High-precision multi-physics field reconstruction method based on nested neural network architecture

Abstract

The invention discloses a high-precision multi-physics field reconstruction method based on a nested neural network architecture. The method comprises the following steps: S1, acquiring open-source multi-physics field data; s2, a multi-scale Fourier operator-driven U-Net physical field reconstruction model is constructed, the physical field reconstruction model comprises an embedding module, a coding module and a decoding module, the coding module comprises at least one coding layer, each coding layer is formed by sequentially connecting a multi-scale Fourier neural operator layer and a down-sampling layer, and the multi-scale Fourier neural operator layer and the down-sampling layer are embedded into the embedding module; the decoding module comprises at least one decoding layer, and each decoding layer is formed by sequentially connecting a multi-scale Fourier neural operator layer and an up-sampling layer; s3, training the physical field reconstruction model based on a training set; and S4, performing quantitative analysis and qualitative analysis on the reconstructed physical field by using the test set.

Description

Technical Field

[0001] This invention relates to the field of neural network technology, and specifically to a high-precision multiphysics field reconstruction method based on a nested neural network architecture. Background Technology

[0002] Physical field reconstruction is a core supporting technology in many fields such as engineering simulation, weather forecasting, fluid dynamics simulation, and industrial process monitoring. Its core objective is to accurately recover the spatial distribution and evolution of the global physical field using limited sensor observation data. The reconstruction accuracy directly determines the reliability of subsequent analysis and decision-making, system optimization, and risk management.

[0003] Traditional physics field reconstruction methods mainly rely on numerical solutions to partial differential equations or classical interpolation algorithms, but these methods have significant limitations. While numerical solutions can guarantee a certain degree of physical consistency, they are computationally extremely complex and time-consuming, especially when dealing with complex physical scenarios of high dimensions and multiple scales (such as turbulent fluids and non-uniform medium temperature fields), making it difficult to meet the requirements of real-time applications. Interpolation algorithms (such as linear interpolation and radial basis function interpolation) are limited by the distribution characteristics of sparse observation data, and can only infer values ​​in unknown regions through local neighborhood information. This easily leads to the loss of global correlation information and local fine features of the physical field, resulting in smooth over-smoothing, low accuracy, and poor robustness. They are also unsuitable for physical scenarios with strong nonlinear and non-local correlation characteristics, such as fluid velocity fields and temperature fields.

[0004] With the development of deep learning technology, physical field reconstruction methods based on neural networks have gradually become a research hotspot. Existing neural network models mostly employ traditional convolutional neural networks (CNNs) or ordinary U-Net architectures. These models rely on local receptive fields to capture features, making it difficult to effectively model long-distance, cross-scale dependencies in the physical field. This results in significant deficiencies in global consistency and detail reproduction of the reconstruction results. Furthermore, existing models have limited processing methods for sparse observation data, lacking targeted feature embedding mechanisms and failing to fully preserve the spatial location information and physical properties of the original data, further restricting the improvement of reconstruction accuracy.

[0005] Furthermore, existing methods do not fully consider the multi-scale feature extraction and fusion of physical fields during model training, resulting in an uneven ability of the model to capture physical information at different scales. This makes it difficult to simultaneously take into account the overall distribution trend and local detail fluctuations of the physical field. Although some models introduce Fourier transform-related techniques, they have not formed a systematic multi-scale Fourier operator-driven architecture, which cannot effectively achieve efficient conversion and interaction of physical field features between the spatial and frequency domains. This makes it difficult for the model's generalization ability and robustness to meet the diverse needs of multi-physics scenarios (such as fluid velocity fields and temperature fields) in practical applications.

[0006] Existing technologies generally suffer from three major problems: First, they fail to capture global context dependencies effectively, limited by the model's receptive field, making it impossible to effectively model cross-regional relationships in the physical field. Second, they have low accuracy in reconstructing local details, weak ability to structurally represent sparse data, and are prone to losing key details such as edges and gradients. Third, they have weak noise robustness and generalization ability, resulting in significant errors when faced with noisy observation data, and are difficult to adapt to different types of physical field scenarios, which seriously restricts the promotion and application of physical field reconstruction technology in practical engineering. Summary of the Invention

