Multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning

Abstract

The invention discloses a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, and the method comprises the steps: obtaining training data which comprises low-fidelity data and high-fidelity data; preprocessing the constructed deep learning agent model by using a Fourier neural operator and low-fidelity data to obtain a low-fidelity model; training the deep learning proxy model by using a Fourier neural operator and taking the network parameters of the low-fidelity model as initial parameters of high-fidelity training to obtain a high-fidelity proxy model; performing fine tuning on the high-fidelity proxy model by using the high-fidelity data; and predicting the physical field by using the fine-tuned high-fidelity proxy model to obtain a prediction result corresponding to the physical field. According to the invention, the training of the deep learning model is completed by using a large amount of low-fidelity data and a small amount of high-fidelity data, so that the constructed deep learning model can ensure the prediction precision of a physical field, the demand of the deep learning model for the high-fidelity data volume is reduced, and the modeling cost is reduced.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of physical field, and particularly relates to a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning. BACKGROUND

[0002] When constructing a deep learning-based proxy model meeting the accuracy requirement, a large number of high-fidelity data samples containing accurate physical information are often needed. For the acquisition of these high-fidelity data, for example, the method of physical experiment or the numerical simulation data acquired by high-precision grid, a large amount of human and computing resources and a long time are needed.

[0003] However, the existing method often ignores the computing cost needed for acquiring these high-fidelity data when constructing a proxy model. In particular, for the physical field prediction task, the acquisition of high-fidelity data needed for model training accounts for a large part of the entire proxy model construction process, so the acquisition cost of high-fidelity data may be one of the costs of constructing a high-precision proxy model.

[0004] In fact, the acquisition of high-fidelity data with effective physical information, such as real physical field information obtained by experimental method or high-precision simulation data obtained by numerical simulation model constructed by precise grid, needs high acquisition cost, for example, the former not only needs funds to purchase experimental equipment, but also needs to spend time of experimental personnel to debug and operate the experimental equipment; and the latter needs high-performance computing resources and a large amount of computing time. Both of them are time-consuming and resource-consuming methods. Therefore, it significantly increases the cost of constructing a high-precision deep learning proxy model, which is contrary to the original intention of reducing the operation cost and improving the operation efficiency by constructing a deep learning proxy model. Therefore, it is necessary to construct a deep learning proxy model construction method using cheap low-fidelity data combined with a small amount of expensive high-fidelity data to reduce the overall cost of constructing a proxy model. SUMMARY

[0005] To solve the above-mentioned technical problems in the prior art, the present application provides a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, which uses a large amount of low-fidelity data and a small amount of high-fidelity data to perform model transfer work based on the grid independence of Fourier neural operator, completes the training of the deep learning model, so that the constructed proxy model can guarantee the modeling accuracy of the predicted physical field, while significantly reducing the demand of the deep learning model for expensive high-fidelity data, and reducing the overall cost of constructing the model.

[0006] The technical scheme of the present application is as follows:

[0007] A multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning is provided, comprising:

[0008] Obtain training data, the training data comprising low-fidelity data and high-fidelity data, wherein the high-fidelity data comprises sample data containing accurate and effective physical information, and the low-fidelity data comprises sample data not containing or partially containing accurate and effective physical information;

[0009] Construct a deep learning agent model;

[0010] Preprocess the constructed deep learning agent model using the Fourier neural operator and the low-fidelity data to obtain a low-fidelity model;

[0011] Train the deep learning agent model using the Fourier neural operator to take the network parameters of the low-fidelity model as initial parameters for high-fidelity training to obtain a high-fidelity agent model;

[0012] Fine-tune the high-fidelity agent model using high-fidelity data;

[0013] Use the fine-tuned high-fidelity agent model to predict the physical field to obtain the prediction result corresponding to the physical field.

[0014] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, preprocessing the constructed deep learning agent model using the Fourier neural operator and the low-fidelity data to obtain a low-fidelity model comprises:

[0015] Input the low-fidelity data to the deep learning agent model;

[0016] Train the deep learning model using the Fourier neural operator and the input low-fidelity data to fit the mapping relationship between the low-fidelity data and the high-fidelity data;

[0017] Construct a loss function for supervised learning;

[0018] Optimize the Fourier neural operator parameters under the supervised paradigm using the loss function for supervised learning to obtain the low-fidelity model.

