Partial differential equation solving method based on star operation and time-frequency dual-channel feature fusion

Abstract

The application discloses a partial differential equation solving method based on star operation and time-frequency dual-channel feature fusion, and relates to the cross field of artificial intelligence and scientific calculation. The method firstly encodes input physical field data and extracts skip connection features; then, the input is input into a space and frequency feature extraction network in parallel, and multi-scale local space patterns and global long-range dependence relationships are captured respectively; then, the fusion features are obtained through the cross-domain nonlinear coupling of a double-branch structure of a star fusion module; then, the fusion features are sent into a deep star interaction network stacked by multiple star modules, and high-dimensional hidden layer features are output through depth convolution, linear expansion and element-by-element multiplication interaction; finally, the prediction physical field data is output by combining the skip connection feature decoding. The application realizes exponential rank approximation of high-dimensional integral operators while maintaining linear computational complexity, and gives consideration to the cooperative prediction of local details and global patterns of physical fields under high resolution.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of artificial intelligence and scientific computing, specifically to a method for solving partial differential equations based on star-shaped operations and the fusion of time-frequency dual-path features. Background Technology

[0002] The statements in this section are provided only as background information in connection with this disclosure and may not constitute prior art.

[0003] Partial differential equations (PDEs) are core mathematical models describing continuous physical processes in nature and engineering systems, and are widely used in fields such as fluid mechanics, weather forecasting, electromagnetic field analysis, materials science, and chemical reaction kinetics. These problems typically involve the coupling relationships between spatial variables, time variables, and various physical parameters, and their analytical solutions are difficult to obtain in most practical scenarios. Therefore, numerical calculation methods have long been relied upon for approximate solutions.

[0004] Traditional numerical methods for solving physical domain equations (PDEs) mainly include the finite difference method (FDM) and the finite element method (FEM). These methods typically require discretizing the physical domain into a mesh and approximating the solution iteratively. While these methods have advantages in terms of accuracy and theoretical completeness, they suffer from the following significant shortcomings in practical applications: 1. Strong mesh dependency: When the geometry, boundary conditions, or physical parameters change, it is often necessary to rebuild the mesh and repeat the calculation. 2. High computational cost: In high-dimensional problems, multi-scale problems, or problems requiring multiple queries, the computation time and resource consumption are extremely high; 3. Difficult to meet real-time or near-real-time requirements: It is difficult to apply directly in scenarios such as weather forecasting and engineering optimization.

[0005] To overcome the aforementioned problems, a class of data-driven partial differential equation (PDE) solving methods has been proposed in recent years, with neural operators being a representative example. Unlike traditional neural networks that directly fit finite-dimensional input-output relationships, neural operators learn mappings from function space to function space, enabling rapid inference under different initial conditions, boundary conditions, or parameter configurations. The theoretical foundation of most neural operators used for solving PDEs stems from the Green's function representation of solutions. Formally, this representation reconstructs the solution by integrating the kernel function with the input activation term. This integral operator framework has spawned diverse architectural systems, ranging from spectral methods such as Physical Information Neural Networks (PINNs) and Fourier Neural Operators (FNOs) to the latest innovations based on Transformer models and state-space models. However, existing paradigms still have some drawbacks: 1. Physical Information Neural Networks (PINNs): These networks approximate PDE solutions by introducing physical constraints into the loss function, but they typically require retraining the model for each specific problem, resulting in high computational costs. 2. Fourier Neural Operator (FNO) and subsequent research: It belongs to the spectral method and uses Fourier transform to perform global modeling in the frequency domain. It has good generalization ability, but it has a smoothing problem in characterizing local high-frequency details and it is difficult to capture local high-frequency details or complex boundary features. 3. Attention-based neural operators or Transformer-like methods: These methods simulate the structure of integral operators through self-attention mechanisms, but their... The computational complexity increases quadratically with the number of discrete points, making it difficult to extend to high-resolution, fine-grid scenes; while convolutional methods, which excel at local features, have extremely limited receptive fields.