[0007] The purpose of this invention is to provide a physical field reconstruction method that can efficiently process sparse observation data, fully capture the multi-scale features and global correlation information of the physical field, and balance reconstruction accuracy and computational efficiency to solve the above problems.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: A high-precision multiphysics reconstruction method based on a nested neural network architecture includes the following steps: S1. Obtain open-source multiphysics data; S2. Construct a physical field reconstruction model of U-Net driven by multi-scale Fourier operators. The physical field reconstruction model includes an embedding module, an encoding module, and a decoding module. The encoding module contains at least one encoding layer. Each encoding layer is composed of a multi-scale Fourier neural operator layer and a downsampling layer connected in sequence. The decoding module contains at least one decoding layer. Each decoding layer is composed of a multi-scale Fourier neural operator layer and an upsampling layer connected in sequence. S3. The physical field reconstruction model is trained based on the training set. The embedding module adopts a preset embedding method to convert the original sparse observation data in the training set into a structured feature representation. The structured feature representation is input into the encoding module, and multi-scale physical field features are extracted through the multi-scale Fourier neural operator layer of each encoding layer. Then, the multi-scale physical field features are compressed in spatial dimension and expanded in feature channel dimension through the downsampling layer to output the encoded features. The encoded features are input into the decoding module. The corresponding scale features from the encoding module and the output features of the previous decoding layer are fused through the multi-scale Fourier neural operator layer of each decoding layer. Then, the physical space resolution is gradually restored through the upsampling layer to achieve the reconstruction of the complete physical field. S4. Quantitative and qualitative analysis of the reconstructed physical field is performed using the test set. The quantitative analysis is achieved by calculating the average absolute error between the reconstructed predicted physical field and the real physical field. The qualitative analysis is achieved by randomly sampling data examples from the test set and inputting them into the trained model, outputting the reconstructed physical field and displaying it visually, and comparing the characteristic consistency and relative error between the reconstructed physical field and the real physical field.

[0009] Preferably, the open-source multiphysics dataset is divided into fluid velocity field datasets and temperature field datasets according to the physical scene type. Each physical scene dataset is further divided into training set, validation set, and test set according to a preset ratio of 80%, 10%, and 10%. Each physical field dataset contains a total of m data samples, and the total number of samples in all datasets is N. Each sample is represented as ( i y i ), where 1≤i≤N, i For sparse observation data and as the initial input to the model, y i The actual distribution of the physical field is used as the training label.

[0010] Preferably, in step S3, the embedding module transforms the original sparse observation data into a structured data representation using any one or more preset embedding methods, such as mask embedding, multilayer perceptron embedding, or Voronoi embedding, and uses this representation as the initial state of the Fourier space. The general expression for the embedding method is: ,in, This is the original sparse observation data. For structured data representation, For embedding mapping functions, The total number of data observations. For the first The spatial coordinates of each observation point are used to locate the position of the original sparse observation data in the physical field.

[0011] Preferably, the mask embedding places the observed values ​​of the physical field at corresponding positions on the grid to preserve spatial relationships and information. This is achieved by concatenating a mask matrix and a coordinate matrix, with the remaining positions filled with 0. The mask embedding formula is: ,in, For data coordinates, The coordinates are two-dimensional coordinates of the mask grid, which is a pre-defined uniformly distributed grid used to carry the spatial location information of the physical field observations; The multilayer perceptron embedding maps the input to high-dimensional hidden layer features through an MLP containing the GLUE activation function, then reshapes the hidden layer features into a single-channel feature map, and then uses 1 1. Convolution generates embedded features, and the embedding formula of the multilayer perceptron is: , ,in, , , , , , For learnable parameters, This is a high-dimensional hidden layer feature vector output by a multilayer perceptron, with dimensions determined by learnable parameters. , Decide, This is an activation function used to enhance the model's nonlinear fitting ability. Used to reshape the feature map; The Voronoi embedding divides the computational region using Voronoi polygons, ensuring that physics observations remain consistent within their corresponding spatial locations. For other spatial locations, the physics values ​​in the unknown region are filled using nearest-neighbor interpolation. The formula for the Voronoi embedding is: ,in, Used to calculate the Euclidean distance between two coordinates. To remove The index of any other observation point besides the observation point is used to determine the physical field value of the unknown region by means of the nearest neighbor rule.

[0012] Preferably, the encoder representation learning expression in the encoding module in step S2 is: , ,in, 1 represents the total number of coding layers in the coding module, used to limit the number of iterations for each coding layer. The total number of layers of the Fourier operator. , For learnable parameters, This is a loop function, indicating that the output features of the previous layer's encoding are input sequentially. Each sublayer uses a Fourier operator, and the output is used as the encoded feature of the current layer. The input features are encoded in the next layer. Encode the output features for the current layer. This is the final output feature of the coding layer after downsampling, used to pass to the next coding layer or decoding module.

[0013] Preferably, the decoder reconstruction learning expression in the decoding module in step S2 is: , ,in, 2 represents the total number of decoding layers in the decoding module, used to limit the number of iterations for each decoding layer. This is a feature concatenation function that concatenates the corresponding scale features output by the encoding module with the features output by the previous decoding layer along the feature channel dimension. , For learnable parameters, The corresponding scale features output by the encoding module. The reconstructed features are the output of the current layer's decoding. These are the reconstructed features after upsampling of the decoding layer, used to gradually restore the spatial resolution of the physical field.