[0019] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, the loss function for supervised learning comprises:

[0020]

[0021] wherein L low represents the loss function for supervised learning, N represents the number of flow field data points, i represents the i-th flow field point, represents the low-fidelity prediction value, indicates a low-fidelity data label value, and l indicates low-fidelity data.

[0022] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, the low-fidelity model comprises:

[0023]

[0024] wherein, wherein, F l indicates a low-fidelity model learning function, indicates a low-fidelity input physical quantity, and in the above formula, N indicates a positive integer.

[0025] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, when training the deep learning model using the Fourier neural operator and the input low-fidelity data to fit the mapping relationship between the low-fidelity data and the high-fidelity data, the low-fidelity data is divided into two parts according to a preset ratio, one part is used for training, and the other part is used for verifying the training result.

[0026] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, fine-tuning the high-fidelity proxy model using the high-fidelity data comprises:

[0027] The network parameters of the low-fidelity model are used as the initial parameters of the high-fidelity training, and the predicted value is calculated;

[0028] The initial parameters are trained, and the error between the predicted value and the label value is minimized using the initial parameters;

[0029] The network parameters of the initial high-fidelity proxy model are determined by the minimum error of the high-fidelity data validation set, and the high-fidelity proxy model is obtained.

[0030] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning described above, the error between the predicted value and the label value is minimized by the following formula:

[0031]

[0032] wherein, indicates a predicted value, F h indicates the initial parameters of the high-fidelity model, indicates a high-fidelity input physical quantity, i indicates the i th flow field point, i∈{1,2,…,M}, M indicates a positive integer, and h indicates high-fidelity data.

[0033] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, the network parameters of the initial high-fidelity proxy model are determined by the minimum error of the high-fidelity data validation set and are calculated by the following formula:

[0034]

[0035] Wherein, L high represents high-fidelity data, M represents the number of flow field data points, and M>>N, represents the high-fidelity data label value.

[0036] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, the high-fidelity proxy model comprises:

[0037]

[0038] Wherein, M represents the final high-fidelity proxy model, represents the network parameters of the final high-fidelity proxy model.

[0039] Further, in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, different initial learning rates are set in the processes of preprocessing the constructed deep learning proxy model by using the Fourier neural operator and the low-fidelity data and fine-tuning the high-fidelity proxy model by using the high-fidelity data, and the initial learning rate of preprocessing the constructed deep learning proxy model by using the Fourier neural operator and the low-fidelity data is set to be greater than the initial learning rate of fine-tuning the high-fidelity proxy model by using the high-fidelity data.

[0040] The main advantages of the technical scheme of the present application are as follows:

[0041] The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning solves the problem of the general deep learning model requiring a large amount of high-fidelity data, processes the input data according to different physical field cases to obtain input data with specific formats, pre-trains a low-fidelity model using a large amount of low-fidelity data, fine-tunes the low-fidelity model using a small amount or zero high-fidelity data, so that the constructed proxy model can guarantee the modeling accuracy of the predicted physical field, significantly reduces the demand of the deep learning model for expensive high-fidelity data, and reduces the overall cost of model construction. BRIEF DESCRIPTION OF DRAWINGS

[0042] The drawings described herein are used to provide further understanding of the embodiments of the present application, and form a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application, and do not constitute an improper limitation on the present application. In the drawings:

[0043] Figure 1 A flowchart of a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 1.

[0044] Figure 2 A flowchart of preprocessing of a deep learning agent model constructed by using a Fourier neural operator and low-fidelity data in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 2.

[0045] Figure 3 A principle diagram of the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 3.

[0046] Figure 4 A network framework diagram for field prediction and field reconstruction by using a Fourier neural operator in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 4.

[0047] Figure 5 A principle diagram of preprocessing of a deep learning agent model constructed by using a Fourier neural operator and low-fidelity data and training of the deep learning agent model by using network parameters of the low-fidelity model as initial parameters for high-fidelity training in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 5.

[0048] Figure 6 A flowchart of fine-tuning of a high-fidelity agent model by using high-fidelity data in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in FIG. 6. DETAILED DESCRIPTION

[0049] To make the objectives, technical solutions, and advantages of the present application clearer, the technical solutions of the present application will be described below in conjunction with specific embodiments of the present application and corresponding drawings. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments of the present application, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of protection of the present application.

