Variable parameter physical field rapid prediction and parameter inversion method based on physical information neural network

Abstract

The invention discloses a variable parameter physical field rapid prediction and parameter inversion method based on a physical information neural network, and relates to the crossing field of machine learning and industrial simulation. In order to enable the neural network to have generalization ability for physical field parameters, a solution function of a field is creatively expressed as a form related to the field parameters, and specifically, the solution function of the field is expressed as a polynomial expression expressed by the field parameters based on a finite difference method. Secondly, a neural network is built, a series of equation parameters and corresponding solutions are used as training data, the neural network is trained to predict coefficients in a polynomial expression, and therefore the model can solve a class of equations within the parameter range; the defect that PINNs can only be trained for one parameter is overcome; the model training process is not limited to a certain specific parameter, the trained model can be fixed when the inversion problem is solved, and only the equation parameters are updated in the training process, so that the solving speed of the inversion problem is improved.

Description

A Fast Prediction and Parameter Inversion Method for Variable-Parameter Physical Fields Based on Physical Information Neural Networks Technical Field

[0001] This invention relates to the intersection of machine learning and industrial simulation, specifically including a method for rapid prediction and parameter inversion of variable-parameter physical fields based on a physical information neural network. This method can be widely applied in engineering fields requiring parametric design and optimization, such as computational fluid dynamics, structural mechanics, and electromagnetic field analysis. Background Technology

[0002] In many engineering design and analysis fields, physical processes can be described by parameterized partial differential equations (PDEs). For example, in the design of aero-engines, the field parameters can be flight speed or angle of attack, and the physical field is the air pressure distribution; in the thermal management of electronic chips, the field parameters can be the thermal conductivity of the material, and the physical field is the temperature distribution. In practical applications, calculations related to physical fields can be divided into two categories: forward problems and inverse problems. The forward problem is to obtain the field simulation results corresponding to the given physical field expression and its parameters. The inverse problem is to invert the field parameters corresponding to the solution data when there are some spatial domain coordinates and their corresponding solutions, given unknown field parameters. Traditional numerical simulation methods based on finite element and finite volume methods, although highly accurate, have huge computational costs per simulation, and require repeated calls to the simulator when parameter scanning, optimization, or inversion is needed, resulting in unacceptable overall time costs and severely slowing down the design cycle. Therefore, developing a general technology that can quickly and accurately predict and invert variable parameter physical fields is key to improving the R&D efficiency of multiple industrial fields.

[0003] With the continuous development of deep learning, the Physics Informed Neural Network (PINN) applies deep learning methods to predict physical fields. It utilizes existing gradient calculation methods in deep learning to approximate the derivative terms in the physical field expression, and uses the expression itself as a loss term in the deep learning model training process for optimization. Specifically, it builds a neural network model and trains it using the initial conditions, boundary conditions, and loss terms represented by the equations of the physical field as the model's loss.

[0004]

[0005] This allows the model to simultaneously satisfy initial conditions, boundary conditions, and PDE constraints. PINN can also be used to solve inverse problems. It trains the model using solution data corresponding to a certain unknown parameter, simultaneously optimizing both network and equation parameters during training. The goal is to make the network fit the solution data as accurately as possible. The equation parameters obtained through this training process are the optimal equation parameters.

[0006] However, the efficiency of the PINN model in physics prediction needs improvement. In the forward problem, the PINN model can only be trained on one parameter at a time, and retraining is required if the parameters of the equation change. In the inverse problem, PINN needs to optimize both network parameters and equation parameters simultaneously, resulting in a long solution time. This makes the traditional PINN method difficult to apply to physics prediction scenarios with variable parameters.