[0006] In summary, existing technologies generally face the following common problems: either they are computationally efficient but lack expressive power, or they have strong expressive power but excessive computational complexity, making it difficult to simultaneously achieve accuracy, efficiency, and scalability in complex PDE problems. Developing neural network operators that can simultaneously achieve computational efficiency, grid independence, and strict approximation guarantees still faces significant challenges. Summary of the Invention

[0007] The purpose of this invention is to provide a partial differential equation solution method based on star-shaped operation and time-frequency dual-path feature fusion, in order to address the above-mentioned shortcomings of the prior art. This method solves the technical problems that existing neural operators cannot simultaneously extract local / global features, have insufficient high-dimensional nonlinear expression capabilities, and whose computational complexity increases quadratically with resolution.

[0008] The technical solution of the present invention is as follows: A partial differential equation solution method based on star-shaped computation and time-frequency dual-path feature fusion, applied to physical field evolution prediction in scientific computing, includes the following steps: Step S1: Obtain the input physical field data of the partial differential equation to be solved at the current time. The input physical field data is processed using a latent coding module. Projecting onto the latent manifold yields the first feature. The first-layer output of the latent coding module is retained as a skip connection feature. ; Step S2: Place the first feature The first feature is input in parallel to a spatial feature extraction network and a frequency feature extraction network, respectively; wherein, the spatial feature extraction network captures the multi-scale local spatial pattern of the first feature to obtain spatial features. The frequency feature extraction network extracts the first feature. After sequentially performing Fast Fourier Transform, frequency domain weighted multiplication, and Inverse Fourier Transform, a star-gated mechanism is introduced to perform element-wise multiplication with the first feature after linear projection to obtain the frequency feature. ; Step S3: Transfer the spatial features and the frequency characteristics The input is fed to a star-shaped fusion module, which employs linear residual branching and nonlinear multiplication branching based on element-wise multiplication to perform cross-domain nonlinear coupling on the spatial features and the frequency features, and outputs fused features. ; Step S4: Merge the features The input features are fed into a deep star-shaped interactive network composed of multiple stacked star-shaped modules. Within each star-shaped module, the input features are depthwise convolutioned and then split into two paths. These paths are then linearly expanded point-by-point and subjected to element-wise multiplication interactions. Finally, the output features of a single star-shaped module are obtained through output projection. The fused features... After processing by the star-shaped modules stacked in multiple layers, high-dimensional hidden layer features are output. ; Step S5: Transfer the high-dimensional hidden layer features With the jump connection feature The data are concatenated and mapped back to the physical field space using a feature decoding module, outputting the predicted physical field data of the partial differential equation at future times. .

[0009] Further, in step S1, the latent coding module is composed of... The convolutional blocks are composed of stacked convolutional blocks, and their processing follows the formula:

[0010] in, This indicates a latent coding operation. Represents a two-dimensional convolution kernel. This represents the convolution operation. GroupNormalization represents group normalization. This represents the LeakyReLU activation function.

[0011] Further, in step S2, the spatial feature extraction network adopts a U-Net network structure based on the Inception module, which includes downsampling encoding and upsampling decoding processes; the spatial features The calculation process satisfies the following formula:

[0012]

[0013]

[0014]

[0015]

[0016] in, These are the initial features input to the spatial feature extraction network; In the first Encoded features after downsampling and feature extraction at each scale This represents the maximum depth of the spatial feature extraction network. This represents the deepest layer of encoded features; This refers to the initial decoded features obtained after processing the deepest encoded features; In the first The previous layer decoded features after upsampling at each scale Features encoded by skip connections at corresponding scales The decoded features are obtained after concatenation and feature extraction. This indicates the Inception module operation used for fusing features from multiple receptive fields; Indicates feature concatenation operation; This indicates a spatial feature extraction operation.

[0017] Further, in step S2, the frequency feature extraction network adopts a star-gated Fourier architecture, and the frequency features... The calculation process satisfies the following formula:

[0018] in, Represents the Fast Fourier Transform. This represents the inverse Fourier transform. This represents a learnable complex weight matrix. This represents the matrix multiplication operation. This represents a linear projection operation. This represents element-wise multiplication. This represents the activation function. This indicates group normalization.

[0019] Furthermore, in step S3, the process by which the star-shaped fusion module performs cross-domain nonlinear coupling between the spatial features and the frequency features satisfies the following formula:

[0020]

[0021]

[0022] in, Represents the spatial features, Represents the frequency characteristics; The output of the linear residual branch, This is the output of the nonlinear multiplication branch; and For different first linear transformation layers and second linear transformation layers, Map a linear layer to the output; This represents the activation function. This represents element-wise multiplication. This is a DropPath regularization operation.