[0050] To make the objectives, technical solutions, and advantages of the present application clearer, the technical solutions of the present application will be described below in conjunction with specific embodiments of the present application and corresponding drawings. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments of the present application, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of protection of the present application.

[0051] Fourier Neural Operators (FNO) are used to solve Partial Differential Equations (PDEs) quickly and accurately. In the present invention, FNO is used as the original algorithm for building a multi-fidelity model, which is crucial for the pre-training of a low-fidelity model and the fine-tuning of a high-fidelity model.

[0052] The physical fields in the present invention include fluid fields and temperature fields, and the multi-fidelity physical field reconstruction method based on Fourier Neural Operator transfer learning in the present invention is aimed at the reconstruction and prediction of fluid fields and temperature fields.

[0053] The technical solutions provided by the embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Figures 1-6 , detailed description of the technical solutions provided by the embodiments of the present invention.

[0054] The present invention builds a deep learning agent model that meets the accuracy requirement based on low-fidelity data containing only a small amount of effective physical information. In the present invention, the multi-fidelity learning method is set to include three parts: data generation with different fidelities, pre-training of a low-fidelity model, and fine-tuning of a high-fidelity model. This allows the present invention to transfer the parameters of a low-fidelity model to the initial parameters of a high-fidelity model without changing the model structure. The relationship is represented as:

[0055] y L =F L (x L );

[0056]

[0057] where y L and y H represent the outputs of low-fidelity data and high-fidelity data, respectively; x L represents the input of low-fidelity data, x H represents the input of high-fidelity data, F L represents the functional relationship between the input x L of low-fidelity data and the corresponding output y L of low-fidelity data, F H represents the functional relationship between the input x H of high-fidelity data and the corresponding output y H of high-fidelity data, and Epoch refers to the training process of passing all samples in the training data set once (and only once) in deep learning. In an Epoch, the training algorithm will input all samples into the model in the specified order for forward propagation, loss calculation, backpropagation, and parameter update.

[0058] The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning of the application is different from the traditional method of using a large amount of expensive high-fidelity data to construct a high-fidelity model. Specifically, the method of the application first uses a large amount of low-fidelity sample data that is easy to obtain to train a low-fidelity model, which is called a pre-training process, and then uses the grid invariance of FNO to directly use the network parameters of the low-fidelity model as the initial parameters of high-fidelity training, so that only a small amount of high-fidelity data is needed to train a high-fidelity proxy model, which can accurately predict high-fidelity flow field data under various working conditions.

[0059] Specifically, the embodiment of the application provides a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, which can train a high-fidelity proxy model with a small amount of high-fidelity data and accurately predict high-fidelity flow field data under various working conditions. The following will be specifically explained.

[0060] Specifically, in order to achieve the above-mentioned purpose and effect, the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning of the application, as shown in Figure 1 and Figure 3 , the method comprises the following steps S1-S6:

[0061] Step S1: obtaining training data, the training data comprising low-fidelity data and high-fidelity data, wherein the high-fidelity data comprises sample data containing accurate and effective physical information, and the low-fidelity data comprises sample data not containing or partially containing accurate and effective physical information.

[0062] Step S2: constructing a deep learning proxy model.

[0063] In some optional implementation manners of the embodiment of the application, the constructed deep learning proxy model comprises an embedding module adapted to different input conditions (such as measurement points, working conditions, and layouts), as shown in Figure 4 (a); and a network architecture stacked by multiple Fourier layers.

[0064] Step S3: preprocessing the constructed deep learning proxy model by using a Fourier neural operator and low-fidelity data to obtain a low-fidelity model.