[0007] To improve the simulation efficiency of variable-parameter physics fields, the AI ​​for Science field currently employs two main approaches: meta-learning and operator learning. Meta-learning uses two networks: one performs the target task, and the other outputs the parameters of the main network. Operator learning utilizes the general approximation theorem of deep learning, employing neural networks to represent operators and training them with a large amount of labeled data. However, these methods violate the physical meaning inherent in partial differential equations, and their accuracy in solving certain complex problems still has room for improvement. Designing a fast prediction method for variable-parameter physics fields that conforms to physical meaning would help improve the accuracy and efficiency of simulations, thereby enabling the solution of batch forward and inverse problems. This is of great significance for improving the efficiency of engineering design and analysis. Summary of the Invention

[0008] The purpose of this invention is to achieve rapid prediction and inversion of variable-parameter physical fields based on neural networks. To enable the neural network to generalize to physical field parameters, this invention innovatively represents the solution function of the field in a form related to the field parameters. Specifically, it uses the finite difference method to represent the solution function of the field as a polynomial expression in terms of the field parameters. Then, a neural network is built, using a series of equation parameters and their corresponding solutions as training data. The neural network is trained to predict the coefficients in the polynomial expression, thus enabling the model to solve a class of equations within the parameter range. This overcomes the limitation of PINNs, which can only be trained on a single parameter. Since the model training process is not limited to a specific parameter, the trained model can be fixed when solving the inversion problem, and only the equation parameters are updated during training, thereby improving the speed of solving the inversion problem.

[0009] This invention mainly includes the following steps:

[0010] 1. Discretize the physical field expression to construct a general representation of the variable parameter physical field, so as to facilitate the subsequent training of neural network models with solution data;

[0011] 2. Build a deep learning model, taking the equation parameters and spatial coordinates as network inputs and the coefficients of the polynomials in the corresponding spatial coordinates as outputs;

[0012] 3. Using multiple sets of known parameters and corresponding solution samples, the MSE error between the true solution and the predicted solution is used as the loss function to update the model parameters through multiple rounds of iteration;

[0013] 4. For field prediction problems, input any new field parameter value β and spatiotemporal coordinates into the trained network, and directly output the physical field distribution under that parameter; for inversion problems, use randomly initialized field parameters and spatial coordinates as network inputs, fix the neural network parameters, iteratively update the input field parameters, and stop training when the MSE error between the network prediction solution and the training set is less than a set value, so as to obtain high-precision field parameter prediction values.

[0014] First, the physical field expression is discretized to construct a general representation of the variable parameter physical field. Using the finite difference method, the derivative terms in the physical field expression are expressed as the relationship between solutions at discrete points in the spatial domain. Through recursive derivation in the time domain, the relationship between any point in the spatial domain and the initial conditions can be obtained. In this relational expression, the coefficients of the equation are treated as independent variables of a polynomial, extracted, and simplified to obtain the expression between the field information and the field parameters.

[0015]

[0016] in Indicates the physical field at position The value at that location, Represents spatial coordinates, Represents the time coordinate; It is a field parameter. It is the degree of the parameter, and its range of values ​​is Integers between 1 and 2. This general form applies to any physical field, the difference being that the number of terms in the polynomial varies depending on the complexity of the physical field.

[0017] A deep learning model is built to learn the mapping between field parameters and field information. The network input includes equation parameters, spatial and temporal coordinates, and the network output consists of all coefficients in the aforementioned polynomial expression. The deep learning model employs the Tanh activation function and uses the Xavier-Normal initialization method to initialize the model parameters.

[0018] After the model is built, the parameter domain is evenly divided within a certain parameter range. For each parameter, the solution of the partial differential equation to be trained is obtained for a series of parameters within the domain range by training the traditional PINN neural network, performing the simulation process through simulation software such as MATLAB or COMSOL, or conducting actual observations.

[0019] The solutions obtained above are randomly shuffled and mixed to form the training dataset;

[0020] Predicting polynomial coefficients using a neural network Then, the network-predicted solution is calculated using the aforementioned polynomial expression, and the MSE loss is calculated by comparing it with the solution data obtained from simulation.

[0021]

[0022] in Represents the spatial coordinates of the i-th point. This represents the time coordinate of the i-th point. These are the parameters of the neural network.

