Multi-domain physical field parameterized prediction method and device based on PINN, and storage medium

Abstract

The invention relates to the technical field of multi-physics field coupling, in particular to a PINN-based multi-domain physics field parameterization prediction method and device and a storage medium, and the method comprises the steps: obtaining a control equation and boundary conditions of a multi-domain physics field and independent feature parameters of each sub-domain; constructing a trunk branch framework model of a multi-domain physical field through a physical information neural network; the space coordinates and the characteristic parameters of the multi-domain physical field are input into the main branch frame model, and predicted values of a velocity field and a pressure field are generated; carrying out training point sampling on the multi-domain physical field, and training a trunk branch frame model by using point data obtained by sampling to obtain a parameterized model; and predicting a multi-domain actual physical field through the parameterized model. According to the method, the PINN, parameterization input and the control equation of the multi-domain physical field are combined, the actual multi-domain physical field can be efficiently and accurately predicted, the calculation complexity of a traditional numerical method is reduced, and the method is suitable for rapid prediction of the complex parameterization physical field.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-physical field coupling, and in particular to a method, device and storage medium for parameterized prediction of multi-domain physical fields based on PINN. Background Art

[0002] When dealing with complex multi-domain physical field problems, traditional numerical methods such as the finite element method (FEM) and the finite volume method (FVM) suffer from high computational complexity and long calculation times. In particular, solving the Darcy-Brinkman equation, the core equation describing porous media flow, often requires significant computational resources and time. In recent years, with the rapid development of deep learning technology, the Physical Information Neural Network (PINN), a new approach combining deep learning with physical governing equations, has been gradually applied to the prediction and simulation of physical fields.

[0003] In porous media flow problems, the Darcy-Brinkman equation combines Darcy's law and the Navier-Stokes equations, effectively simulating low-speed flows and flow phenomena in high-porosity media. However, traditional numerical methods for solving the Darcy-Brinkman equation require dealing with complex boundary conditions and multi-physics coupling, as well as refined meshing, resulting in low computational efficiency. PINN, by directly embedding the governing physical equations into the neural network's loss function, can bypass the complex numerical discretization process and utilize automatic differentiation to automatically perform derivatives, significantly improving the efficiency of case solutions. However, existing PINN methods mostly predict single-domain physical fields, and the parametric prediction of multi-domain physical fields remains challenging, preventing the full utilization of PINN's advantages. The governing equations of multi-domain physical fields often involve complex boundary conditions and multi-physics coupling, and the parametric input further increases the complexity of problem solving. Using PINN to efficiently and accurately predict parametric multi-domain physical fields remains an urgent problem. Summary of the Invention

[0004] The purpose of the embodiments of the present invention is to solve the problem of low computational efficiency in multi-domain physical field prediction in the prior art by combining parameterized PINN with the governing equations of multi-domain physical fields.

[0005] In order to achieve the above-mentioned objectives, an embodiment of the present invention provides a method for parametric prediction of multi-domain physical fields based on PINN, including: obtaining the control equations, boundary conditions and independent characteristic parameters of the multi-domain physical field; constructing a trunk-branch framework model of the multi-domain physical field through a physical information neural network; inputting the spatial coordinates and characteristic parameters of the multi-domain physical field into the trunk-branch framework model to generate predicted values ​​of the velocity field and the pressure field; sampling training points of the multi-domain physical field, using the sampled point data to train the trunk-branch framework model to obtain a parameterized model; and predicting the actual multi-domain physical field through the parameterized model.

[0006] Preferably, the governing equation of the multi-domain physical field is the multi-domain Darcy-Brinkman equation.

[0007] Preferably, the trunk-branch framework model includes an input layer, a trunk network, a branch network and an output layer. The input layer is used to receive the parameters of the multi-domain physical field, the trunk network is used to extract the global physical characteristics of the multi-domain physical field, the branch network is used to process the local physical characteristics of each subdomain, and the output layer is used to output the predicted multi-domain actual physical field.

