Resampling region differential physics neural network and method for complex turbulent flow prediction

Abstract

A resampling region differential physical neural network and method for predicting complex turbulence, belonging to the interdisciplinary field of fluid mechanics and deep learning, addresses the problems of insufficient high-frequency structure characterization, weak sampling strategy targeting, low training efficiency, and unstable physical constraints in existing flow field prediction methods for complex turbulence. This paper proposes to enhance the expression of high-frequency features by frequency mapping of spatial and temporal coordinates, constructing regional-scale features based on regional aggregation calculations and embedding physical constraints, dynamically filtering samples by combining the deviation of physical constraints with the flow field energy intensity, modeling features using an encoding-decoding structure and mapping them to the flow field physical quantity space, and constructing a composite constraint including data fitting and physical conservation for iterative training. This achieves high-precision prediction of complex turbulent flow fields and is suitable for complex flow field analysis, engineering fluid simulation, and related intelligent computing scenarios.

Description

Technical Field

[0001] This involves the interdisciplinary field of fluid mechanics and deep learning, specifically a resampled region differential physics neural network and prediction method for predicting complex turbulence. Background Technology

[0002] Computational fluid dynamics (CFD) is a core technology for flow field prediction and analysis. Currently, it primarily uses numerical solutions to the Navier-Stokes equations to predict physical quantities such as velocity and pressure fields. In engineering practice, discretization methods such as the finite difference method, finite volume method, and finite element method are often employed, combined with turbulence modeling methods such as the Reynolds-averaged Navier-Stokes model, large eddy simulation, and direct numerical simulation, to approximate solutions for complex flow processes. For example, in scenarios such as aerospace aerodynamic design, ship hydrodynamic analysis, and cabin flooding, these methods can accurately characterize the overall evolution of the flow field. However, their computational processes typically rely on high-precision mesh generation and large-scale iterative solutions, resulting in long computation times and high storage costs. Under multi-scale coupled or strongly nonlinear flow conditions, numerical stability and convergence efficiency remain significantly limited.

[0003] With the development of deep learning technology, flow field prediction methods based on the fusion of data-driven and physical constraints have gradually become a research hotspot. Among them, models represented by physical information neural networks (PINs) achieve approximate solutions to flow fields with limited labeled data by embedding governing equations, initial conditions, and boundary conditions into a loss function. For example, in problems such as one-dimensional Burgers equations, two-dimensional incompressible flows, and flow around a cylinder, PINs can, to some extent, replace traditional numerical solutions, significantly reducing computational costs. Meanwhile, improved models combining operator learning networks or Transformer structures have also made progress in global feature modeling capabilities and cross-scene generalization abilities.

[0004] However, existing methods still face several technical bottlenecks in complex turbulent scenarios. On the one hand, traditional physical information neural networks suffer from significant frequency bias, lacking the ability to accurately represent high-frequency, small-scale turbulent structures and failing to accurately capture secondary flows and localized dramatic changes. On the other hand, existing sampling strategies often rely on static or residual-based single indicators, failing to incorporate key physical features such as flow field energy distribution. This results in insufficient sampling in critical regions and excessive samples in redundant regions, impacting training efficiency and prediction accuracy. Furthermore, in the joint prediction of multiple physical quantities, there is a supervision imbalance between output channels, making secondary flow components easily ignored and reducing overall prediction stability. Simultaneously, traditional point-based differential calculation methods are sensitive to data noise, easily introducing fluctuations in differential results and weakening the effectiveness of physical constraints. Although some methods have improved by introducing attention mechanisms or adaptive sampling strategies, they have not yet achieved coordinated optimization of high-frequency feature modeling, key region sampling, and physical constraint stability.