[0023] Point The network parameters are optimized by minimizing the MSE loss using optimizers commonly used in deep learning, such as Adam. Following this step, PINN loss can be used for fine-tuning the model parameters to make them more closely resemble physical properties.

[0024] After training the model, fix the optimal network parameters θ. In field prediction problems, input any new field parameter value β and spatiotemporal coordinates into the trained network. After obtaining the polynomial coefficients of the network output, the physical field distribution under that parameter can be directly output through the polynomial calculation mentioned above. In the scenario of solving inversion problems, given an unknown parameter and its corresponding solution, set the parameter to be solved to any initial value, fix the network parameters, and use gradient descent to optimize the field parameters in the expression in the direction of minimizing loss. :

[0025]

[0026] To obtain the optimal field parameter value β, the solution process for the parameter inversion problem is completed. The meanings of the symbols in this formula are consistent with those in the expression above. This invention only requires optimization of the input parameters when solving the inverse problem, which significantly reduces training time compared to traditional PINN. Attached Figure Description

[0027] Figure 1: Complete flowchart of the present invention

[0028] Figure 2: Spatiotemporal grid partitioning diagram of the finite difference method

[0029] Figure 3: Schematic diagram of the method for solving forward problems according to the present invention

[0030] Figure 4: Schematic diagram of the method for solving the inverse problem according to the present invention

[0031] Figure 5: Application of the present invention and other methods on the typical Burgers equation. Comparison chart of prediction results Detailed Implementation

[0032] This invention primarily achieves rapid prediction of variable parameter physical fields and accelerated solution of their inversion problems. It is implemented on a Windows 10 operating system, using Python 3.8 as the programming language and PyTorch 1.13.0 as the deep learning framework, with model training performed on an NVIDIA GeForce RTX 3090 (16GB) graphics card. In the implementation process, firstly, based on the finite difference method, the physical field expression is derived and rewritten to form a general polynomial expression. Then, simulation and scientific computing software such as COMSOL and MATLAB are used to obtain solutions for the physical field on several representative parameters within the range of parameters to be solved. These solution data are then used to train a fully connected neural network model, resulting in a model with parameter generalization capabilities and enabling rapid solution of the inversion problem.

[0033] The Burgers equation is an important nonlinear partial differential equation in fluid mechanics and mathematical physics, describing both nonlinear convection and diffusion mechanisms, which closely matches the characteristics of actual traffic flow. Therefore, it is widely used in traffic flow modeling. Its expression is:

[0034] In traffic flow modeling scenarios, the Burgers equation... This describes the propagation speed of traffic disturbances at a specific location and time on the road. A negative value for traffic disturbance indicates that congestion has occurred at that location. It is the rate of change of traffic flow propagation speed with respect to time, representing the degree of drastic change in traffic conditions perceived by a roadside monitoring point. When its value is large, it indicates that the road conditions at this location have changed drastically. This represents the rate of change of traffic flow propagation speed with respect to spatial coordinates. It describes the inherent characteristics of traffic shock waves, namely that once congestion occurs at one point, the congestion will spread rapidly along the road. It is the second derivative of the traffic flow propagation speed with respect to spatial coordinates. The term is a viscous term, which makes the traffic wave smoother and can effectively simulate the change of traffic flow from smooth to congested. It is a variable parameter in the Burgers equation. In traffic flow modeling, it quantifies the driver's reaction speed and the degree of their ability to predict congestion on the road. A higher value indicates a faster driver reaction and a higher degree of predictability regarding road congestion, thus mitigating the propagation speed of traffic disturbances to some extent. In practice, this is typically achieved by observing and analyzing traffic participants or road traffic videos, and then using established numerical prediction models to apply field parameters to the actual performance of traffic participants. Quantify it.