[0008] Preferably, the prediction method provided by the embodiment of the present invention further includes optimizing the model parameters of the trunk-branch framework model through a loss function, wherein the loss function includes the residuals of the control equations of the physical field and the residuals of the boundary conditions.

[0009] Preferably, the prediction method provided by the embodiment of the present invention also includes adjusting the model parameters of the trunk-branch framework model through a hybrid optimization strategy, and the hybrid optimization strategy includes using the ADAM algorithm for global search and using the L-BFGS algorithm for fine optimization.

[0010] Preferably, sampling is performed by a data collocation strategy, which includes adopting Gaussian distribution sampling at the interface and Latin hypercube sampling in each subdomain.

[0011] Preferably, the L-BFGS algorithm reconfigures the optimizer before starting each batch of training.

[0012] Preferably, the branch network supports knowledge transfer, which includes: pre-training the trunk-branch framework model to obtain an initialization model; freezing the trunk network of the initialization model; and fine-tuning the branch network of the initialization model through data training to obtain a new parameterized model.

[0013] The present invention also provides a control device, comprising: a memory, a processor, and a computer program stored in the memory and runnable on the processor, wherein the processor executes the computer program to implement a method for multi-domain physical field parameterization prediction according to any one of the above items of the present application.

[0014] The present invention also provides a machine-readable storage medium, characterized in that the machine-readable storage medium stores instructions, which enable the machine to execute the method for multi-domain physical field parameterization prediction according to any one of the above applications.

[0015] Through the above technical solution, the parameters of multi-domain physical fields can be predicted efficiently and accurately, the computational complexity of traditional numerical methods can be reduced, and a new solution can be provided for the rapid prediction of complex physical fields.

[0016] Other features and advantages of the embodiments of the present invention will be described in detail in the subsequent detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The accompanying drawings are used to provide a further understanding of the embodiments of the present invention and constitute a part of the specification. Together with the following detailed description, they are used to explain the embodiments of the present invention, but do not constitute a limitation of the embodiments of the present invention. In the accompanying drawings: Figure 1 This is a flow chart of the method for multi-domain physical field parameterization prediction based on PINN; Figure 2 is a schematic diagram of different flow conditions in the gradient porous medium in two subdomains; Figure 3 This is a diagram of a dual-domain gradient porous media model; Figure 4 It is a trunk-branch framework model diagram; Figure 5 It is the distribution diagram of sampling points of different data matching strategies; Figure 6 This is a comparison chart of loss values ​​under different data matching strategies; Figure 7 is an absolute error result graph of the prediction results provided by an embodiment of the present invention; and Figure 8 3 is a diagram showing the corresponding streamlines and prediction results of the velocity amplitude U and FVM solution provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0018] The following describes the specific implementation of the embodiment of the present invention in detail with reference to the accompanying drawings. It should be understood that the specific implementation described herein is only used to illustrate and explain the embodiment of the present invention and is not used to limit the embodiment of the present invention.

[0019] Figure 1 The process of multi-domain physical field parameterization prediction method based on PINN is demonstrated. Figure 1 Describe the implementation steps of the present invention in detail: Step S100: Obtain the control equations, boundary conditions and independent characteristic parameters of each subdomain of the multi-domain physical field.

[0020] Construct a gradient porous media model, such as Figure 2 As shown. Interface Γ in the model int Ω is divided into two subdomains Ω1 and Ω2, each with different porosity. Figure 2 Where ∂Ω1 and ∂Ω2 represent impermeable boundaries, Γ1 and Γ2 represent the interfaces between subdomains Ω1 and Ω2 and the adjacent media, respectively, which are usually used as inlets and outlets. With interface Γ int Orthogonal, such as Figure 2 As shown in (a), Nearby Γ int Nearby It remains constant due to conservation of flow.

