with shared parallel features according to claim 1, characterized in that: The neural network with shared parallel features utilizes the relationship between high-precision and low-precision data, specifically including: Both low-fidelity and high-fidelity solutions can be mapped to a set of basis functions The function space spanned by them, where their representation on this space and Representing coefficients by limiting and The relative distance between where d f is the relative distance to strengthen the close relationship, α H =α L ⊙(1+d f λ), where , ⊙ is a point-wise multiplication operator.

5. A multi-fidelity physical information neural network modeling method with shared parallel features according to claim 1, characterized in that: The low-precision data and high-precision physical information input are used to jointly constrain the high-precision and low-precision information, specifically including: The LF output matrix and HF output matrix of the MF network are used to approximate the LF solution and HF physical solution of the system respectively. The loss function of the MF network training includes the low- and high-fidelity parts defined by the previous low- and high-fidelity information: The loss function of the multi-fidelity physical information neural network with specific shared parallel features is defined as follows:

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