[0023] Furthermore, in step S4, the calculation process for a single star-shaped module satisfies the following formula:

[0024]

[0025] in, Features input to the current star topology module; It is a spatial mixing feature; Indicates a depthwise convolution operation; and These represent the first linear pointwise expansion operation and the second linear pointwise expansion operation, respectively. Indicates the activation function; This represents element-wise multiplication; Indicates a linear projection layer; Indicates the output depth of the convolutional layer; This is a DropPath regularization operation.

[0026] Furthermore, in step S4, the deep star-shaped interactive network is stacked. The star topology module described in the layer enables effective interaction order With network depth The increase of exhibits a doubly exponential growth, and its recursive relationship satisfies: ,in For the interaction rank of a single-layer star topology module, in order to maintain Approximating a high-dimensional nonlinear Green's function integral operator with linear computational complexity.

[0027] Furthermore, in step S5, the process by which the feature decoding module maps the high-dimensional hidden layer features back to the physical field space satisfies the following formula:

[0028]

[0029] in, For the transposed convolution operation, For output mapping layer, For feature splicing operations, This is for upsampling and reconstructing features.

[0030] Furthermore, the method also includes a model training phase, in which mean squared error is used as the loss function. And based on the autoregressive training mode, end-to-end parameter updates are performed by minimizing the loss function, and the calculation formula is as follows:

[0031] in, For sample size, Indicates the start time of the current prediction block. This represents a set that defines which points in the entire time series will be used as the starting point for prediction. The value in is The step size increases incrementally. The parameter is The partial differential equation solution network, Indicates the predicted sequence of physical fields. Represents a sequence of real physical fields. This indicates the calculation of the square of the L2 norm.

[0032] Furthermore, the input physical field data Represented as dimension The data tensor, where For batch size, For time steps, For the number of channels, For height, For width.

[0033] Compared with existing technologies, the advantages of this invention are: 1. Achieving efficient approximation of high-dimensional integral operators while maintaining linear computational complexity: This invention introduces element-wise multiplication (star-shaped operation) as the basic computational unit of the neural operator, utilizing depthwise convolution, linear pointwise expansion, and element-wise multiplication interactions within the star-shaped module, and performing multi-level stacking. This structure can implicitly simulate the feature interaction relationships in high-dimensional integral operators without explicitly calculating the kernel function matrix, theoretically achieving a double exponential growth of the feature interaction rank with network depth. This overcomes the limitations of traditional linear or convolutional neural operators in terms of expressive power, while strictly maintaining the overall computational complexity at a constant level. This completely avoids the secondary complexity problem caused by the attention mechanism and is suitable for high-resolution physics modeling.

[0034] 2. Time-Frequency Dual-Path Architecture for Collaborative Capture of Local Structure and Global Patterns of Physical Fields: This invention innovatively employs a space-frequency dual-path architecture. The spatial feature extraction network utilizes a U-Net structure including an Inception module to extract multi-scale local spatial features (such as boundary changes, local continuity, and other geometric features); the frequency feature extraction network uses star-gated Fourier layers to extract global long-range dependencies (such as global spectral structure). This design leverages the complementarity principle of function classes to solve the modeling bias problem in existing technologies that cannot simultaneously achieve both local detail accuracy and global smoothness consistency.

[0035] 3. A specific feature fusion mechanism enables high-order nonlinear coupling of cross-domain features: The star-shaped fusion module of this invention designs a dual-branch structure with a linear superposition residual path and a nonlinear multiplication branch path. Unlike simple additive fusion, this mechanism can dynamically weight and modulate frequency features according to spatial context, thereby explicitly modeling the nonlinear coupling relationship of cross-domain features during the feature fusion stage. This effectively avoids the information loss caused by a single feature path and significantly improves the prediction accuracy of the model for complex nonlinear dynamic systems. Attached Figure Description

[0036] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments recorded in the embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0037] Figure 1 This is a schematic diagram of the overall and local structure of the partial differential equation solving network in an embodiment of the present invention; wherein, Figure 1 The left side shows a detailed internal structure diagram of a single "StarBlock" and "StarFuse" in the deep star-shaped interactive network in an embodiment of the present invention; the right side shows a schematic diagram of the overall process architecture of the end-to-end partial differential equation solving network in an embodiment of the present invention. Figure 2 This is a schematic diagram illustrating the principle of achieving effective interactive rank double exponential growth in hierarchical star schema operations in an embodiment of the present invention; Figure 3This is a table comparing the prediction error (RMSE and nRMSE) performance of embodiments of the present invention on multiple standard PDE benchmark tasks (Darcy flow, shallow water equation, diffusion reaction equation, Navier-Stokes equation); Figure 4 This is a visual comparison diagram of the prediction residuals on four equation (dataset) benchmark tasks according to an embodiment of the present invention. Detailed Implementation

[0038] It should be noted that relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0039] The features and performance of the present invention will be further described in detail below with reference to embodiments.