[0021] Darcy's law is expressed as follows: (1) Where p represents the static pressure, μ represents the dynamic viscosity of the fluid, and K represents the permeability of the porous medium, as shown in Figure 2 As shown in (b), the distribution trend of pressure p can be obtained.

[0022] When the Darcy velocity at the interface With tangential components, such as Figure 2 As shown in (c), the flow behavior becomes more complicated in the subdomain due to the rapid changes in the parameters of the momentum equation. Figure 2 d), the direction of the inlet velocity and Γ int parallel, resulting in lateral flow perpendicular to the boundary.

[0023] More preferably, the governing equation of the multi-domain physical field in step S100 may be the multi-domain Darcy-Brinkman equation. In the concept of average volume, the macroscopic momentum equation of incompressible flow in porous media is modified by Einstein's formula, thereby obtaining the following Darcy-Brinkman equation: (2) Among them, d p represents the particle size of the porous medium, and ε represents the porosity.

[0024] like Figure 3 As shown, the dual-domain gradient porous media model provided by the embodiment of the present invention converts the entire domain Γ int It is divided into two subdomains Ω1 and Ω2, with respective porosities ε1 and ε2. The boundary conditions of the mathematical model are: (3) Wherein, the superscript d represents the inlet Γ in and exit Γ out, u and v correspond to the Dirichlet conditions for the horizontal and vertical velocity components, v corresponds to the horizontal and vertical velocity components, represents the impermeable boundary between the two subdomains. The boundary Ω = ∂Ω1 ∪ ∂Ω2 is regarded as the Neumann condition, that is, a completely sliding boundary.

[0025] Step S200: Constructing a trunk-branch framework model of a multi-domain physical field through a physical information neural network.

[0026] The framework model used in this embodiment is the TB-net PINN framework model, which inherits the gridless nature of the vanilla PINN method, namely, the use of automatic differentiation to calculate numerical gradients. At the same time, the original dense fully connected neural network is replaced by a trunk-branch structure, which not only enhances the extraction of global and specific features but also reduces parameter complexity, thereby facilitating deeper training.

[0027] refer to Figure 4 In a preferred embodiment, the trunk-branch framework model includes an input layer, a trunk network, a branch network and an output layer. The input layer is used to receive the parameters of the multi-domain physical field, the trunk network is used to extract the global physical characteristics of the multi-domain physical field, the branch network is used to process the local physical characteristics of each subdomain, and the output layer is used to output the predicted characteristic parameters of the multi-domain physical field.

[0028] The trunk-branch framework model can be expressed as follows: (4) Where M represents the number of layers in the backbone network, N represents the number of layers in a single branch network, i represents the index of the current branch network, k represents the total number of branch networks, and T o Represents the output of the backbone network, and the symbol ◦ represents recursion. represents a single-layer operator in the backbone network, B o represents the output of the branch network, O represents the output of the entire network, represents a single-layer operator in a branch network, represents the operator of the output layer, represents the neural network parameters.

[0029] Step S300: inputting the spatial coordinates and characteristic parameters of the multi-domain physical field into the trunk-branch framework model to generate predicted values ​​of the velocity field and the pressure field.

[0030] like Figure 4 As shown, an embodiment of the present invention provides a parameterized solution for the flow field in a trunk-branch dual-domain gradient porous medium. For a specific process, you can choose to omit the parameter input parameters ε1 and ε2, or you can choose to perform knowledge migration on a specific porosity configuration with a branch network after parameterized training.

[0031] By introducing characteristic parameters into the mathematical model of the flow, the actual values ​​of the physical quantities u, p and v can be calculated after training using the following formulas: (5) Where V represents the characteristic velocity, P represents the characteristic pressure, 、 and is the output of the trunk-branch framework model. By applying the above formula, the actual value can be recovered after training.

[0032] Step S400: sampling training points of the multi-domain physical field, and using the sampled point data to train the trunk-branch framework model to obtain a parameterized model.

