Improved Fourier neural operator-based fluid mechanics optimization simulation method

Abstract

The invention discloses a fluid mechanics optimization simulation method based on an improved Fourier neural operator, which comprises the following steps: constructing and training a neural operator model with a multilayer perceptron to carry out parameterization on a kernel integral operator, and averaging parameters required by the model to process different frequency components, so that a fixed number of parameters are adapted to any number of frequency components; the accuracy of the Fourier neural operator in fluid mechanics simulation at different frequency components is improved, and meanwhile the parameter quantity needed by the model is reduced. 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

Technical Field

[0001] The present invention relates to a technology in the field of neural networks, in particular to a fluid mechanics optimization simulation method based on an improved Fourier neural operator. Background Art

[0002] Existing fluid mechanics simulations rely on numerical methods, such as the finite element method, but these methods require huge computing time for simulation; while deep learning-based methods currently lack accuracy, and the models used require a large number of parameters, resulting in high hardware requirements. Summary of the invention

[0003] In view of the shortcomings of the existing Fourier neural operator, that is, the independent parameters are required to process different frequency components, the number of parameters is too large during simulation, and the simulation accuracy is poor, the present invention proposes a fluid mechanics optimization simulation method based on an improved Fourier neural operator. The parameters required for different frequency components are processed by a neural network amortized model, so that the Fourier neural operator can maintain a low number of parameters while reducing the simulation error of the model under different frequency components. An amortized parameterized neural operator model is implemented, and improvements in the field of fluid mechanics simulation are achieved based on the model.

[0004] The present invention is achieved through the following technical solutions:

[0005] The present invention relates to a fluid mechanics optimization simulation method based on an improved Fourier neural operator. By constructing and training a neural operator model with a multi-layer perceptron to parameterize a kernel integral operator and amortize the parameters required by the model to process different frequency components, a fixed number of parameters can be used to adapt to any number of frequency components, thereby improving the accuracy of the Fourier neural operator in fluid mechanics simulation at different frequency components and reducing the number of parameters required by the model.

[0006] The neural operator model fits the mapping between two function spaces through a neural network, thereby obtaining partial differential equation solutions for different functions through a set of model parameters.

[0007] The parameterized kernel integral operator means that when the kernel function satisfies the form of K(xy), the convolution theorem is used to transform the complex integral operation into a more easily calculated multiplication in Fourier space, specifically: (T(f))(x)=F -1 (R·(Ff))(x), where: F is the Fourier transform, R is the kernel function of frequency after the Fourier transform, K(x,y) is the kernel function, and f(x) is the function; in the discrete case, R is obtained by parameterizing the values ​​of different frequency components.

[0008] The parameterization is to use a multi-layer perceptron to fit the discrete frequency coordinates to the mapping of the corresponding kernel function on the coordinates, specifically: the value of a given frequency coordinate on a set of basis functions And two multilayer perceptrons N1 and N2, calculate the complex-valued kernel function of the frequency after Fourier transformation Where: N is the number of basis functions, k is the number of frequency coordinates. Technical Effects

[0009] The present invention utilizes a multi-layer perceptron and an orthogonal basis function to embed the kernel function in the fitting kernel integral operator, and has a lower fluid mechanics simulation error than the prior art, including a lower error in all frequency domains. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 It is a flow chart of the present invention;

[0011] Figure 2 It is a schematic diagram of the system of the present invention;

[0012] Figure 3 It is a schematic diagram of the effect of the embodiment. DETAILED DESCRIPTION

[0013] like Figure 1 As shown, the present invention relates to a fluid mechanics optimization simulation method based on an improved Fourier neural operator, comprising:

[0014] Step 1: Obtain the training data set: After randomly generating a certain number of different initial vorticity values ​​using the Gaussian process, use the numerical solution method to solve it: first, obtain the velocity field by solving the Poisson equation, and then use the Crank-Nicholson method to solve the vorticity values ​​at subsequent moments under a fixed time step.

[0015] Step 2: Construct a neural operator model and use the data set obtained in step 1 for training: input the vorticity of the previous few moments (the first 10 in the experiment) into the neural operator model, calculate the loss function based on the predicted vorticity of the next moment, and update the parameters of the neural operator model.