[0065] Specifically, as shown in Figure 2 , preprocessing the constructed deep learning proxy model by using a Fourier neural operator and low-fidelity data to obtain a low-fidelity model comprises the following steps S31-S34:

[0066] Step S31: inputting low-fidelity data to the deep learning proxy model;

[0067] Step S32: training the deep learning model using the Fourier neural operator and the input low-fidelity data to fit the mapping relationship between the low-fidelity data and the high-fidelity data;

[0068] Specifically, according to the specific circumstances of the selected different physical fields, the original FNO is utilized and improved to predict and reconstruct the flow field and the temperature field in the Fourier space. As shown in Figure 4 , the improved framework contains an embedding module and multiple Fourier layers. The embedding module is divided into three categories of MLP-CNN, mask and 2D image data, corresponding to the processing methods of the inputs of the three physical fields. The input module should be used as the initial state of the multiple Fourier layers. Fast Fourier transform is performed on each Fourier layer, and then nonlinear mapping training is performed on the network parameters. The superposition of multiple Fourier layers improves the accuracy of the reconstruction of the physical field.

[0069] As shown in Figure 4 , Figure 4 The network framework for field prediction and field reconstruction using the Fourier neural operator in the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning provided by an embodiment of the present application is shown in Figure 4 , where (a) is input embedding, (b) is a Fourier layer, and (c) is a network architecture taking case I as an example. Specifically, in (a) of Figure 4 , cases I-III represent the embedding modules of the laminar single-cylinder wake, the airfoil flow field and the temperature field, respectively. For FNO modeling, the input parameters of these three engineering examples are one-dimensional discrete velocity, one-dimensional working condition combination and two-dimensional heat source distribution field.

[0070] As an example, for example, MLP-CNN embedding: one-dimensional vector v contains a series of discrete velocities, which are as follows:

[0071] v={v1,v2,…,v m};

[0072] where v m m represents the mth discrete point velocity.

[0073] As another example, in the deep learning agent model proposed in the application, as shown in Figure 4 , Figure 4 , (a) in (a) represents different forms of input data; Figure 4 (b) in (b) represents the main network architecture stacked by multiple Fourier layers; Figure 4 (c) in (c) represents the specific network architecture corresponding to the first form in (a) in (a). As shown in Figure 5As shown in (c) in the figure, a Multilayer Perceptron (MLP) expands the one-dimensional velocity vector into one-dimensional unstructured data. The unstructured data is then reshaped into a 2D low-resolution array, which is equivalent to obtaining a coarse grid. Then a series of convolution layers, instance norm layers, activation functions and interpolation are embedded to generate a high-resolution field P(x) as the initial input of the Fourier layer.

[0074] As another example, for example Mask embedding: one-dimensional physical quantities including inflow velocity and airfoil angle of attack can be combined to calculate the velocity in x and y directions (i.e. ux and uy). These two velocities connected to the airfoil can be mapped to three two-dimensional mask matrices (Mu, Mv, Ms) respectively, forming a modeling input a(x) with three-way channels. These matrices correspond to discrete calculation domains (i.e. Du, Dv, Ds) with size (nx, ny). The coordinates of position (i, j) are The three mask matrices are divided into two parts, including the inside and outside of the airfoil. For the value of Ms, the inside and outside of the airfoil are filled with 1 and 0 respectively. For Mu or Mv, the values inside the airfoil are fixed 0 values, placed at positions The remaining positions are filled by ux and uy, which are expressed as follows:

[0075]

[0076] where M i,j The value of the flow field mask representing the airfoil at (i, j) point, x k represents the x-direction coordinate, y k represents the y-direction coordinate.

[0077] As yet another example, a two-dimensional heat source distribution is taken as two-dimensional image data, which can be directly taken as the initial input state of the Fourier layer by embedding a convolution layer with a kernel size of 1x1.

[0078] It should be noted that the Fourier layer is an application of the prior art, and will not be described in detail in the embodiments of the present application.

[0079] Step S33: constructing a loss function of supervised learning;

[0080] Specifically, the loss function of supervised learning includes:

[0081]

[0082] where L low represents the loss function of supervised learning, N represents the number of flow field data points, i represents the i-th flow field point, represents a low-fidelity predicted value, represents a low-fidelity data label value, and l represents a low-fidelity data.

[0083] Step S34: Optimize the Fourier neural operator parameters under the supervised paradigm using the loss function of supervised learning to obtain a low-fidelity model.

[0084] Specifically, low-fidelity models include:

[0085]

[0086] Among them, F l represents the low-fidelity model learning function, Represents the low-fidelity input physical quantity. In the above formula, i∈{1,2,…,N}, and N represents a positive integer.