[0033] More preferably, the model parameters of the trunk-branch framework model can be optimized by a loss function, which includes the residuals of the control equations of the physical field and the residuals of the boundary conditions. The loss function can be expressed as follows: (6) in, represents the total loss, represents the boundary condition loss, Indicates physical loss, represents the wall loss, represents the inlet loss, Indicates export losses.

[0034] Specifically for the dual-domain darcy-brinkman, the formula is further split: (7) in, represents the loss contribution in the specified domain Ω, and denote the loss contributions in subdomains Ω1 and Ω2, respectively. Specifically, represents the loss of the continuity assumption, and and They represent the losses of the corresponding momentum equations. Each basic loss term is associated with a specific weight parameter, denoted by λ.

[0035] The loss function is constructed by the following formula, with the goal of finding the optimal parameter set θ ∗ , to minimize overall losses : (8) Here, θ represents the collective parameter.

[0036] More preferably, when training a trunk-branch framework model, the model parameters of the trunk-branch framework model can be adjusted using a hybrid optimization strategy. This strategy can include global search using the ADAM algorithm and refined optimization using the L-BFGS algorithm. The ADAM algorithm combines the benefits of the momentum gradient descent (MGD) algorithm and the root mean square propagation (RMSProp) algorithm. Its core concept is to perform an unbiased correction on the estimates of the first- and second-order derivatives of the gradient, and then use these corrected estimates to update the parameters. Unlike the ADAM algorithm, the L-BFGS algorithm uses the second-order derivative to refine the search direction. This gradient generally provides a more accurate representation of the target curvature than the first-order derivative, thereby accelerating the objective function. This process can be described as a two-step iterative approximation of the search direction, followed by a line search to determine the appropriate update step size. L-BFGS is sensitive to the initial starting point and may become trapped in a local minimum for complex models. Therefore, a global search is first performed using the ADAM algorithm, followed by the L-BFGS algorithm for subsequent optimization.

[0037] A reference solution was generated using the finite volume method (FVM) to ensure the accuracy of predictions obtained with the parameterized model. The selection of training and collocation points is crucial, given that even slight differences in porosity can lead to significant differences in permeability. A limited number of training points hinders the model from learning boundary conditions, while an excessive number of training points hinders effective problem solving. Near interfaces, changes in multi-pore porosity can lead to abrupt changes in macroscopic quantities, mimicking the effects of discontinuities and thus increasing the complexity of the solution process.

[0038] More preferably, the sampling in step S400 may be performed by a data matching strategy, which may include adopting Gaussian distribution sampling on the interface and adopting Latin hypercube sampling in each subdomain.

[0039] Figure 5 is the sampling point distribution diagram of different data matching strategies. For example, Figure 5 As shown in the figure, targeting the discontinuous characteristics of multi-domain physical field interfaces, five data matching strategies are proposed to optimize the spatial distribution of training points and improve model prediction accuracy. Specifically, they include uniform sampling strategy, adding fewer clusters strategy, keeping fewer clusters unchanged strategy, adding more clusters strategy, and keeping more clusters unchanged strategy.

[0040] The optimization process is divided into two stages. In the first stage, the ADAM algorithm is used for 1×10 5 epochs, with a learning rate α = 1 × 10 −4 , momentum parameters β1 = 0.9 and β2 = 0.99. Subsequently, the L-BFGS optimizer was applied to a maximum of 5×10 4 Iterations. Figure 6 This is a comparison chart of loss values ​​under different data matching strategies. Figure 6 As shown in Figure 3, after training, the loss curves of all strategies steadily decrease during the iteration process and eventually reach a low level. In particular, the use of the L-BFGS optimizer can significantly accelerate the reduction of the loss value.