[0005] In summary, existing technologies suffer from several drawbacks in predicting complex turbulence, including insufficient ability to characterize high-frequency structures, weak targeting of sampling strategies, low efficiency in utilizing training resources, imbalance among multiple output targets, and susceptibility of differential calculations to noise interference. Summary of the Invention

[0006] To address the shortcomings of existing technologies, such as insufficient high-frequency structure characterization capabilities, weak sampling strategy targeting, low efficiency in training resource utilization, imbalance among multiple output targets, and susceptibility to noise interference in differential calculations during complex turbulence prediction, the technical solution provided by this invention is as follows: A resampled region differential physics neural network for predicting complex turbulence includes: It includes a feature mapping structure, a region feature processing structure, an encoding / decoding structure, and an output mapping structure connected in sequence, and a sample filtering structure interactively connected to the encoding / decoding structure; The feature mapping structure is used to perform frequency mapping on the input spatial-temporal coordinates to obtain a high-dimensional feature representation; The region feature processing structure is used to perform region aggregation calculations based on the high-dimensional feature representation to obtain region-scale features and embed physical constraints to form a constraint feature representation. The sample selection structure is used to evaluate and select samples based on the constraint feature representation to obtain an optimized sample set; The encoding / decoding structure is used to perform feature modeling on the feature representation corresponding to the optimized sample set to obtain decoded features; The output mapping structure is used to map the decoded features to the flow field physical quantity space to obtain the flow field prediction value.

[0007] Furthermore, a preferred embodiment is provided, wherein the feature mapping structure is used to transform the spatial-temporal coordinates through multi-frequency periodic mapping and concatenate them with the original input, and then obtain a high-dimensional feature representation through linear transformation.

[0008] Furthermore, a preferred embodiment is provided, wherein the regional feature processing structure is used to divide local regions according to proximity relationships and to perform cumulative calculation and averaging processing on the features within the regions to obtain regional scale features.

[0009] Furthermore, a preferred embodiment is provided in which the sample screening structure is used to jointly evaluate and screen key samples based on the degree of deviation from physical constraints and the energy intensity of the flow field to update the sample set.

[0010] Furthermore, a preferred embodiment is provided, wherein the encoding and decoding structure is a multi-layer encoding and decoding network structure built based on a self-attention mechanism.

[0011] Furthermore, a preferred embodiment is provided in which the encoding and decoding structure introduces a nonlinear activation method that adapts to the changing characteristics of the flow during the feature modeling process.

[0012] The method for predicting complex turbulence, based on the aforementioned neural network, includes: Scale the flow field data to obtain spatiotemporal feature data; Frequency mapping is performed on spatiotemporal feature data to obtain high-dimensional feature representations; Region aggregation calculation is performed based on high-dimensional feature representation, and physical constraints are embedded to obtain constraint feature representation; Sample selection is performed based on constraint feature representation to obtain an optimized sample set; Encode and decode the feature representations corresponding to the optimized sample set to obtain decoded features; The decoded features are mapped to the flow field physical quantity space to obtain the flow field prediction value.

[0013] A computer storage medium for storing a computer program, which, when read by the computer, is executed by the computer using the method described thereon.

[0014] A computer, including a processor and a storage medium, executes the method when the processor reads a computer program stored in the storage medium.

[0015] A computer program product, which, as a computer program, implements the method when the computer program is executed.

[0016] Compared with the prior art, the advantages of the technical solution provided by the present invention are as follows: By mapping the spatiotemporal coordinates to a high-dimensional frequency space at the input end and concatenating them with the original input, high-frequency feature representation is introduced, which enables the model to significantly enhance its ability to represent high-frequency and small-scale structures in multi-scale turbulence. Compared with the frequency bias problem caused by traditional physical information neural networks modeling only based on low-dimensional coordinates, this feature can more accurately capture secondary flows and local drastic change regions, thereby improving the prediction accuracy of complex flow field details.

[0017] By integrating and accumulating local regions and then averaging to obtain regional differential features, the traditional differential form based on single-point differentiation is transformed into regional-scale differentiation. This effectively reduces the impact of single-point data noise on the differential results, making the physical constraints more stable and reliable. Compared with existing point differentiation methods, which are prone to high-frequency fluctuations and error amplification, this feature can more realistically reflect the local continuous change law of the flow field, thereby improving the physical consistency of the model in complex turbulence.

[0018] By constructing an encoding and decoding structure based on a self-attention mechanism and embedding fluctuation activation functions in each layer, the model can take into account both global dependencies and local nonlinear feature representations during the modeling process. Compared with the problem that traditional feedforward networks or convolutional networks cannot capture long-distance correlations and periodic fluctuations at the same time, this feature can improve the ability to jointly model the overall structure and local dynamics of complex flow fields, thereby improving the overall accuracy and stability of prediction.