[0016] like Figure 2As shown, the neural operator model includes: a linear layer unit, an activation function unit, a multi-layer perceptron operator unit and a kernel integral operator unit, wherein: the linear layer unit performs linear change processing according to the input vorticity function information to obtain a function result of a higher or lower dimension, the activation function unit performs nonlinear change processing according to the function information to obtain a function result after nonlinear mapping, the multi-layer perceptron unit performs orthogonal basis function calculation and nonlinear change processing according to the frequency coordinate information to obtain a kernel function result in the kernel integral operator, and the kernel integral operator unit performs integral operator operation according to the kernel function and the input vorticity function to obtain a function result after linear mapping.

[0017] The multi-layer perceptron unit includes linear changes at the beginning and end, an intermediate kernel integral operator, and an activation function, wherein: The activation function uses the Gaussian error linear unit function.

[0018] The kernel function result is obtained by calculating the multi-layer perceptron, and specifically includes:

[0019] Step a: after obtaining the frequency coordinates of each discrete vorticity value after Fourier transformation, calculate the values ​​of a finite set of orthogonal basis functions on this set of coordinates.

[0020] Step b: Input the values ​​obtained in step a into two multi-layer perceptrons to fit the real and imaginary parts of the kernel function, and obtain the final kernel function value, which is: the value of the given frequency coordinate on a set of basis functions And two multi-layer perceptrons N1 and N2, through the formula Use multi-layer perceptron to perform nonlinear changes to obtain the kernel function after Fourier transform

[0021] The loss function is: the difference between the vorticity predicted by the neural operator model and the vorticity value of the actual data set is calculated using the second norm.

[0022] Step 3, fluid mechanics simulation: Input the observed actual vorticity into the neural operator model trained in step 2 to obtain the predicted value of the vorticity over time. Specifically, the actual observed vorticity data from 0s to T-1 seconds is input into the neural operator model trained in step 2 to obtain the predicted value of the vorticity over time. Input the neural operator model trained in step 2 to obtain the vorticity data of the Tth second Then, the obtained vorticity data of the Tth second and the data from 1 to T-1 seconds are input into the neural operator model to obtain the vorticity result of the T+1th second, and the vorticity data of the subsequent time are obtained by iteration. Among them, the time period input during training is [0, T-1] seconds, and the spatial resolution is a×a.

[0023] After specific practical experiments, under the hyperparameter settings of 4-layer kernel integral operator and hidden dimension of 32, using the method of randomly initializing model parameters and AdamW optimizer, on the standard two-dimensional vorticity numerical simulation data set, this method reduces the relative error by 35% on average, and has lower relative errors in all frequency ranges.

[0024] As shown in Table 1, the relative errors of the present invention and similar methods on the standard data set are shown. From the experimental results, it can be seen that the relative error of the model proposed by the present invention is significantly reduced compared with other methods, reaching the most advanced level at present.

[0025] Table 1

[0026] As shown in Table 2, this method can achieve a smaller error value in each frequency domain while having fewer parameters than the existing deep learning technology.

[0027] Table 2

[0028] like Figure 3 As shown in the figure, it is a pseudo-color map of vorticity simulation prediction based on the above data set. Different colors represent different values. The upper row is the actual vorticity, and the lower row is the simulation result. From left to right, they are the vorticity of 11-20s respectively. After model training, this method can more accurately predict the change of vorticity within a certain time range and achieve better fluid mechanics simulation effect.

[0029] Compared with the prior art, the present method uses two multi-layer perceptrons to parameterize the kernel integral operator, so that only fixed learnable parameters (on the multi-layer perceptron) are needed to obtain the value of the kernel function on each frequency coordinate, and the separate learnable parameters originally required to parameterize each frequency coordinate are amortized to the two multi-layer perceptrons, thereby reducing the number of required parameters. Other Fourier neural operator methods perform frequency truncation and directly use learnable parameters to fit the kernel function of the kernel integral operator. Therefore, the present invention has a lower relative error in the high-frequency domain. At the same time, due to the powerful function fitting ability of the multi-layer perceptron, the present invention also has a lower relative error in the low-frequency domain.

[0030] The above-mentioned specific implementation can be partially adjusted in different ways by those skilled in the art without departing from the principle and purpose of the present invention. The protection scope of the present invention shall be based on the claims and shall not be limited by the above-mentioned specific implementation. Each implementation scheme within its scope shall be subject to the constraints of the present invention.