[0014] Preferably, the calculation process of the multi-scale Fourier neural operator layer is as follows: Taking the hidden layer feature state of the physical field as input, we first perform a multi-scale discrete Fourier transform on the physical field features to convert them to Fourier space. The formula for this process is as follows: , ,in, The imaginary unit satisfies 2 =−1, used for complex number operations in the Discrete Fourier Transform. This represents the hidden layer characteristic state of the physical field. , For the Fourier pattern number, The number of Fourier operator layers. and These are the height and width of the feature map, respectively. This represents a multi-scale feature interpolation merging operator, which will... Features at different scales output by each Fourier operator sublayer are interpolated to the same dimension and then merged. For the first In each Fourier operator sublayer, the feature map is in ( ) Eigenvalues ​​of the location; In Fourier space, the kernel integral operator is executed through a randomly initialized learnable parameter matrix to achieve nonlocal information interaction and global regularization. The calculation process is as follows: , ,in, For learnable parameter matrix, This is a uniform distribution function used to initialize the learnable parameter matrix. Ensure that the initial value of the parameter is within Random distribution within the interval and These are the feature input and output dimensions, respectively. The spatial dimension of the Fourier kernel, and the number of Fourier modes. , Matching For vector product, For the first In the sublayer of the Fourier operator, the first... The first batch of samples Feature values ​​of each input feature dimension For learnable parameter matrix The Middle Line number The elements of the column are used to implement the kernel integration operation in Fourier space; Performing a multi-scale inverse Fourier transform converts the feature space from Fourier space to Euclidean space. Then, by averaging the features at each Fourier scale and applying activation functions, the final hidden layer features are obtained. The formula for this process is as follows: ,in, After the inverse Fourier transform, in physical space ( The final hidden layer feature value at position ) For the first In a sublayer of Fourier operators, the Fourier space ( ) Eigenvalues ​​of the location; The final hidden layer features obtained in the first step are compared with the original input features by step 1. The results of the linear transformations of the first convolutional layer are summed to obtain the input features for the next layer. The formula for this process is as follows: ,in, As the input features for the next layer, 1 1. Learnable parameters of a convolutional layer for The original input features of the layer, The spatial location coordinates of the feature For the first The output features of the layer after processing by the multi-scale Fourier neural operator have the same dimension as the original input features.

[0015] Preferably, in step S3, the model training uses average loss as the loss function to calculate the average absolute error between the predicted value and the true value, as shown in the formula: ,in, The average loss function is... The total number of data observations. For the model to the first The predicted physical field values ​​for each observation point For the first The true physical field values ​​at each observation point correspond to the labeled data in the training set.

[0016] Preferably, the hyperparameters for model training are set as follows: batch size of 8, number of training epochs of 500, initial learning rate of 0.001, Adam optimizer used as the optimizer, ExponentialLR exponential learning rate scheduler used as the scheduler, and learning rate decay multiplication factor of 0.98.

[0017] Preferably, the number of encoding layers in the encoding module and the number of decoding layers in the decoding module are both 3, and each multi-scale Fourier neural operator layer contains 3 Fourier operator sub-layers.

[0018] By adopting the above technical solution, the present invention has the following advantages compared with the prior art: 1. This invention provides a high-precision multiphysics field reconstruction method based on a nested neural network architecture. It deeply integrates the U-Net architecture with a multi-scale Fourier neural operator, forming a complementary advantage. The encoding and decoding structure of U-Net, combined with skip connections, can accurately capture local detail features and edge information of the physical field, achieving refined reconstruction. The multi-scale Fourier neural operator, through frequency domain global linear transformation, efficiently models the long-distance dependencies and global distribution patterns of the physical field, effectively suppressing high-frequency noise. The synergistic effect of these two methods enables the model to both restore the overall structural consistency of the physical field and accurately reproduce local detail fluctuations, significantly reducing reconstruction errors. Compared to traditional models, the mean absolute error is significantly reduced.

[0019] 2. This invention provides a high-precision multiphysics reconstruction method based on a nested neural network architecture. Employing an end-to-end model structure, it enables the model to learn the common laws and differentiated features of different physical fields through joint training and feature transfer on multiphysics scene datasets such as fluid velocity fields and temperature fields. The efficient processing of continuous function spaces by multi-scale Fourier neural operators makes the reconstruction results more robust to the input data distribution. High-quality reconstruction across physical scenes can be achieved without redesigning the model structure for specific physical fields, significantly expanding the application scope of the method and reducing the development cost of multi-scene applications.

[0020] 3. This invention provides a high-precision multiphysics field reconstruction method based on a nested neural network architecture. The model architecture has a clear hierarchy, with the encoding module responsible for feature extraction and compression, and the decoding module responsible for feature fusion and reconstruction. Each module has a clearly defined function, facilitating subsequent optimization and improvement. Furthermore, the model supports flexible adjustment of hyperparameters such as the number of encoding layers, decoding layers, and Fourier operator layers, allowing for targeted adaptation to the accuracy requirements and hardware resources of different application scenarios. It possesses strong scalability and can be further extended to reconstruction tasks of other types of physical fields such as pressure fields and electromagnetic fields.

[0021] 4. This invention provides a high-precision multiphysics field reconstruction method based on a nested neural network architecture. It sets up three differentiated feature embedding methods: mask embedding, multilayer perceptron embedding, and Voronoi embedding. These methods can be flexibly selected according to the distribution characteristics and smoothness of sparse observation data, achieving efficient structured representation of sparse data. Voronoi embedding is naturally robust to missing or shifted measurement points, while multilayer perceptron embedding effectively handles sparse and non-smooth data scenarios. Combined with the frequency domain low-pass filtering characteristics of Fourier neural operators, the model maintains stable reconstruction performance even with noisy sparse observation data input, significantly improving its adaptability to complex data scenarios.