[0035] In the forward problem, given the reaction speed and predictability of traffic participants to road conditions, the goal is to determine the direction of traffic flow propagation at different times and locations. The inverse problem, on the other hand, involves knowing the road traffic flow conditions and working backward to deduce the participants' behavior, thus helping managers understand the causes of congestion. Quickly solving the forward and inverse problems of the Burgers equation is helpful in predicting traffic conditions and guiding road traffic management in practice. Given the importance of the Burgers equation in actual traffic flow simulation, the following uses a simulation of the Burgers equation as an example to illustrate and verify the technical effects achieved by this invention.

[0036] 1. Discretize the physical field expression using the finite difference method to obtain a polynomial expression between the solution function and the equation parameters, which facilitates subsequent neural network modeling.

[0037] In the derivation, it is assumed that the initial conditions are fixed and that periodic boundary conditions exist. When discretizing the partial derivatives using the finite difference method, a uniform grid needs to be generated in the spatiotemporal domain. The grid generation method is shown in Figure 2. It is the time step. It is the spatial step size.

[0038] Considering the stability conditions of the equation, the central difference method is selected, using two adjacent time steps.

[0039] and Discretizing the partial derivatives of the equation using the coordinates on the coordinates, we get:

[0040]

[0041] in Indicates the first Each time step Indicates the first One space step. Spacetime coordinates The solution to the physical field at that location. Spacetime coordinates The solution to the physical field at that location. Spacetime coordinates The solution to the physical field at that location.

[0042] Coordinates The solution to the physical field at point . This formula shows that the ... The solution of the physical field at the time step and the first The relationship between the solution of the physical field at each time step and the field parameters is... The relevant linear polynomial has coefficients that are related to the equation parameters and the function values ​​of other points adjacent to the current grid point.

[0043] By recursively deriving this formula, we can obtain that for any two time steps... and The relationship between them is also a polynomial related to the coefficients of the equation. Further abstracting the expression obtained from finite differences, we can obtain the polynomial expression for the solution function:

[0044] in Indicates the physical field at position The value at that location, These are field parameters, and are important as mentioned above. These are the coefficients of the polynomial. Represents spatial coordinates, Represents the time coordinate. It is the degree of the parameter, and its range of values ​​is Integers between [a certain value] and [a certain value]. This invention uses a neural network to predict a series of coefficients in the aforementioned polynomial. The specific structure of the neural network is described below. It is the highest power in the polynomial. During the experimental process of the Burgers equation, It has already achieved satisfactory solution accuracy.

[0045] 2. Build a deep learning model and incorporate field parameters. Time coordinates Spatial coordinates As network input, the coefficients of the above polynomial in the corresponding spatiotemporal coordinates are... As output, the deep learning model used in this invention is a fully connected neural network model, and the network input is field parameters. Spatial coordinates Time coordinates The network output is a series of polynomial coefficients from the network predictions. The neural network has four hidden layers, each with 100 neurons. It uses the Tanh activation function and initializes its parameters using the Xavier-Normal method during initialization. The optimizer used during training is Adam, with a learning rate of 1e-4. The number of training epochs ranges from 100,000 to 150,000, depending on the complexity of the problem. In the Burgers equation experiment, it was trained for 100,000 epochs. Several different equation parameters and their corresponding solution data were selected to train the neural network. The loss function used during training is the mean squared error between the model's predicted solution and the true solution at the corresponding parameter and location. The specific expression of the loss function is as follows:

[0046]

[0047] in It is the number of field parameters selected during training. Indicates the first There are several field parameters; N is the number of data points selected for each field parameter. Indicates the first Data points, Represents the spatial coordinates of the i-th data point. This represents the time coordinate of the i-th data point; It is the highest power in the polynomial. This represents the power of the parameter, and its value range is... . The field parameters are At that time, the spacetime coordinates are The actual solution data is obtained, and the method for obtaining the solution data will be described later. Therefore, this invention is based on a supervised data-driven approach.