[0087] The above-mentioned use of Fourier neural operators and low-fidelity data to pre-process the constructed deep learning agent model and use Fourier neural operators to train the deep learning agent model with the network parameters of the low-fidelity model as the initial parameters of high-fidelity training, and the principle diagram of the high-fidelity agent model is as follows: Figure 5 As shown, Figure 5 The workflow for FNO training on multi-fidelity data is shown in Figure 5 The workflow shown in includes two stages: pre-training and fine-tuning, where Figure 5 The left part is a schematic diagram of the preprocessing principle of the constructed deep learning agent model using Fourier neural operators and low-fidelity data. Figure 5 The middle right side shows the principle diagram of using the Fourier neural operator to train the deep learning proxy model using the network parameters of the low-fidelity model as the initial parameters for high-fidelity training to obtain a high-fidelity proxy model. Figure 3 The displayed content is as follows:

[0088] Large amounts of cheap, low-fidelity data acquired using coarse grids Used to train low-fidelity models. For FNO's low-fidelity deep learning proxy model, predictions can be obtained.

[0089]

[0090] Where F l Represents a neural operator with learnable parameters. L1 loss function is used to optimize the neural operator parameters under the supervision paradigm, and the loss function L low Described as:

[0091]

[0092] in, Represents the corresponding low-fidelity label value.

[0093] To avoid overfitting, when training the deep learning model with the Fourier neural operator and the input low-fidelity data to fit the mapping relationship between the low-fidelity data and the high-fidelity data, the low-fidelity data is divided into two parts according to a preset ratio, one part is used for training, and the other part is used for verifying the training result.

[0094] As an example, the ratio of dividing the above low-fidelity data into two parts is set to 8:1.

[0095] It can be understood that the above preset ratio is set according to actual needs, and the above-mentioned ratio is only exemplarily illustrated as an optional implementation manner, and is not the only limitation of the present application.

[0096] Therefore, the network parameters are optimized The minimum error of the verification set is relied on.

[0097] As an example, under the low-fidelity data pre-training, the neural operator learns the mapping from pl to Ul, which is not accurate but close to the target mapping of the high-fidelity data. Since the trained can directly provide high-resolution predictions of zero high-fidelity samples, it is determined to use a small amount of high-fidelity data as the initial network to improve the accuracy of the multi-fidelity model.

[0098] As Figure 3 shown, Figure 5 the left column d in the middle represents the low-fidelity input and output data; the last side e represents the high-fidelity input and output data; the middle a-c modules, from bottom to top, are data preparation, model training, and model generation; if the a-c modules are regarded as left and right parts, the left is a low-fidelity model obtained by pre-training, and the right is a high-fidelity model obtained by fine-tuning. Combined with Figure 5 , Figure 3 the middle Figure 5 a-c modules are further explained and described in detail, and in Figure 5 from bottom to top, it includes data preparation, model training, and model training completion. If Figure 6 is divided into left and right parts for explanation and description, the left side represents the training and construction principle of the low-fidelity model, and the right side represents the fine-tuning principle of the high-fidelity model training.

[0099] Step S4: training the deep learning agent model with the Fourier neural operator using the network parameters of the low-fidelity model as the initial parameters of the high-fidelity training to obtain a high-fidelity agent model;

[0100] Step S5: fine-tuning the high-fidelity agent model with high-fidelity data;

[0101] Specifically, as ​As shown, fine-tuning the high-fidelity proxy model with high-fidelity data includes the following steps S51-S53:

[0102] Step S51: The network parameters of the low-fidelity model are used as the initial parameters for high-fidelity training, and the predicted values are calculated;

[0103] Step S52: The initial parameters are trained using a small amount of high-fidelity data, and the error between the predicted values and the label values is minimized using the initial parameters;

[0104] Specifically, the error between the predicted values and the label values is determined by the following formula:

[0105]

[0106] wherein, represents the predicted value, F h represents the initial parameters of the high-fidelity model, represents the high-fidelity input parameters, i represents the i-th flow field data point, M represents a positive integer, and h represents high-fidelity data.

[0107] Step S53: The network parameters of the initial high-fidelity proxy model are determined by the minimum error of the high-fidelity data validation set, and the high-fidelity proxy model is obtained.