[0022] 5. This invention provides a high-precision multiphysics reconstruction method based on a nested neural network architecture. The model adopts a course-based learning approach to guide the training process, using average L1 loss as the loss function to ensure rapid convergence and stability at the optimal state. Simultaneously, the model training process does not require complex manual feature engineering, supports direct output of complete physical fields from sparse observation data, and the trained model has fast inference speed, meeting the real-time requirements of engineering applications. Through a verification method combining quantitative and qualitative analysis, the reconstruction effect can be intuitively evaluated, providing reliable results support for engineering applications. Attached Figure Description

[0023] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a diagram of the model architecture of the present invention; Figure 3 This is a diagram of the U-Net model of the present invention; Figure 4 This is a diagram of the Fourier neural operator model of the present invention; Figure 5 This invention provides a visualization of the physical field reconstruction results. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0025] Example Please refer to Figures 1 to 5 As shown, this invention discloses a high-precision multiphysics field reconstruction method based on a nested neural network architecture. This embodiment takes the reconstruction of fluid velocity and temperature fields as the target scenario and verifies the effectiveness of the method based on an open-source multiphysics field dataset. The hardware environment used in this embodiment is: CPU Intel Core i9-13900K, GPU NVIDIA RTX 4090 (24GB VRAM), 64GB RAM; the software environment is: Python 3.9, PyTorch 2.0, CUDA 11.8, NumPy 1.24, Matplotlib 3.7 (for visualization analysis). The specific steps include: S1. Obtain open-source multiphysics data. The open-source multiphysics data is divided into fluid velocity field dataset and temperature field dataset according to the physical scene type. The fluid velocity field dataset comes from the cylinder flow simulation results of the CFD numerical simulation open-source library OpenFOAM. The temperature field dataset comes from the two-dimensional unsteady temperature distribution data of the Heat Equation open-source dataset.

[0026] Fluid velocity field dataset: Each sample contains a velocity component in a two-dimensional plane (u x u y The spatial resolution is 112×192. The sparse observation data is generated through uniform sampling, and each sample retains the fluid velocity values ​​of 16 observation points.

[0027] Temperature field dataset: Each sample is a temperature value T in a two-dimensional plane with a spatial resolution of 200×200. Sparse observation data is generated by uniform sampling, and each sample retains the temperature values ​​of 36 observation points.

[0028] Based on a ratio of 80%, 10%, and 10%, the fluid velocity field dataset and temperature field dataset in the physical scene dataset are divided into training set, validation set, and test set, respectively. Each physical field dataset contains a total of m data samples, and the total number of samples in all datasets is N. Each sample is represented as ( i y i ), where 1≤i≤N, i Sparse observation data (including the coordinates of observation points and their corresponding physical field values) is used as the initial input to the model, y i The actual distribution of the physical field is used as the training label.

[0029] S2. Construct a physical field reconstruction model of U-Net driven by multi-scale Fourier operators. The physical field reconstruction model includes an embedding module, an encoding module, and a decoding module. The encoding module contains at least one encoding layer. Each encoding layer is composed of a multi-scale Fourier neural operator layer and a downsampling layer connected in sequence. The decoding module contains at least one decoding layer. Each decoding layer is composed of a multi-scale Fourier neural operator layer and an upsampling layer connected in sequence. S3. The physical field reconstruction model is trained based on the training set. The embedding module adopts a preset embedding method to convert the original sparse observation data in the training set into a structured feature representation. The structured feature representation is input into the encoding module, and multi-scale physical field features are extracted through the multi-scale Fourier neural operator layer of each encoding layer. Then, the multi-scale physical field features are compressed in spatial dimension and expanded in feature channel dimension through the downsampling layer to output the encoded features. The encoded features are input into the decoding module. The corresponding scale features from the encoding module and the output features of the previous decoding layer are fused through the multi-scale Fourier neural operator layer of each decoding layer. Then, the physical space resolution is gradually restored through the upsampling layer to achieve the reconstruction of the complete physical field. Preferably, in step S3, the embedding module transforms the original sparse observation data into a structured data representation using any one or more preset embedding methods, such as mask embedding, multilayer perceptron embedding, or Voronoi embedding, and uses this representation as the initial state of the Fourier space. The general expression for the embedding method is: ,in, This is the original sparse observation data. For structured data representation, For embedding mapping functions, The total number of data observations. For the first The spatial coordinates of each observation point are used to locate the position of the original sparse observation data in the physical field.

