Two-dimensional vorticity prediction method based on improved Fourier neural operator

Abstract

The invention discloses a two-dimensional vorticity prediction method based on an improved Fourier neural operator, and the method comprises the steps: constructing and training a neural operator model with a multilayer perceptron to carry out the parameterization of a kernel integral operator, and carrying out the averaging of parameters needed by the model for processing different frequency components; therefore, a fixed number of parameters are adapted to any number of frequency components to realize two-dimensional vorticity prediction. The method has a lower relative error in a high-frequency domain, and meanwhile, due to the powerful function fitting capability of the multi-layer perceptron, the method also has a lower relative error in a low-frequency domain.

Description

[0001] This application is a divisional application of the following application: Application No. [202411825731.X] Application Date [2024 / 12 / 12] Title [A Fluid Dynamics Optimization Simulation Method Based on Improved Fourier Neural Operator] Applicant [Shanghai Jiao Tong University]. Technical Field

[0002] This invention relates to a technique in the field of neural networks, specifically a two-dimensional vortex prediction method based on an improved Fourier neural operator. Background Technology

[0003] Existing two-dimensional vortex prediction techniques rely on numerical methods, such as the finite element method, but these methods require a huge amount of computation time when performing simulations; while deep learning-based methods currently lack accuracy and the models they use require a large number of parameters, resulting in high hardware requirements. Summary of the Invention

[0004] This invention addresses the shortcomings of existing Fourier neural operators, which require independent parameters to handle different frequency components, as well as the problems of excessive parameter count and poor simulation accuracy during simulation. It proposes a two-dimensional vorticity prediction method based on an improved Fourier neural operator. By using a neural network amortized model to handle the parameters required for different frequency components, the Fourier neural operator can maintain a low parameter count while reducing simulation errors at different frequency components. This results in an amortized parameterized neural operator model, and improvements are achieved in the field of fluid dynamics simulation based on this model.

[0005] This invention is achieved through the following technical solution:

[0006] This invention relates to a two-dimensional vortex prediction method based on an improved Fourier neural operator. By constructing and training a neural operator model with a multilayer perceptron, parameterized kernel integral operators and amortized models are used to process the parameters required for different frequency components, thereby achieving two-dimensional vortex prediction with a fixed number of parameters to adapt to any number of frequency components.

[0007] The neural operator model described above uses a neural network to fit the mapping between two function spaces, thereby obtaining solutions to partial differential equations in different functions through a set of model parameters.

[0008] The parameterized kernel integral operator mentioned above refers to the operator that, when the kernel function satisfies In this form, the convolution theorem is used in Fourier space to transform the complex integration operation into a more computationally scalable multiplication, specifically: ,in: Fourier transform, The kernel function with respect to frequency after Fourier transform. For kernel function, For functions; in the discrete case, The values ​​are obtained by parameterizing the values ​​of different frequency components.

[0009] The parameterization mentioned refers to the mapping of discrete frequency coordinates to the corresponding kernel function values ​​at those coordinates using a multilayer perceptron. Specifically, it involves finding the values ​​of a given frequency coordinate on a set of basis functions. and two multilayer perceptrons and Calculate the complex numerical kernel function with respect to frequency after Fourier transform. , where N is the number of basis functions and k is the number of frequency coordinates. Technical effect

[0010] This invention utilizes a multilayer perceptron and orthogonal basis function embedding to fit the kernel function in the kernel integral operator. It achieves lower fluid dynamics simulation errors compared to existing techniques, including lower errors across all frequency domains. Attached Figure Description

[0011] Figure 1 This is a flowchart of the present invention;

[0012] Figure 2 This is a schematic diagram of a neural operator model;

[0013] Figure 3 This is a schematic diagram illustrating the effect of an example. Detailed Implementation

[0014] like Figure 1 As shown, this invention relates to a two-dimensional vorticity prediction method based on an improved Fourier neural operator, comprising:

[0015] Step 1: Training Dataset Acquisition: After randomly generating a certain number of different initial vorticity values ​​using a Gaussian process, the numerical solution method is used to solve the vorticity values ​​at each subsequent time step using the Crank-Nicholson method.