[0040] Example 1 Before elaborating on the specific steps of this method, let's first combine... Figure 1 The right side describes the overall architecture of the partial differential equation solving network (i.e., the neural operator model) that implements the method of this invention.

[0041] This invention takes the input physical field of the partial differential equation (PDE) to be solved as input, and outputs the corresponding PDE solution field through spatial-frequency dual-path feature extraction, star-based feature fusion, and deep star-shaped neural operator modeling. From a system structure perspective, the technical solution constituting the end-to-end partial differential equation solving network of this invention mainly includes the following sequentially connected functional modules: 1. Latent coding module (corresponding to) Figure 1 The Encoder in the PDE is used to receive discrete input physical field data from the PDE and to perform feature projection and encoding on the input physical field data. 2. Dual-path feature extraction module: includes a parallel spatial feature extraction network (corresponding to...) Figure 1 SpatialNet (in the network) and frequency feature extraction network (corresponding to Figure 1 FreqNet (in the context of FreqNet) is used to capture multi-scale local spatial patterns and global long-range dependencies; 3. Star-shaped fusion module (corresponding to) Figure 1 StarFuse (in the context of StarFuse) is used to perform cross-domain unified nonlinear fusion of the extracted spatial and frequency features. 4. Deep star-shaped interactive network (corresponding to) Figure 1 StarNet in the context of StarNet consists of multiple star-shaped modules (corresponding to...) Figure 1 The StarBlock hierarchy is used to approximate operators for solving complex partial differential equations. 5. Feature decoding module (corresponding to) Figure 1 The Decoder in the model maps the features of the high-dimensional latent space back to the physical field space and outputs the predicted physical field data of the partial differential equations at future times.

[0042] The aforementioned functional modules are connected sequentially to form an end-to-end PDE solution network structure. Based on this network structure, the complete end-to-end execution flow of the method in this embodiment includes the following steps S1 to S5: Step S1: Obtain the input physical field data of the partial differential equation to be solved at the current time. The input physical field data is processed using a latent coding module. Projecting onto the latent manifold yields the first feature. The first-layer output of the latent coding module is retained as a skip connection feature. .

[0043] In specific implementation, in order to adapt to the solution of multidimensional physical fields, the input physical field data Represented as dimension The data tensor, its discretization in At each of the grid points, Batch size. For time steps. For the number of channels (e.g., the fluid velocity field) (Quantity) The height of the physics grid. For width.

[0044] To project the high-dimensional input physical field onto a compact feature space to extract effective features, the latent coding module consists of... It consists of stacked convolutional blocks, each block containing a 2D convolutional layer, group normalization (GN), and a Leaky ReLU activation function. Its processing follows the formula:

[0045] in, This indicates a latent coding operation. Represents a two-dimensional convolution kernel. This represents the convolution operation. This represents group normalization (used to stabilize the feature distribution of high-dimensional tensors). This represents the LeakyReLU activation function. It should be noted that if the partial differential equation to be solved is for one-dimensional time series prediction or three-dimensional volumetric data (such as 3D flow fields), the aforementioned two-dimensional convolutional layer can be directly and equivalently replaced by a one-dimensional or three-dimensional convolutional layer.

[0046] In this process, the output features of the first convolutional block are deliberately extracted as... The principle behind this is that local high-frequency boundary details of physical fields (such as turbulence) are easily lost during downsampling in subsequent deep networks. By using skip connections, the most original high-frequency geometric features can be directly passed to the decoder.

[0047] Step S2: Place the first feature The first feature is input in parallel to a spatial feature extraction network and a frequency feature extraction network, respectively; wherein, the spatial feature extraction network captures the multi-scale local spatial pattern of the first feature to obtain spatial features. The frequency feature extraction network extracts the first feature. After sequentially performing Fast Fourier Transform, frequency domain weighted multiplication, and Inverse Fourier Transform, a star-gated mechanism is introduced to perform element-wise multiplication with the first feature after linear projection to obtain the frequency feature. .