[0030] Preferably, the mask embedding places the observed values ​​of the physical field at corresponding positions on the grid to preserve spatial relationships and information. This is achieved by concatenating a mask matrix and a coordinate matrix, with the remaining positions filled with 0. The mask embedding formula is: ,in, For data coordinates, The coordinates are two-dimensional coordinates of the mask grid, which is a pre-defined uniformly distributed grid used to carry the spatial location information of the physical field observations; The multilayer perceptron embedding maps the input to high-dimensional hidden layer features through an MLP containing the GLUE activation function, then reshapes the hidden layer features into a single-channel feature map, and then uses 1 1. Convolution generates embedded features, and the embedding formula of the multilayer perceptron is: , ,in, , , , , , For learnable parameters, This is a high-dimensional hidden layer feature vector output by a multilayer perceptron, with dimensions determined by learnable parameters. , Decide, This is an activation function used to enhance the model's nonlinear fitting ability. Used to reshape the feature map; In this embodiment, the MLP contains three fully connected layers. The input dimension is the number of effective observation points (16 sampling points for the fluid velocity field, i.e., the input dimension is 16; 36 sampling observation points for the temperature field, i.e., the input dimension is 36), the output dimension is 128, the second hidden layer has a dimension of 256, and the third output layer has a dimension of 64. A convolutional layer has 1 input channel and 64 output channels, and can learn parameters. , Generated through random initialization.

[0031] The Voronoi embedding divides the computational region using Voronoi polygons, ensuring that physics observations remain consistent within their corresponding spatial locations. For other spatial locations, the physics values ​​in the unknown region are filled using nearest-neighbor interpolation. The formula for the Voronoi embedding is: ,in, Used to calculate the Euclidean distance between two coordinates. To remove The index of any other observation point besides the observation point is used to determine the physical field value of the unknown region by means of the nearest neighbor rule.

[0032] In this embodiment, the Voronoi embedding is performed by dividing the physical space using the Voronoi function in the Scipy library. Each observation point corresponds to a Voronoi polygon, and the physical field values ​​at unobserved locations within the polygon are filled by nearest neighbor interpolation.

[0033] The encoder representation learning expression in the encoding module described in step S2 is as follows: , ,in, 1 represents the total number of coding layers in the coding module, used to limit the number of iterations for each coding layer. The total number of layers of the Fourier operator. , For learnable parameters, This is a loop function, indicating that the output features of the previous layer's encoding are input sequentially. Each sublayer uses a Fourier operator, and the output is used as the encoded feature of the current layer. The input features are encoded in the next layer. Encode the output features for the current layer. This is the final output feature of the coding layer after downsampling, used to pass to the next coding layer or decoding module.

[0034] The total number of encoding layers in the encoding module in this embodiment = 3. Each coding layer consists of a multi-scale Fourier neural operator layer (FNO_layer) and a downsampling layer connected sequentially, as shown in the following structure: First encoding layer: 64 input feature channels, FNO_layer contains 3 Fourier operator sublayers (M=3), number of Fourier modes =32、 =32; the downsampling layer uses 2×2 max pooling, with 128 output feature channels and spatial resolutions of 100×100 (temperature field) and 56×96 (velocity field).

[0035] Second coding layer: 128 input feature channels, FNO_layer structure is the same as the first coding layer; downsampling layer uses 2 2. Max pooling, output feature channels 256, spatial resolution 50×50 (temperature field) and 28 48 (Velocity field).

[0036] Third coding layer: 256 input feature channels, FNO_layer structure is the same as the first coding layer; downsampling layer uses 2 2. Max pooling, 512 output feature channels, 25×25 spatial resolution (temperature field) and 14 24 (Velocity field).

[0037] loop function Implemented through loop iteration in PyTorch, parameters can be learned. , Initialize using the Xavier normal distribution.

[0038] The decoder reconstruction learning expression in the decoding module described in step S2 is: , ,in, 2 represents the total number of decoding layers in the decoding module, used to limit the number of iterations for each decoding layer. This is a feature concatenation function that concatenates the corresponding scale features output by the encoding module with the features output by the previous decoding layer along the feature channel dimension. , For learnable parameters, The corresponding scale features output by the encoding module. The reconstructed features are the output of the current layer's decoding. These are the reconstructed features after upsampling of the decoding layer, used to gradually restore the spatial resolution of the physical field.

[0039] The total decoding layer of the decoding module in this embodiment 2=3, each decoding layer consists of a multi-scale Fourier neural operator layer (FNO_layer) and an upsampling layer connected sequentially, as shown in the following structure: Decoding layer 1: Input feature channels 512, FNO_layer contains 3 Fourier operator sublayers (M=3), Fourier modes =32、 =32; the upsampling layer uses transposed convolution (2 2, step size 2), output feature channels 256, spatial resolution 50×50 (temperature field) and 28×48 (velocity field); through feature splicing function The output features of the second coding layer (256 channels) are fused and then compressed to 256 channels by a 1×1 convolution.

[0040] Second decoding layer: 256 input feature channels, FNO_layer structure is the same as the first decoding layer; the upsampling layer uses transposed convolution (2^2 / 2^3). 2, step size 2), output feature channel count 128, spatial resolution 100 100 (temperature field) and 56×96 (velocity field); the output features of the first coding layer (128 channels) are fused, and then processed by 1 1. Convolution is compressed to 128.

[0041] The third decoding layer has 128 input feature channels and the FNO_layer structure is the same as the first decoding layer; the upsampling layer uses transposed convolution (2^32)^42. 2, step size 2), output feature channel count 64, spatial resolution 200 200 (temperature field) and 112×192 (velocity field); the output features of the fused embedded module (64 channels) are then processed by 1... 1. Convolution is compressed to 64.