[0108] Specifically, the network parameters of the initial high-fidelity proxy model are determined by the minimum error of the high-fidelity data validation set, which is determined by the following formula:

[0109]

[0110] wherein, L high represents high-fidelity data, M represents the number of flow field data points, and M>>N, represents the high-fidelity data label value.

[0111] Specifically, the high-fidelity proxy model includes:

[0112]

[0113] wherein, M represents the final high-fidelity proxy model, represents the network parameters of the final high-fidelity proxy model.

[0114] As an example, a small amount of high-fidelity data wherein M<<N, for adjusting the high-fidelity proxy model includes:

[0115] First, the pre-trained model parameters are set as the initial parameters F h of the high-fidelity model.

[0116] Then, the predicted value is calculated The expression is:

[0117]

[0118] Likewise, F can be trained h To minimize the error between the predicted value And the label value predicted value Between the error:

[0119]

[0120] The final network parameters Can be determined by the minimum error (the ratio is 1 / 9) of the high-fidelity data validation set. After training on limited high-fidelity data, a multi-fidelity model M can be obtained, where

[0121] Step S6: using the fine-tuned high-fidelity proxy model to predict the physical field, and obtaining the predicted result corresponding to the physical field.

[0122] In the process of using the Fourier neural operator and low-fidelity data to preprocess the deep learning proxy model constructed and using high-fidelity data to fine-tune the high-fidelity proxy model, different initial learning rates are set respectively, and the initial learning rate of using the Fourier neural operator and low-fidelity data to preprocess the deep learning proxy model constructed is set to be greater than the initial learning rate of using high-fidelity data to fine-tune the high-fidelity proxy model.

[0123] As an example, the initial learning rates of pre-training and fine-tuning are lr1 and lr2 respectively. Generally, the relationship of lr1>lr2 is fixed, which can avoid that the higher the value of lr2 is, the greater the change of the parameters of the deep learning proxy model is, and the features learned from the low-fidelity data are destroyed.

[0124] Therefore, the multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning of the application adopts a deep learning method, uses the meshless features of the Fourier neural operator, adopts transfer learning, and uses a large amount of low-fidelity data and a small amount (zero) of high-fidelity data to construct a physical field reconstruction and prediction model, thereby significantly reducing the amount of high-fidelity data required to construct a deep learning proxy model, and simplifying the structure of a multi-fidelity model.

[0125] The prior art deep learning agent construction method often needs a large amount of high-fidelity data containing accurate and effective physical information to train the model, however, obtaining a large amount of high-fidelity data will greatly increase the modeling cost, and since most of the existing agent models based on a large amount of low-fidelity data and a small amount of high-fidelity data introduce an 'additional layer' or 'bridge function' between the low-fidelity model and the high-fidelity model, or usually contain three or more fidelity models, there are cumbersome and time-consuming problems, and for the added 'bridge function' of the deep model such as the convolutional neural network, the low-fidelity and high-fidelity data used for modeling are generated by the coarse grid and the fine grid on the basis of simulation calculation, resulting in different network weights of the two-dimensional data, making the neural network structure complex, and the training process depends on a large amount of high-fidelity training samples, the present application builds a multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, which is based on the grid independence of the Fourier neural operator, and migrates the low-fidelity model pre-trained by a large amount of low-fidelity data to fine-tune the model by a small amount or zero high-fidelity data, to realize multi-fidelity model training.

[0126] In summary, the method of the present application solves the problem of the demand for a large amount of high-fidelity data for general deep learning models, reduces the cost of constructing deep learning models, and simplifies the multi-fidelity network structure by using the Fourier neural operator. In the process of solving the above technical problems, the present application first processes the input data according to different physical field cases to obtain input data with characteristic format, then pre-trains a low-fidelity model using a large amount of low-fidelity data, and fine-tunes the low-fidelity model using a small amount or zero high-fidelity data, and finally obtains a multi-fidelity model that meets the accuracy requirements.

[0127] It should be noted that in this paper, relational terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between the entities or operations. Moreover, the term "include", "contain" or any other variant thereof is intended to cover non-exclusive inclusion, so that the process, method, article or equipment including a series of elements not only includes those elements, but also includes other elements not explicitly listed or inherent to such process, method, article or equipment. In addition, "front", "back", "left", "right", "up", "down" in this paper are with reference to the placement state shown in the drawings.