[0016] Step 2: Construct a neural operator model and train it using the dataset obtained in Step 1: Input the vorticity of the first few time steps (the first 10 in the experiment) into the neural operator model, calculate the loss function based on the predicted vorticity of the next time step, and update the parameters of the neural operator model.

[0017] like Figure 2As shown, the neural operator model includes: a linear layer unit, an activation function unit, a multilayer perceptron operator unit, and a kernel integral operator unit. Specifically: the linear layer unit performs linear transformation processing based on the input vortex function information to obtain a function result with higher or lower dimensions; the activation function unit performs nonlinear transformation processing based on the function information to obtain a nonlinearly mapped function result; the multilayer perceptron unit performs orthogonal basis function calculation and nonlinear transformation processing based on frequency coordinate information to obtain the kernel function result in the kernel integral operator; and the kernel integral operator unit performs integral operator operations based on the kernel function and the input vortex function to obtain the linearly mapped function result.

[0018] The multilayer perceptron unit includes a linear transformation at the beginning and end, an intermediate kernel integral operator, and an activation function, wherein the activation function uses a Gaussian error linear unit function.

[0019] The kernel function result is obtained through multilayer perceptron calculation, specifically including:

[0020] Step a: After obtaining the frequency coordinates of the discrete vorticity values ​​after Fourier transformation, calculate the values ​​of a finite set of orthogonal basis functions on this set of coordinates.

[0021] Step b: Input the values ​​obtained in step a into two multilayer perceptrons respectively, allowing them to fit the real and imaginary parts of the kernel function to obtain the final kernel function values. Specifically, these values ​​are the values ​​of a given frequency coordinate on a set of basis functions. and two multilayer perceptrons and Through formula The kernel function after Fourier transform is obtained by using a multilayer perceptron to perform nonlinear transformation. .

[0022] The loss function is calculated using the L2 norm, which is the difference between the vorticity predicted by the neural operator model and the vorticity value in the actual dataset.

[0023] Step 3, Fluid Dynamics Simulation: Input the observed actual vorticity into the neural operator model trained in Step 2 to obtain the predicted value of vorticity over time. Specifically, input the observed vorticity data from 0s to T-1 seconds. Input the neural operator model trained in step 2 to obtain the vortex data at second T. The obtained vortex data at second T and the data from second 1 to T-1 are then input into the neural operator model to obtain the vortex result at second T+1. This process is repeated iteratively to obtain the vortex data for subsequent time periods. The training input time period is [0, T-1] seconds, and the spatial resolution is... .

[0024] Through specific practical experiments, with a hyperparameter setting of 4-layer kernel integral operator and hidden dimension of 32, and by randomly initializing model parameters and using the AdamW optimizer, this method reduces the relative error by an average of 35% on a standard two-dimensional vortex numerical simulation dataset, while also exhibiting lower relative errors across all frequency ranges.

[0025] Table 1 shows the relative errors of the present invention and similar methods on a standard dataset. The experimental results demonstrate that the relative error of the present invention is significantly lower than that of other methods, reaching the state-of-the-art level to date.

[0026] Table 1

[0027] As shown in Table 2, this method can achieve smaller error values ​​in each frequency domain while requiring fewer parameters than existing deep learning techniques.

[0028] Table 2

[0029] like Figure 3 As shown, this is a pseudo-color image of vorticity simulation prediction based on the above dataset. Different colors represent different values. The top row is the actual vorticity, and the bottom row is the simulation result. From left to right, the vorticity is from 11 to 20 seconds. After model training, this method can predict the change of vorticity within a certain time range with relatively high accuracy, achieving a good fluid dynamics simulation effect.

[0030] Compared with existing technologies, this method utilizes two multilayer perceptrons to parameterize the kernel integral operator, thus requiring only fixed learnable parameters (on the multilayer perceptrons) to obtain the kernel function values ​​at each frequency coordinate. This amortizes the separate learnable parameters originally needed for parameterizing each frequency coordinate across the two multilayer perceptrons, thereby reducing the number of parameters required. Other Fourier neural operator methods truncate frequencies and directly fit the kernel function of the kernel integral operator using learnable parameters. Therefore, this invention has a lower relative error in the high-frequency domain. Furthermore, due to the powerful function fitting capability of the multilayer perceptron, this invention also has a lower relative error in the low-frequency domain.