[0042] Learnable parameters in this embodiment , Initialize using a Xavier normal distribution.

[0043] It also includes an output layer, which uses a 1×1 convolutional layer to map the 64-channel features output by the decoding module to the target 1-channel physical field features, and outputs the final reconstructed physical field.

[0044] The calculation process of the multi-scale Fourier neural operator layer is as follows: Taking the hidden layer feature state of the physical field as input, we first perform a multi-scale discrete Fourier transform on the physical field features to convert them to Fourier space. The formula for this process is as follows: , ,in, The imaginary unit satisfies 2 =−1, used for complex number operations in the Discrete Fourier Transform. This represents the hidden layer characteristic state of the physical field. , For the Fourier pattern number, The number of Fourier operator layers. and These are the height and width of the feature map, respectively. This represents a multi-scale feature interpolation merging operator, which will... Features at different scales output by each Fourier operator sublayer are interpolated to the same dimension and then merged. For the first In each Fourier operator sublayer, the feature map is in ( ) Eigenvalues ​​of the location; In Fourier space, the kernel integral operator is executed through a randomly initialized learnable parameter matrix to achieve nonlocal information interaction and global regularization. The calculation process is as follows: , ,in, For learnable parameter matrix, This is a uniform distribution function used to initialize the learnable parameter matrix. Ensure that the initial value of the parameter is within Random distribution within the interval and These are the feature input and output dimensions, respectively. The spatial dimension of the Fourier kernel, and the number of Fourier modes. , Matching For vector product, For the first In the sublayer of the Fourier operator, the first... The first batch of samples Feature values ​​of each input feature dimension For learnable parameter matrix The Middle Line number The elements of the column are used to implement the kernel integration operation in Fourier space; Performing a multi-scale inverse Fourier transform converts the feature space from Fourier space to Euclidean space. Then, by averaging the features at each Fourier scale and applying activation functions, the final hidden layer features are obtained. The formula for this process is as follows: ,in, After the inverse Fourier transform, in physical space ( The final hidden layer feature value at position ) For the first In a sublayer of Fourier operators, the Fourier space ( ) Eigenvalues ​​of the location; The final hidden layer features obtained in the first step are compared with the original input features by step 1. The results of the linear transformations of the first convolutional layer are summed to obtain the input features for the next layer. The formula for this process is as follows: ,in, As the input features for the next layer, 1 1. Learnable parameters of a convolutional layer for The original input features of the layer, The spatial location coordinates of the feature For the first The output features of the layer after processing by the multi-scale Fourier neural operator have the same dimension as the original input features.

[0045] Preferably, in step S3, the model training uses average loss as the loss function to calculate the average absolute error between the predicted value and the true value, as shown in the formula: ,in, The average loss function is... The total number of data observations. For the model to the first The predicted physical field values ​​for each observation point For the first The true physical field values ​​at each observation point correspond to the labeled data in the training set.

[0046] The hyperparameters for model training were set as follows: batch size of 8, number of training epochs of 500, initial learning rate of 0.001, and the Adam optimizer was used. The scheduler uses the ExponentialLR exponential learning rate scheduler, and the multiplication factor for learning rate decay is 0.98 (learning rate after each training round = current learning rate × 0.98).

[0047] The hyperparameters for model training in this embodiment are set as follows: batch size of 8, number of training epochs of 500, initial learning rate of 0.001, and the Adam optimizer is used. The scheduler uses the ExponentialLR exponential learning rate scheduler, and the multiplication factor for learning rate decay is 0.98 (learning rate after each training round = current learning rate × 0.98).

[0048] The training set is randomly shuffled and input into the model in batches of 8 samples each time. The validation set is used to evaluate the model performance every 10 rounds. If the validation set loss does not decrease for 20 consecutive rounds, the training is terminated by early stopping, and the model weights with the minimum validation set loss are retained.

[0049] During model training, multi-scale discrete Fourier transform is used: a two-dimensional discrete Fourier transform is implemented using PyTorch's fft2 function, transforming the input features of each Fourier operator sublayer and retaining the first 32 Fourier modes. =32、 =32).

[0050] Fourier space kernel integral: learnable parameter matrix Through uniform distribution Initialization, dimension is (in =64、 = =32), kernel integration is implemented through matrix multiplication.

[0051] Multi-scale inverse Fourier transform: The inverse transform is implemented using the ifft2 function of PyTorch, which converts the frequency domain features back to the spatial domain. Then, the output features of the three Fourier operator sub-layers are arithmetically averaged and processed by the GLUE activation function to obtain the final hidden layer features.

[0052] Feature fusion: The final hidden layer features are combined with the original input features through a process 1 The results of the convolution transformation are added together to obtain the input features for the next layer, ensuring the continuity of feature propagation.

[0053] S4. Quantitative and qualitative analysis of the reconstructed physical field is performed using the test set. The quantitative analysis is achieved by calculating the mean absolute error (MAE) between the reconstructed predicted physical field and the real physical field. The qualitative analysis is achieved by randomly sampling data examples from the test set and inputting them into the trained model, outputting the reconstructed physical field and displaying it visually, and comparing the characteristic consistency and relative error between the reconstructed physical field and the real physical field.