[0128] Finally, it should be noted that the above examples are only used to illustrate the technical solutions of the present application, and are not intended to limit the present application; although the present application has been described in detail with reference to the foregoing examples, those skilled in the art should understand that the technical solutions recorded in the foregoing examples can be modified, or some technical features can be replaced by equivalents; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present application.

Claims

1. A multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning, characterized in that: include: Acquire training data, where the training data includes low-fidelity data and high-fidelity data, wherein the high-fidelity data includes sample data that completely contains accurate and valid physical information, and the low-fidelity data includes sample data that does not contain or partially contains accurate and valid physical information; Build deep learning agent models; The constructed deep learning agent model is preprocessed using Fourier neural operators and low-fidelity data to obtain a low-fidelity model; The network parameters of the low-fidelity model are used as the initial parameters for high-fidelity training using the Fourier neural operator to train the deep learning proxy model, thus obtaining a high-fidelity proxy model. Fine-tune high-fidelity proxy models using high-fidelity data; The fine-tuned high-fidelity proxy model is used to predict the physical field and obtain the corresponding prediction results of the physical field.

2. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 1 is characterized in that: The constructed deep learning agent model is preprocessed using Fourier neural operators and low-fidelity data to obtain a low-fidelity model including: Input low-fidelity data to the deep learning agent model; Using Fourier neural operators and the input low-fidelity data to train the deep learning model to fit the mapping relationship between low-fidelity data and high-fidelity data; Constructing loss functions for supervised learning; The loss function of supervised learning is used to optimize the parameters of the Fourier neural operator under the supervised paradigm to obtain a low-fidelity model.

3. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 2 is characterized in that: Loss functions for supervised learning include: Among them, L low Represents the loss function of supervised learning, N represents the number of flow field data points, i represents the i-th flow field point, represents the low-fidelity prediction value, Represents the low-fidelity data label value, and l represents low-fidelity data.

4. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 2 is characterized in that: Low-fidelity mockups include: Among them, F l represents the low-fidelity model learning function, Represents the low-fidelity input physical quantity. In the above formula, N represents a positive integer.

5. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 2 is characterized in that: When using Fourier neural operators and input low-fidelity data to train a deep learning model to fit the mapping relationship between low-fidelity data and high-assurance data, the low-fidelity data is divided into two parts according to a preset ratio, one part is used for training and the other part is used to verify the training results.

6. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 1 is characterized in that: Fine-tuning a high-fidelity proxy model using high-fidelity data involves: Use the network parameters of the low-fidelity model as the initial parameters for high-fidelity training and calculate the predicted value; Train the initial parameters and use the initial parameters to minimize the error between the predicted value and the label value error prediction value; The network parameters of the initial high-fidelity proxy model are determined by minimizing the error of the high-fidelity data validation set to obtain a high-fidelity proxy model.

7. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 6 is characterized in that: The error between the minimized predicted value and the label value error predicted value is calculated and determined by the following formula; in, represents the predicted value, F h represents the initial parameters of the high-fidelity model, Represents the high-fidelity input physical quantity, i represents the i-th flow field point, M represents a positive integer, and h represents high-fidelity data.

8. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 6 is characterized in that: The network parameters of the initial high-fidelity proxy model are determined by the minimum error of the high-fidelity data validation set and calculated using the following formula: Among them, L high represents high-fidelity data, M represents the number of flow field data points, and M>>N, U i h Represents high-fidelity data label values.

9. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 6, characterized in that: High-fidelity proxy models include: Among them, M represents the final high-fidelity proxy model, Represents the network parameters of the final high-fidelity proxy model.

10. The multi-fidelity physical field reconstruction method based on Fourier neural operator transfer learning according to claim 1, characterized in that: Different initial learning rates are set in the processes of preprocessing the constructed deep learning proxy model using Fourier neural operators and low-fidelity data and fine-tuning the high-fidelity proxy model using high-fidelity data, and the initial learning rate for preprocessing the constructed deep learning proxy model using Fourier neural operators and low-fidelity data is set to be greater than the initial learning rate when fine-tuning the high-fidelity proxy model using high-fidelity data.

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