[0031] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.

Claims

1. A two-dimensional vorticity prediction method based on an improved Fourier neural operator, characterized in that, By constructing and training a neural operator model with a multilayer perceptron, parameterized kernel integral operators and amortized models are used to process the parameters required for different frequency components, thereby achieving two-dimensional vorticity prediction with a fixed number of parameters to adapt to any number of frequency components. The neural operator model described above uses a neural network to fit the mapping between two function spaces, thereby obtaining solutions to partial differential equations in different functions through a set of model parameters.

2. The two-dimensional vorticity prediction method based on the improved Fourier neural operator according to claim 1, characterized in that, The parameterized kernel integral operator mentioned above refers to the operator that, when the kernel function satisfies In this form, the convolution theorem is used in Fourier space to transform the complex integration operation into a more computationally scalable multiplication, specifically: ,in: Fourier transform, The kernel function with respect to frequency after Fourier transform. For kernel function, For functions; in the discrete case, The values ​​are obtained by parameterizing the values ​​of different frequency components.

3. The two-dimensional vorticity prediction method based on the improved Fourier neural operator according to claim 1, characterized in that, The parameterization mentioned refers to: using a multilayer perceptron to fit discrete frequency coordinates to the corresponding kernel function values ​​at those coordinates; specifically, it involves: given the values ​​of frequency coordinates on a set of basis functions. and two multilayer perceptrons and Calculate the complex numerical kernel function with respect to frequency after Fourier transform. , where N is the number of basis functions and k is the number of frequency coordinates.

4. The two-dimensional vorticity prediction method based on the improved Fourier neural operator according to claim 1, characterized in that, The neural operator model includes: linear layer units, activation function units, multilayer perceptron operator units, and kernel integral operator units. Specifically: the linear layer units perform linear transformation processing based on the input vorticity function information to obtain function results of higher or lower dimensions; the activation function units perform nonlinear transformation processing based on the function information to obtain the nonlinearly mapped function result; the multilayer perceptron units perform orthogonal basis function calculations and nonlinear transformation processing based on frequency coordinate information to obtain the kernel function result in the kernel integral operator; and the kernel integral operator units perform integral operator operations based on the kernel function and the input vorticity function to obtain the linearly mapped function result.

5. The two-dimensional vorticity prediction method based on the improved Fourier neural operator according to claim 4, characterized in that, The kernel function result is obtained through multilayer perceptron calculation, specifically including: Step a: After obtaining the frequency coordinates of each discrete vortex value after Fourier transform, calculate the values ​​of a finite set of orthogonal basis functions on this set of coordinates. Step b: Input the values ​​obtained in step a into two multilayer perceptrons respectively, allowing them to fit the real and imaginary parts of the kernel function to obtain the final kernel function values. Specifically, these values ​​are the values ​​of a given frequency coordinate on a set of basis functions. and two multilayer perceptrons and Through formula The kernel function after Fourier transform is obtained by using a multilayer perceptron to perform nonlinear transformation. .

6. The two-dimensional vorticity prediction method based on the improved Fourier neural operator according to any one of claims 1-5, characterized in that, specifically... include: Step 1: Training Dataset Acquisition: After randomly generating several different initial vorticity values ​​using a Gaussian process, the vorticity values ​​at subsequent times are solved using numerical methods, i.e., by solving the Poisson equation to obtain the velocity field and using the Crank-Nicholson method to solve for the vorticity values ​​at a fixed time step. Step 2: Construct a neural operator model and train it using the dataset obtained in Step 1: Input the vorticity of the previous few time steps into the neural operator model, calculate the loss function based on the predicted vorticity of the next time step, and update the parameters of the neural operator model. Step 3, Fluid Dynamics Simulation: Input the observed actual vorticity into the neural operator model trained in Step 2 to obtain the predicted value of vorticity over time. Specifically, input the observed vorticity data from 0s to T-1 seconds. Input the neural operator model trained in step 2 to obtain the vortex data at second T. The obtained vortex data at second T and the data from second 1 to T-1 are then input into the neural operator model to obtain the vortex result at second T+1, and this process is iterated until the vortex data for subsequent times is obtained. The input time period during training is [0, T-1] seconds, and the spatial resolution is... .

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