[0054] Quantitative analysis: MAE was calculated for 2000 samples in the test set (1000 fluid velocity fields and 1000 temperature fields). Quantitative testing and evaluation were performed on each physical field test set. The model evaluation results are shown in Table 1 below.

[0055] Table 1: Quantitative test evaluation results for each physics test set

[0056] In the fluid velocity field test set, the prediction error MAE was 0.404, achieving the minimum error; in the temperature field test set, the prediction error MAE was 0.004781, also achieving the minimum error. Quantitative analysis results show that the reconstruction accuracy of the method in this invention is significantly better than that of the single U-Net network model and the single Fourier neural operator FNO method.

[0057] Qualitative analysis: One fluid velocity field sample and one temperature field sample are randomly sampled from the test set. After embedding, sparse observational physical field features are obtained, which are then input into the trained model to obtain the reconstruction results. These results are then visualized, such as... Figure 5 As shown, the consistency and relative error of the reconstructed physical field with the real physical field are compared: Fluid velocity field: The reconstruction results can accurately restore the wake vortex structure of the flow around the cylinder. The position and size of the vortex are basically consistent with the real field. The relative error is concentrated in the boundary layer region (maximum relative error < 5%).

[0058] Temperature field: The reconstruction results can accurately capture the frontal contour of temperature diffusion. The gradient changes between high-temperature and low-temperature regions are in good agreement with the real field, with a relative error of < 4%.

[0059] This invention presents a high-precision multiphysics field reconstruction method based on a nested neural network architecture. With its core advantages of efficiently processing sparse data, accurately capturing multi-scale features, and having strong cross-scenario generalization ability, it can be widely applied in multiple fields such as engineering simulation, industrial monitoring, weather forecasting, and scientific research. Specific application scenarios are as follows: Weather forecasting, environmental monitoring, and other fields require the reconstruction of the global physical field based on sparse observation station data. This invention can improve the prediction accuracy and timeliness, and support disaster early warning and environmental governance.

[0060] Regional meteorological field reconstruction: Based on sparse temperature and wind speed observation data from meteorological stations, this invention can quickly reconstruct the two-dimensional / three-dimensional temperature field and wind field distribution within a region, accurately capture the movement patterns of weather systems such as fronts and cyclones, improve the accuracy of short-term weather forecasts, and provide technical support for early warning of disasters such as rainstorms and strong winds.

[0061] Atmospheric pollution diffusion simulation: Based on sparse pollutant concentration monitoring data around pollution sources, this invention can reconstruct the concentration field distribution of pollutant diffusion. Combined with wind field and temperature field information, it can predict the diffusion path and impact range of pollutants, providing a basis for decision-making in emergency response to air pollution (such as pollution source shutdown and personnel evacuation).

[0062] Marine environmental monitoring: Based on sparse data obtained from marine observation buoys and satellite remote sensing, the ocean temperature field and ocean current velocity field are reconstructed, and the evolution trends of red tides and ocean current eddies are analyzed to support the development of marine fishery resources and the safety of maritime shipping.

[0063] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A high-precision multiphysics reconstruction method based on a nested neural network architecture, characterized in that, Includes the following steps: S1. Obtain open-source multiphysics data; S2. Construct a physical field reconstruction model of U-Net driven by multi-scale Fourier operators. The physical field reconstruction model includes an embedding module, an encoding module, and a decoding module. The encoding module contains at least one encoding layer. Each encoding layer is composed of a multi-scale Fourier neural operator layer and a downsampling layer connected in sequence. The decoding module contains at least one decoding layer. Each decoding layer is composed of a multi-scale Fourier neural operator layer and an upsampling layer connected in sequence. S3. The physical field reconstruction model is trained based on the training set. The embedding module adopts a preset embedding method to convert the original sparse observation data in the training set into a structured feature representation. The structured feature representation is input into the encoding module, and multi-scale physical field features are extracted through the multi-scale Fourier neural operator layer of each encoding layer. Then, the multi-scale physical field features are compressed in spatial dimension and expanded in feature channel dimension through the downsampling layer to output the encoded features. The encoded features are input into the decoding module. The corresponding scale features from the encoding module and the output features of the previous decoding layer are fused through the multi-scale Fourier neural operator layer of each decoding layer. Then, the physical space resolution is gradually restored through the upsampling layer to achieve the reconstruction of the complete physical field. S4. Quantitative and qualitative analysis of the reconstructed physical field is performed using the test set. The quantitative analysis calculates the average absolute error between the reconstructed predicted physical field and the real physical field. The qualitative analysis inputs randomly sampled data examples from the test set into the trained model, outputs the reconstructed physical field, and displays it visually. The consistency of features and relative error between the reconstructed physical field and the real physical field are compared.

2. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 1, characterized in that: The open-source multiphysics dataset is divided into fluid velocity field datasets and temperature field datasets based on the physical scene type. Each physical scene dataset is further divided into training, validation, and test sets according to a preset ratio of 80%, 10%, and 10%. Each physical scene dataset contains a total of m data samples, and the total number of samples in all datasets is N. Each sample is represented as (…). i y i ), where 1≤i≤N, i For sparse observation data and as the initial input to the model, y i The actual distribution of the physical field is used as the training label.

3. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 2, characterized in that, The embedding module in step S3 transforms the original sparse observation data into a structured data representation using any one or more preset embedding methods, such as mask embedding, multilayer perceptron embedding, or Voronoi embedding, and uses this representation as the initial state of the Fourier space. The general expression for the embedding method is: ,in, This is the original sparse observation data. For structured data representation, For embedding mapping functions, The total number of data observations. For the first The spatial coordinates of each observation point are used to locate the position of the original sparse observation data in the physical field.

4. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 3, characterized in that: The mask embedding places the physical field observations at corresponding grid positions to preserve spatial relationships and information. This is achieved by concatenating a mask matrix and a coordinate matrix, with the remaining positions filled with 0. The mask embedding formula is as follows: ,in, For data coordinates, The coordinates are two-dimensional coordinates of the mask grid, which is a pre-defined uniformly distributed grid used to carry the spatial location information of the physical field observations; The multilayer perceptron embedding maps the input to high-dimensional hidden layer features through an MLP containing the GLUE activation function, then reshapes the hidden layer features into a single-channel feature map, and then uses 1 1. Convolution generates embedded features, and the embedding formula of the multilayer perceptron is: , ,in, , , , , , For learnable parameters, This is a high-dimensional hidden layer feature vector output by a multilayer perceptron, with dimensions determined by learnable parameters. , Decide, This is an activation function used to enhance the model's nonlinear fitting ability. Used to reshape the feature map; The Voronoi embedding divides the computational region using Voronoi polygons, ensuring that physics field observations remain consistent within their corresponding spatial locations. For other spatial locations, the physics field values ​​in the unknown region are filled using nearest-neighbor interpolation. The formula for the Voronoi embedding is: ,in, Used to calculate the Euclidean distance between two coordinates. To remove The index of any other observation point besides the observation point is used to determine the physical field value of the unknown region by means of the nearest neighbor rule.

5. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 4, characterized in that: The encoder representation learning expression in the encoding module described in step S2 is as follows: , ,in, 1 represents the total number of coding layers in the coding module, used to limit the number of iterations for each coding layer. The total number of layers of the Fourier operator. , For learnable parameters, This is a loop function, indicating that the output features of the previous layer's encoding are input sequentially. Each sublayer uses a Fourier operator, and the output is used as the encoded feature of the current layer. The input features are encoded by the next layer. Encode the output features for the current layer. This is the final output feature of the coding layer after downsampling, used to pass to the next coding layer or decoding module.

6. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 5, characterized in that: The decoder reconstruction learning expression in the decoding module described in step S2 is: , ,in, 2 represents the total number of decoding layers in the decoding module, used to limit the number of iterations for each decoding layer. This is a feature concatenation function that concatenates the corresponding scale features output by the encoding module with the features output by the previous decoding layer along the feature channel dimension. , For learnable parameters, The corresponding scale features output by the encoding module. The reconstructed features are the output of the current layer's decoding. These are the reconstructed features after upsampling of the decoding layer, used to gradually restore the spatial resolution of the physical field.

7. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 6, characterized in that, The calculation process of the multi-scale Fourier neural operator layer is as follows: Using the hidden layer feature state of the physical field as input, we first perform a multi-scale discrete Fourier transform on the physical field feature to transform it into Fourier space. In Fourier space, nonlocal information interaction and global regularization are achieved by executing kernel integral operators through randomly initialized learnable parameter matrices. Perform multi-scale inverse Fourier transform to transform the feature space from Fourier space to Euclidean space. Then, by averaging the features at each Fourier scale and applying activation functions, the final hidden layer features are obtained. The final hidden layer features obtained in the first step are compared with the original input features by step 1. The results of the linear transformations of the first convolutional layer are summed to obtain the input features for the next layer. The formula for this process is as follows: ,in, As the input features for the next layer, 1 1. Learnable parameters of a convolutional layer for The original input features of the layer, The spatial location coordinates of the feature For the first The output features of the layer after processing by the multi-scale Fourier neural operator have the same dimension as the original input features.

8. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 7, characterized in that: In step S3, the model training uses average loss as the loss function to calculate the average absolute error between the predicted and true values. The formula is as follows: ,in, The average loss function is... The total number of data observations. For the model to the first The predicted physical field values ​​for each observation point For the first The true physical field values ​​at each observation point correspond to the labeled data in the training set.

9. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 1, characterized in that, The hyperparameters for model training were set as follows: batch size of 8, number of training epochs of 500, initial learning rate of 0.001, Adam optimizer used as the optimizer, ExponentialLR exponential learning rate scheduler used as the scheduler, and learning rate decay multiplication factor of 0.

98.

10. The high-precision multiphysics reconstruction method based on a nested neural network architecture as described in claim 1, characterized in that: The encoding module has 3 encoding layers and the decoding module has 3 decoding layers. Each multi-scale Fourier neural operator layer contains 3 Fourier operator sublayers.

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