[0048] This step is one of the core innovations of this invention. Existing technologies either rely solely on convolution (which has a limited receptive field) or solely on Fourier transform (which loses local details). This invention designs a dual-path parallel architecture that utilizes the complementarity principle of function classes for collaborative modeling.

[0049] Regarding the spatial pathway, the spatial feature extraction network adopts a U-Net network structure based on the Inception module (in alternative embodiments, a ResNet or DenseNet backbone network with the ability to extract local features may also be used). It includes downsampling encoding and upsampling decoding processes, calculated using the following formula:

[0050]

[0051]

[0052]

[0053]

[0054] in, These are the initial features input to the spatial feature extraction network; In the first Encoded features after downsampling and feature extraction at each scale Indicates the maximum depth. This represents the deepest layer of encoded features; These are the initial decoding features; In the first The previous layer decoded features after upsampling at each scale Features encoded by skip connections at corresponding scales The decoded features are obtained after concatenation and feature extraction. The Inception module operation is used to fuse features from multiple receptive fields; Concat represents the feature stitching operation. The deep structure of this module ensures that local boundary conditions and subtle turbulent structures in the fluid physics field are effectively captured.

[0055] Regarding the frequency path, the frequency feature extraction network adopts a star-gated Fourier architecture, and the calculation formula is as follows:

[0056] in, This represents the Fast Fourier Transform (FFT). This represents the inverse Fourier transform (iFFT). This represents a learnable complex weight matrix (used to retain only low-frequency modes that contribute to long-range dependencies in the frequency domain). This represents the matrix multiplication operation. This represents a linear projection operation. This represents element-wise multiplication. This represents the activation function, and GN represents group normalization. This module uses... The implemented "star-shaped gating mechanism" effectively captures the global long-range dependence of the physical field by dynamically weighting frequency components based on local statistical data.

[0057] The specific operation is as follows: 1. Input Perform a Fast Fourier Transform (FFT) to obtain the frequency domain representation. ; 2. In the frequency domain and with learnable complex weight matrices Multiply (only retain low-frequency mode); 3. Perform inverse Fourier transform (iFFT) to restore the spatial domain; 4. Introduce a star-gated control mechanism to combine the above results with those obtained through linear projection. The original features are multiplied element-wise (i.e., star-shaped operation) to dynamically modulate the frequency components.

[0058] Step S3: Transfer the spatial features and the frequency characteristics Input to the star fusion module, as shown in the attached diagram. Figure 1 As shown on the left, the star-shaped fusion module employs linear residual branching and nonlinear multiplication branching based on element-wise multiplication to perform cross-domain nonlinear coupling on the spatial features and the frequency features, outputting fused features. .

[0059] Unlike the simple matrix addition and fusion in existing technologies, the star-shaped fusion module in this embodiment is designed with a specific feature fusion mechanism, which combines linear superposition residual paths and nonlinear multiplication branch paths, as shown in the following formula:

[0060]

[0061]

[0062] in, , ; For the output of the linear residual branch, This is the output of the nonlinear multiplication branch; and For different first linear transformation layers and second linear transformation layers (which can be replaced by multilayer perceptrons (MLPs) in alternatives). Map a linear layer to the output; This represents the activation function. This represents element-wise multiplication. This is the DropPath regularization operation (which can also be replaced with Dropout depending on training requirements). This mechanism can dynamically weight and modulate frequency features according to spatial context, explicitly modeling the nonlinear coupling of cross-domain features.

[0063] Step S4: Merge the features It is fed into a deep star-shaped interaction network composed of multiple stacked star-shaped modules (StarBlock); combined with Figure 1 The detailed internal structure diagram shown on the left illustrates that within each star-shaped module, the input features are depthwise convolutioned and then split into two paths. These paths are then linearly expanded point-by-point and subjected to element-wise multiplication before being projected back to the output of the individual star-shaped module. The fused features... After processing by the star-shaped modules stacked in multiple layers, high-dimensional hidden layer features are output. .