Claims

1. A fluid mechanics optimization simulation method based on an improved Fourier neural operator, characterized in that: By constructing and training a neural operator model with a multi-layer perceptron to parameterize the kernel integral operator and the parameters required by the amortized model to process different frequency components, a fixed number of parameters can be used to adapt to any number of frequency components, thereby improving the accuracy of the Fourier neural operator in fluid mechanics simulation at different frequency components while reducing the number of parameters required by the model; The neural operator model fits the mapping between two function spaces through a neural network, thereby obtaining partial differential equation solutions for different functions through a set of model parameters.

2. The fluid mechanics optimization simulation method based on the improved Fourier neural operator according to claim 1 is characterized in that: The parameterized kernel integral operator means that when the kernel function satisfies the form of K(xy), the convolution theorem is used to transform the complex integral operation into a more easily calculated multiplication in Fourier space, specifically: (T(f))(x)=F -1 (R·(Ff))(x), where: F is the Fourier transform, R is the kernel function of frequency after the Fourier transform, K(x,y) is the kernel function, and f(x) is the function; in the discrete case, R is obtained by parameterizing the values ​​of different frequency components.

3. The fluid mechanics optimization simulation method based on the improved Fourier neural operator according to claim 1 is characterized in that: The parameterization is to use a multi-layer perceptron to fit the discrete frequency coordinates to the mapping of the corresponding kernel function on the coordinates, specifically: the value of a given frequency coordinate on a set of basis functions And two multilayer perceptrons N1 and N2, calculate the complex-valued kernel function of the frequency after Fourier transformation Where: N is the number of basis functions, k is the number of frequency coordinates.

4. The fluid mechanics optimization simulation method based on the improved Fourier neural operator according to claim 1 is characterized in that: The neural operator model includes: a linear layer unit, an activation function unit, a multi-layer perceptron operator unit and a kernel integral operator unit, wherein: the linear layer unit performs linear change processing according to the input vorticity function information to obtain a function result of a higher or lower dimension, the activation function unit performs nonlinear change processing according to the function information to obtain a function result after nonlinear mapping, the multi-layer perceptron unit performs orthogonal basis function calculation and nonlinear change processing according to the frequency coordinate information to obtain a kernel function result in the kernel integral operator, and the kernel integral operator unit performs integral operator operation according to the kernel function and the input vorticity function to obtain a function result after linear mapping.

5. The fluid mechanics optimization simulation method based on the improved Fourier neural operator according to claim 4 is characterized in that: The kernel function result is obtained by calculating the multi-layer perceptron, and specifically includes: Step a, after obtaining the frequency coordinates of each value of the discrete vorticity value after Fourier transformation, 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 multi-layer perceptrons to fit the real and imaginary parts of the kernel function, and obtain the final kernel function value, which is: the value of the given frequency coordinate on a set of basis functions And two multi-layer perceptrons N1 and N2, through the formula Use multi-layer perceptron to perform nonlinear changes to obtain the kernel function after Fourier transform 6. The fluid mechanics optimization simulation method based on the improved Fourier neural operator according to any one of claims 1 to 5, characterized in that: include: Step 1: Obtaining the training data set: After randomly generating a number of different initial vorticity values ​​using the Gaussian process, the numerical solution method is used to solve the velocity field by solving the Poisson equation and using the Crank-Nicholson method to solve the vorticity values ​​at subsequent moments under a fixed time step; Step 2: construct a neural operator model and use the data set obtained in step 1 for training: input the vorticity of the previous several moments into the neural operator model, calculate the loss function based on the predicted vorticity of the next moment, and update the parameters of the neural operator model; Step 3, fluid mechanics simulation: Input the observed actual vorticity into the neural operator model trained in step 2 to obtain the predicted value of the vorticity over time. Specifically, the actual observed vorticity data from 0s to T-1 seconds is input into the neural operator model trained in step 2 to obtain the predicted value of the vorticity over time. Input the neural operator model trained in step 2 to obtain the vorticity data of the Tth second Then, the obtained vorticity data of the Tth second and the data from 1 to T-1 seconds are input into the neural operator model to obtain the vorticity result of the T+1th second and iterate to obtain the vorticity data of the subsequent time, where: the time period input during training is [0, T-1] seconds, and the spatial resolution is a×a.

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