[0048] 3. Determine the interval of the variable field parameters and divide the interval evenly to obtain the true solutions for certain field parameters within the interval. This can be achieved through simulation software such as MATLAB and COMSOL, training with the traditional PINN model, or actual observation. When using COMSOL simulation software to obtain solution data, after building the .mph model according to the software manual, the COMSOL LiveLink for MATLAB plugin can be used to batch generate solution data for different field parameters. In the experiment of the Burgers equation, it is necessary to achieve rapid solution for the variable field parameters. The range is [0.001, 1). This parameter interval is divided, and field parameters of 0.001, 0.002, 0.005, 0.105, 0.205, 0.305, 0.405, 0.505, 0.605, 0.705, 0.805, and 0.905, along with their corresponding solutions, are selected for model training. In the Burgers equation, smaller equation parameters indicate higher problem complexity; therefore, a larger number of parameters are selected for model training within the smaller parameter range.

[0049] 4. Before quickly solving the forward and inverse problems of variable parameter physical fields, a neural network model needs to be trained first. From the solution data of multiple parameters obtained in step 3, a portion of each parameter is randomly selected for model training. In the Burgers equation, 2000 data points are selected for each parameter. The model is trained using the loss function from step 2 for 100,000 training rounds. After training, the model can be fixed and used to solve the forward and inverse problems of generalization parameters. To verify the effectiveness of this invention, five parameters—0.0015, 0.0025, 0.003, 0.004, and 0.1—were selected for testing the model's generalization ability. In the forward problem, several classic operator learning and meta-learning methods, including traditional PINN, DATS+Hyper, DATS+MAD, GPT-PINN, DeepONet, PI-DeepONet, P2INN, and FNO, were compared; in the inverse problem, the traditional PINN method was selected for comparison.

[0050] 5. When solving the forward problem, first input the field parameters and spatiotemporal coordinates into the model and perform a forward inference to obtain the polynomial coefficients of the network prediction. Then, use the polynomial expression given in step 1 to calculate the solution predicted by the network. The L2 error of the generalization of parameters for each method is shown in the table below:

[0051] Traditional PINNDATS + HyperDATS + MADGPT - PINNDeepONet 0.319±0.171 0.192±0.095 0.207±0.093 0.476±0.159 0.027±0.013 PI-DeepONet P2INNFNO This invention 0.511±0.102 0.384±0.016 0.168±0.123 0.007±7e-4 surface

[0052] As can be seen, the method of the present invention achieves the lowest error compared to other methods in positive problems.

[0053] The inference times for each method are shown in the table below (unit: seconds):

[0054] Traditional PINNDATS+HyperDATS+MADGPT-PINNDeepONet2402e-32e-32.962e-3PI-DeepONetP2INNFNO This invention 2e-311.050.150.41 surface

[0055] It is evident that this invention can also achieve fast and real-time parameter generalization, with a calculation time of less than 1 second, meeting the requirements for real-time solution.

[0056] 6. When solving the parameter inversion problem, given the solution data corresponding to the unknown field parameters, it is necessary to obtain the optimal parameters corresponding to the solution data. This invention achieves efficient solution of the parameter inversion problem by reusing the model trained in step 4. The specific implementation steps are shown in Figure 4. The network parameters are fixed, and the unknown parameters are initialized to 0, which are then used as trainable parameters for optimization. The optimizer is Adam, and the number of optimization rounds is 20000. During the optimization process, the MSE loss function between the network-predicted solution and the true solution is also used, and the specific expression is:

[0057]

[0058] in Indicates the number of data points selected. It is a field parameter. Represents the spatial coordinates of the i-th data point.

[0059] Let j represent the time coordinate of the i-th data point, and j represent the power of the field parameter in the polynomial, with a value range of [0, ..., j]. Integers between ] The field parameters are At that time, the spacetime coordinates are The actual solution data is used. In the Burgers equation experiment, five parameters—0.0015, 0.0025, 0.003, 0.004, and 0.1—were selected to test the inverse problem solving capability, with 500 data points used for optimization for each parameter. The L2 relative error between the predicted and actual parameters is as follows:

[0060] Traditional PINN, this invention: 2.535±2.576 0.039±0.026 surface

[0061] It is evident that this invention can achieve high-precision inverse problem solving.