[0041] The absolute error of the prediction results relative to the FVM solution using different data collocation strategies is shown in the following figure: Figure 7 As shown in Figure 2, except for the interface and inlet regions, where the sharp gradients lead to slightly higher errors, the errors are still relatively low in the entire flow domain. ∞ Quantitatively analyze the prediction performance of different data concatenation strategies and normalize L2: (9) where φ pred represents the physical quantity being evaluated, N FVM Refers to the number of spatial points used in the FVM calculation. Evaluated by the prediction error metric, the prediction accuracy of the above strategies is relatively high.

[0042] Figure 8 The corresponding streamlines and prediction results of velocity amplitude U and FVM solutions using different data matching strategies are shown in Figure 2. Figure 8 As shown in the figure, due to the porosity difference, the fluid flowing from the left is deflected to the right upon entering the gradient porous medium. Once it crosses the interface, it merges with the fluid on the right, with the tangential direction of the streamlines gradually becoming parallel to the outlet normal. Due to the diversion of the fluid to the right, the velocity amplitude on the left tends to be lower than that on the right. Clearly, all strategies produce streamline distributions consistent with the FVM solution.

[0043] Step S500: Predicting the multi-domain actual physical field through the parameterized model.

[0044] Training a parameterized model with a single batch of data can encounter training load and convergence issues, so the L-BFGS strategy can be used to perform sequential training, which includes multiple batches of random data. The data configuration is different for each batch, and the optimization process requires gradient information from the previous iteration to approximate the Hessian Matrix. Therefore, it is preferable that the L-BFGS algorithm reset the optimizer before starting training for each batch to prevent the accumulation of bias in subsequent optimization caused by data retained from previous batches.

[0045] More preferably, the branch network can support knowledge transfer, which includes: pre-training the trunk-branch framework model to obtain an initialized model; freezing the trunk network of the initialized model; and fine-tuning the branch network of the initialized model through data training to obtain a new parameterized model. For example, once a parameterized model is obtained, targeted training can be performed for a given porosity configuration to improve the prediction accuracy of a specific operating scenario. During this period, the values ​​of ε1 and ε2 are fixed. They are fixed rather than randomly generated. Knowledge transfer aims to further reduce the loss function. , thereby improving the overall flow field prediction performance.

[0046] An embodiment of the present invention provides a machine-readable storage medium having instructions stored thereon, which enable a machine to execute the above-mentioned PINN-based multi-domain physical field parameterization prediction method.

[0047] At the same time, an embodiment of the present invention also provides a control device, which includes: a memory, a processor, and a computer program stored in the memory and executable on the processor. The processor executes the computer program to implement the above-mentioned PINN-based multi-domain physical field parameterization prediction method.

[0048] Through the above technical solution, the present invention achieves the following advantages: 1. Combined with the TB-net PINN framework, the computational complexity of traditional numerical methods is significantly reduced; 2. Improve the accuracy of multi-domain physical field prediction through hybrid optimization strategy and loss function design; 3. Suitable for rapid prediction of complex multi-domain physical field parameterization, such as porous media flow, heat conduction and other problems.

[0049] In summary, the present invention provides a new solution for the parametric prediction of multi-domain physical fields, and has broad application prospects in engineering computing, fluid mechanics, environmental science and other fields.

[0050] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. Thus, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0051] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowcharts and / or block diagrams, as well as combinations of processes and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowcharts and / or block diagrams. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0052] These computer program instructions may also be stored in a computer readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 a process or multiple processes and / or boxes Figure 1 The function specified in one or more boxes.

[0053] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing the instructions executed on the computer or other programmable device for implementing the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0054] In a typical configuration, a computing device includes one or more processors (CPUs), input / output interfaces, network interfaces, and memory.

[0055] The memory may include non-permanent memory in a computer-readable medium, random access memory (RAM) and / or non-volatile memory in the form of read-only memory (ROM) or flash RAM. The memory is an example of a computer-readable medium.