[0064] This step is the core of approximating the partial differential equation solution operator (Green's function integral operator). The computation process of a single star-shaped module satisfies the following formula:

[0065]

[0066] in, Input features; This refers to spatial mixing features obtained by mixing spatial information using very few parameters. and These represent the first and second linear point-by-point expansion operations, respectively. Indicates the activation function; This represents element-wise multiplication; Indicates a linear projection layer; Indicates the output depth of the convolutional layer; This is a DropPath regularization operation.

[0067] The internal calculation steps for each StarBlock are as follows: 1. Depth-wise Convolution: It is used to mix spatial information with very few parameters; 2. Star-shaped operation interaction: The features are divided into two paths, each undergoing linear point-wise expansion. Then perform element-wise multiplication; 3. Output projection: after passing through a linear projection layer and depthwise convolution Output.

[0068] In deep star-shaped interaction networks, stacked The star-shaped module described in the layer. This structure, based on element-wise multiplication and its hierarchical stacking, can be rigorously proven mathematically to be effective: although a single-layer star-shaped operation only provides a finite rank... The integral kernel approximates the result, but the stacking of multiple layers makes the effective interactive rank... With network depth The increase of exhibits a doubly exponential growth, and its recursive relationship satisfies: This means that by stacking only a small number of networks (e.g., 8 layers), one can achieve more than Theoretical interaction rank. This architecture completely avoids the problems caused by traditional attention mechanisms. The quadratic complexity problem allows the model to maintain linear computational complexity. Under the premise of [missing information], it not only effectively approximates the high-dimensional nonlinear Green's function integral operator experimentally, but also possesses rigorous rationality and generalizability in theory.

[0069] Step S5: Transfer the high-dimensional hidden layer features With the jump connection feature The data are concatenated and mapped back to the physical field space using a feature decoding module, outputting the predicted physical field data of the partial differential equation at future times. .

[0070] The specific calculation process satisfies the following formula:

[0071]

[0072] in, The transposed convolution operation is used to project a high-dimensional hidden manifold back into physical space to obtain upsampled reconstructed features. ; For the output mapping layer, Concat is the feature concatenation operation. It uses the features extracted in concatenation step S1. This effectively compensates for the complex boundaries and high-frequency details that are smoothed out during network forward propagation, greatly improving the prediction accuracy of the model in scenarios such as fluid mechanics.

[0073] To further theoretically demonstrate the effectiveness of the deep star-shaped interaction network (corresponding to a stack of multiple StarBlocks) in this invention, please refer to [link to relevant documentation]. Figure 2 The diagram illustrates the principle of rank growth in a hierarchical star schema. The detailed derivation of achieving double exponential growth of interactive rank through multi-level stacked star schemas is presented below: Although single-level star schema operations only provide a finite rank While the integral kernel approximates the original, this invention significantly enhances the nonlinear expressive power of features by stacking multiple star-shaped modules. Assuming the output of the first-layer star-shaped operation is... (here) This refers to the aforementioned element-wise multiplication operation. (representing the feature branches after the first and second linear pointwise expansion operations, respectively), their interaction rank is... .

[0074] When performing the second-level star schema operation, assume the weights of the new input feature branches are... and Then the output of the second-level operation can be expressed as:

[0075] Expanding the above product will result in a double summation term:

[0076] By constructing index mappings The above double summation is mathematically equivalent to a rank of A single-layer unfolding.

[0077] And so on, for a network depth index of... (in The intermediate network layers of ) have an effective interaction order Strictly adhere to the following recursive relationship:

[0078] This means that, with the network layer index As the linear increase in density increases, the rank of the integral kernel that the model of this invention can approximate exhibits a doubly exponential growth. Therefore, when the deep star-shaped interactive network reaches the total number of stacked layers... When the effective interaction rank of the overall output is... For example, in a practical implementation, only 8 layers need to be stacked (i.e., If the rank of a single layer is set to an appropriate initial value, then more than 100 rank can be obtained. Theoretical interaction rank. This theoretical derivation fully demonstrates that the present invention can maintain the theoretical interaction rank. While maintaining linear computational complexity, it possesses a powerful ability to simulate the interaction of extremely high-dimensional nonlinear physical fields, giving this model extremely high theoretical interpretability.