[0062] The time required for optimization by the two methods is shown in the table below (unit: seconds):

[0063] Traditional PINN, this invention 1831.81 surface

[0064] Because traditional PINN requires updating network parameters while solving inverse problems, while this invention only requires updating equation parameters, the efficiency of this invention in solving inverse problems is higher than that of traditional PINN, while still meeting the real-time and accuracy requirements in practical applications.

Claims

1. A method for fast prediction and parameter inversion of variable parameter physical fields based on a physical information neural network, characterized in that: The physical field expression is discretized to construct a general representation of the variable parameter physical field. A deep learning model is built, taking the physical field parameters and spatial coordinates as network inputs and the coefficients of the above polynomials in the corresponding spatial coordinates as outputs. The technical function is to learn the general mapping relationship from field parameters to field distribution. Using multiple sets of known parameters and corresponding solution samples, the MSE error between the true solution and the predicted solution is used as the loss function, and the model parameters are updated through multiple rounds of iteration. For field prediction problems, any new field parameter value β and spatiotemporal coordinates are input into the trained network, and the physical field distribution under that parameter is directly output, which can be used for rapid evaluation of engineering design schemes. For inversion problems, randomly initialized field parameters and spatial coordinates are used as network inputs, the neural network parameters are fixed, and the input field parameters are iteratively updated until the MSE error between the network predicted solution and the training set is less than a set value, at which point training stops, and the predicted field parameter value is obtained, which can be used for system state identification, fault diagnosis, or physical parameter calibration.

2. The method for fast prediction and parameter inversion of variable parameter physical fields based on a physical information neural network according to claim 1, characterized in that, The method for discretizing physical field expressions to construct a general representation of variable parameter physical fields includes: using forward difference, backward difference, and central difference forms in the finite difference method to express the derivative terms in the physical field expression as the relationship between solutions at discrete points in the spatial domain; through recursive derivation, the relationship between any point in the spatial domain and the initial conditions can be obtained, and the equation coefficients can be extracted as independent variables of a polynomial for simplification; during the simplification process, since the initial conditions of the equation are determined, they are treated as constants, and the discretized equation at this time only contains the equation parameters as variables, and the form is polynomial; the obtained expression is further abstracted to obtain an expression applicable to different forms of parameterized partial differential equations. Where x and t are the coordinates of any point in space; β is a parameter of the equation. It is the highest power of the polynomial, j is the degree of the parameter, and its range is... 。 3. The method for rapid prediction and parameter inversion of variable parameter physical fields based on a physical information neural network according to claim 1, characterized in that, Methods for building deep learning models include: building a deep learning model where the network inputs include field parameters, spatial and temporal coordinates, and the network output is all the coefficients in a polynomial expression.

4. The method for fast prediction and parameter inversion of variable parameter physical fields based on a physical information neural network according to claim 1, characterized in that, Methods for updating model parameters include: uniformly dividing the parameter domain within a certain parameter range, training with a PINN neural network, or performing simulation using simulation software to obtain simulation solution data of the physical field within its domain for certain typical parameters in the parameter range to be simulated; and randomly shuffling and mixing the solutions obtained above as a training dataset.

5. The method for fast prediction and parameter inversion of variable parameter physical fields based on a physical information neural network according to claim 3, characterized in that, The network output is obtained, and then the output is calculated according to a polynomial expression to obtain the network prediction solution. The mean squared error (MSE) loss is then calculated between the output and the simulation solution. The loss function is expressed as: Based on the above losses, the network parameters are optimized using an optimizer in deep learning.

6. The method for fast prediction and parameter inversion of variable parameter physical fields based on a physical information neural network according to claim 1, characterized in that, The methods for solving the inversion problem include: after training the model, fixing the optimal network parameters θ; when solving the inversion problem, given an unknown parameter and its corresponding solution, randomly initializing the parameters to be solved, and optimizing the equation parameters by minimizing the following loss function: In order to obtain the optimal field parameter value β.

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