[0056] Computer-readable media includes both permanent and non-permanent, removable and non-removable media that can be implemented using any method or technology to store information. Information can be computer-readable instructions, data structures, program modules, or other data. Examples of computer storage media include, but are not limited to, phase-change RAM (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, compact disc read-only memory (CD-ROM), digital versatile disc (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transmission media that can be used to store information that can be accessed by a computing device. As defined herein, computer-readable media does not include transitory computer-readable media such as modulated data signals and carrier waves.

[0057] It should also be noted that the terms "comprises," "includes," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, commodity, or apparatus that includes a series of elements includes not only those elements, but also other elements not expressly listed, or elements inherent to such process, method, commodity, or apparatus.

[0058] The above are merely embodiments of the present invention and are not intended to limit the present invention. It will be apparent to those skilled in the art that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention are intended to be included within the scope of the claims of the present invention.

Claims

1. A method for multi-domain physical field parameterization prediction based on PINN, characterized in that: The method for multi-domain physical field parameterization prediction based on PINN includes: Obtain the governing equations, boundary conditions, and independent characteristic parameters of each subdomain of multi-domain physical fields; Construct a trunk-branch framework model of multi-domain physical fields through physical information neural network; Inputting the spatial coordinates and characteristic parameters of the multi-domain physical field into the trunk-branch framework model to generate predicted values ​​of the velocity field and the pressure field; Sampling training points of the multi-domain physical field, and using the sampled point data to train the trunk-branch framework model to obtain a parameterized model; The multi-domain real physical fields are predicted by the parameterized model.

2. The method for multi-domain physical field parameterization prediction based on PINN according to claim 1, characterized in that: The control equation of the multi-domain physical field is the multi-domain Darcy-Brinkman equation.

3. The method for multi-domain physical field parameterization prediction based on PINN according to claim 1, characterized in that: The trunk-branch framework model includes an input layer, a trunk network, a branch network and an output layer. The input layer is used to receive various parameters of the multi-domain physical field, the trunk network is used to extract the global physical characteristics of the multi-domain physical field, the branch network is used to process the local physical characteristics of each subdomain of the multi-domain physical field, and the output layer is used to output the predicted parameters of the multi-domain physical field.

4. The method for multi-domain physical field parameterization prediction based on PINN according to claim 1, characterized in that: The PINN-based multi-domain physical field parameterized prediction method also includes optimizing the model parameters of the trunk-branch framework model through a loss function, wherein the loss function includes the residual of the control equation of the physical field and the residual of the boundary condition.

5. The method for multi-domain physical field parameterization prediction based on PINN according to claim 1, characterized in that: The PINN-based multi-domain physical field parameterized prediction method also includes adjusting the model parameters of the trunk-branch framework model through a hybrid optimization strategy, wherein the hybrid optimization strategy includes using the ADAM algorithm for global search and using the L-BFGS algorithm for fine optimization.

6. The method for multi-domain physical field parameterization prediction based on PINN according to claim 1, characterized in that: The sampling is performed using a data collocation strategy, which includes adopting Gaussian distribution sampling at the interface and Latin hypercube sampling in each subdomain.

7. The method for multi-domain physical field parameterization prediction based on PINN according to claim 5, characterized in that: The L-BFGS algorithm reconfigures the optimizer before each batch of training starts.

8. The method for multi-domain physical field parameterization prediction based on PINN according to claim 3, characterized in that: The branch network supports knowledge transfer, and the knowledge transfer includes: Pre-training the trunk-branch framework model to obtain an initialization model; Freeze the backbone network of the initialization model; The branch network of the initialization model is fine-tuned through data training to obtain a new parameterized model.

9. A control device, characterized in that: The control device includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method for parameterized prediction of multi-domain physical fields according to any one of claims 1 to 8.

10. A machine-readable storage medium, characterized in that The machine-readable storage medium stores instructions, which enable the machine to execute the method for multi-domain physical field parameterization prediction according to any one of claims 1-8.

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