[0079] Model training phase: This embodiment employs an end-to-end autoregressive training model. Mean squared error (MSE) is defined as the loss function. The parameters are updated by minimizing this loss function based on the autoregressive model. The calculation formula is:

[0080] in, For sample size, Indicates the start time of the current prediction block. This represents a set that defines which points in the entire time series will be used as the starting point for prediction. The value in is The step size increases incrementally. The parameter is The partial differential equation solution network, The representation model is based on the past The sequence of future physical fields predicted at each time step This represents the corresponding sequence of real physical fields. This represents the calculation of the square of the L2 norm. Under this strategy, the model can complete weight updates without explicitly constructing a high-dimensional kernel matrix, significantly reducing memory usage and runtime.

[0081] In this embodiment, it should be noted that the model training is block autoregressive. The start time of the current prediction block is a variable taken from the set. ,for example At that time, the model uses 0 to 1 Data prediction between arrive Data; It represents the length of the prediction block, indicating the length of the input window and the length of the output prediction, which is represented by the length of the block each time the model "looks over". Data from each time step, and attempts to "predict" the future. Data at each time step; It is a set that defines which points in the entire time series will be used as the starting point for prediction, that is... , The value in is Incrementing by step size, for example , Wait, to ensure that the prediction blocks are continuous.

[0082] Experimental verification and effect analysis: To verify the actual technical effect of the method in this embodiment, a comparative experiment was conducted with existing mainstream neural operator models.

[0083] Please see Figure 3 , Figure 3 This paper presents a comparison of the prediction errors (RMSE and nRMSE) of the model of this invention (labeled StarSolver in the figure) with other comparative models (including PINN, Unet, MgNO, FNO, FFNO, Geo-FNO, Transolver, LSM, etc.) on four standard PDE benchmark tasks (Darcy Flow, Shallow-Water equation, Diffusion-Reaction equation, and Navier-Stokes equation). Figure 3 The experimental data show that the method of this invention achieves the lowest prediction error on all four benchmark tasks, achieving the best performance. In particular, it achieves an error reduction of more than 50% in some tasks, significantly improving the solution accuracy of partial differential equations.

[0084] Please see Figure 4 , Figure 4This presentation displays a visual comparison of the prediction residuals of different models on four benchmark tasks (datasets), including the target output (Ground Truth), the prediction results and residuals of the present invention embodiment (StarSolver), the prediction results and residuals of FNO, and the prediction results and residuals of Transolver. Figure 4 Intuitively, compared to existing FNO and Transolver models, the method (star solver) in this embodiment of the invention can capture the local boundary conditions and high-frequency ripple details of the physical field with extremely high accuracy, exhibiting a clearer ripple structure and significantly lower error residuals (the residual plot is lightest in color / closest to zero). This fully demonstrates that the invention successfully achieves a synergistic characterization of the global pattern of the physical field and local high-frequency details through time-frequency dual-path feature fusion and deep star-shaped interactive design.

[0085] The embodiments described above merely illustrate specific implementation methods of this application, and while the descriptions are detailed and specific, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the technical solution of this application, and these modifications and improvements all fall within the scope of protection of this application.

[0086] This background section is provided to generally present the context of the invention. The work of the currently named inventors, the work to the extent described in this background section, and aspects of this section that did not constitute prior art at the time of application are neither expressly nor impliedly acknowledged as prior art to the invention.

Claims

1. A method for solving partial differential equations based on star-shaped operations and the fusion of time-frequency dual-path features, characterized in that, The method is applied to predicting the evolution of physical fields in scientific computing, and includes the following steps: Step S1: Obtain the input physical field data of the partial differential equation to be solved at the current time. The input physical field data is processed using a latent coding module. Projecting onto the latent manifold yields the first feature. The first-layer output of the latent coding module is retained as a skip connection feature. ; Step S2: Place the first feature The first feature is input in parallel to a spatial feature extraction network and a frequency feature extraction network, respectively; wherein, the spatial feature extraction network captures the multi-scale local spatial pattern of the first feature to obtain spatial features. The frequency feature extraction network extracts the first feature. After sequentially performing Fast Fourier Transform, frequency domain weighted multiplication, and Inverse Fourier Transform, a star-gated mechanism is introduced to perform element-wise multiplication with the first feature after linear projection to obtain the frequency feature. ; Step S3: Transfer the spatial features and the frequency characteristics The input is fed to a star-shaped fusion module, which employs linear residual branching and nonlinear multiplication branching based on element-wise multiplication to perform cross-domain nonlinear coupling on the spatial features and the frequency features, and outputs fused features. ; Step S4: Merge the features The input features are fed into a deep star-shaped interactive network composed of multiple stacked star-shaped modules. Within each star-shaped module, the input features are depthwise convolutioned and then split into two paths. These paths are then linearly expanded point-by-point and subjected to element-wise multiplication interactions. Finally, the output features of a single star-shaped module are obtained through output projection. The fused features... After processing by the star-shaped modules stacked in multiple layers, high-dimensional hidden layer features are output. ; Step S5: Transfer the high-dimensional hidden layer features With the jump connection feature The data are concatenated and mapped back to the physical field space using a feature decoding module, outputting the predicted physical field data of the partial differential equation at future times. .

2. The partial differential equation solution method based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S1, the latent coding module is composed of The convolutional blocks are composed of stacked convolutional blocks, and their processing follows the formula: in, This indicates a latent coding operation. Represents a two-dimensional convolution kernel. This represents the convolution operation. GroupNormalization represents group normalization. This represents the LeakyReLU activation function.

3. The partial differential equation solution method based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S2, the spatial feature extraction network adopts a U-Net network structure based on the Inception module, which includes downsampling encoding and upsampling decoding processes; the spatial features The calculation process satisfies the following formula: in, These are the initial features input to the spatial feature extraction network; In the first Encoded features after downsampling and feature extraction at each scale This represents the maximum depth of the spatial feature extraction network. This represents the deepest layer of encoded features; This refers to the initial decoded features obtained after processing the deepest encoded features; In the first The previous layer decoded features after upsampling at each scale Features encoded by skip connections at corresponding scales The decoded features are obtained after concatenation and feature extraction. This indicates the Inception module operation used for fusing features from multiple receptive fields; Indicates feature concatenation operation; This indicates a spatial feature extraction operation.

4. The partial differential equation solution method based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S2, the frequency feature extraction network adopts a star-gated Fourier architecture, and the frequency features The calculation process satisfies the following formula: in, Represents the Fast Fourier Transform. This represents the inverse Fourier transform. This represents a learnable complex weight matrix. This represents the matrix multiplication operation. This represents a linear projection operation. This represents element-wise multiplication. This represents the activation function. This indicates group normalization.

5. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S3, the process by which the star-shaped fusion module performs cross-domain nonlinear coupling between the spatial features and the frequency features satisfies the following formula: in, Represents the spatial features, Represents the frequency characteristics; The output of the linear residual branch, This is the output of the nonlinear multiplication branch; and For different first linear transformation layers and second linear transformation layers, Map a linear layer to the output; This represents the activation function. This represents element-wise multiplication. This is a DropPath regularization operation.

6. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S4, the calculation process for a single star-shaped module satisfies the following formula: in, Features input to the current star topology module; It is a spatial mixing feature; Indicates a depthwise convolution operation; and These represent the first linear pointwise expansion operation and the second linear pointwise expansion operation, respectively. Indicates the activation function; This represents element-wise multiplication; Indicates a linear projection layer; Indicates the output depth of the convolutional layer; This is a DropPath regularization operation.

7. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 6, characterized in that, In step S4, the deep star-shaped interactive network is stacked. The star topology module described in the layer enables effective interaction order With network depth The increase of exhibits a doubly exponential growth, and its recursive relationship satisfies: ,in For the interaction rank of a single-layer star topology module, in order to maintain Approximating a high-dimensional nonlinear Green's function integral operator with linear computational complexity.

8. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, In step S5, the process by which the feature decoding module maps the high-dimensional hidden layer features back to the physical field space satisfies the following formula: in, For the transposed convolution operation, For output mapping layer, For feature splicing operations, This is for upsampling and reconstructing features.

9. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion as described in claim 1, characterized in that, The method also includes a model training phase, which uses mean squared error as the loss function. And based on the autoregressive training mode, end-to-end parameter updates are performed by minimizing the loss function, and the calculation formula is as follows: in, For sample size, Indicates the start time of the current prediction block. This represents a set that defines which points in the entire time series will be used as the starting point for prediction. The value in is The step size increases incrementally. The parameter is The partial differential equation solution network, Indicates the predicted sequence of physical fields. Represents a sequence of real physical fields. This indicates the calculation of the square of the L2 norm.

10. The method for solving partial differential equations based on star-shaped operation and time-frequency dual-path feature fusion according to any one of claims 1-9, characterized in that, The input physical field data Represented as dimension The data tensor, where For batch size, For time steps, For the number of channels, For height, For width.

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