[0019] By introducing a joint scoring mechanism based on physical residuals and flow field energy during the training process, samples are dynamically screened and updated, giving priority to key area samples with high residuals and high energy, while eliminating redundant samples. Compared with strategies that rely solely on residuals or random sampling, this feature enables adaptive redistribution of training data, significantly reducing computational resource consumption and improving training efficiency while ensuring prediction accuracy.

[0020] By mapping the decoded features to the physical quantity space and combining it with inverse normalization, a direct conversion from the feature space to the real physical space is achieved, enabling the prediction results to maintain consistency in physical dimensions and have interpretability. Compared with some methods that only output normalized results or require post-processing to restore physical meaning, this feature can improve the engineering usability and consistency of the model output.

[0021] By assigning differentiated weights to different output physical quantities to enhance the learning of secondary flow components, the supervision intensity among physical quantities in the multi-output prediction process is more balanced. Compared with the problem that secondary flow components are easily masked by dominant variables in existing methods, this feature can significantly improve the prediction accuracy of weak components, thereby improving the overall flow field prediction quality.

[0022] By constructing a composite loss function that includes supervised loss, continuity constraints, and momentum conservation constraints, and combining it with a training method that introduces physical constraints in stages, the model gradually transitions from data-driven to physics-driven, avoiding gradient instability caused by excessive physical constraints in the early stages of training. Compared with a training strategy that introduces all physical constraints at once, this feature can improve the stability of the training process and enhance the physical consistency of the final result.

[0023] By maintaining stable frequency mapping parameters during training and prediction, and combining the synergistic effect of dynamic sampling and regional differential constraints, the model can maintain low prediction error under different flow scenarios and input disturbance conditions. Compared with the existing models' insufficient cross-scenario generalization and noise resistance, this feature can significantly improve the robustness and engineering applicability of the model.

[0024] It is suitable for high-precision prediction and analysis of complex turbulent flow fields. Attached Figure Description

[0025] Figure 1 This is a schematic diagram of the overall architecture of the Resampled Region Differential Physical Neural Network (RARP-Net). Figure 2 This is a flowchart of the RE adaptive resampling module.

[0026] Figure 3 This is a visualization of the prediction results of RARP-Net for the uvp of the inlet water flow field in the chamber at time t=0.0 in the embodiment.

[0027] Figure 4 This is a visualization comparing the prediction results of different models for the uvp of the water inlet flow field in the compartment at time t=1.25 in the example.

[0028] Figure 5 This is a comparison chart of the relative L2 errors of the models in each experimental group on the flooded chamber dataset during the ablation experiment.

[0029] Figure 6 This is a graph showing the relative L2 error of the model's uvp prediction under different noise levels during the robustness experiment.

[0030] Figure 7 This is a visualization of the model's prediction results for the uvp of the water inflow field in the cabin at different noise levels at time t=0.76 during the robustness experiment. Detailed Implementation

[0031] To make the advantages and benefits of the technical solution provided by the present invention clearer, the technical solution provided by the present invention will now be described in further detail with reference to the accompanying drawings, specifically: Implementation Method 1: This implementation method provides a resampled region differential physics neural network for predicting complex turbulence, including: It includes a feature mapping structure, a region feature processing structure, an encoding / decoding structure, and an output mapping structure connected in sequence, and a sample filtering structure interactively connected to the encoding / decoding structure; The feature mapping structure is used to perform frequency mapping on the input spatial-temporal coordinates to obtain a high-dimensional feature representation; The region feature processing structure is used to perform region aggregation calculations based on the high-dimensional feature representation to obtain region-scale features and embed physical constraints to form a constraint feature representation. The sample selection structure is used to evaluate and select samples based on the constraint feature representation to obtain an optimized sample set; The encoding / decoding structure is used to perform feature modeling on the feature representation corresponding to the optimized sample set to obtain decoded features; The output mapping structure is used to map the decoded features to the flow field physical quantity space to obtain the flow field prediction value.

[0032] The feature mapping structure is used to transform the spatial-temporal coordinates through multi-frequency periodic mapping, concatenate them with the original input, and then perform a linear transformation to obtain a high-dimensional feature representation.

[0033] The regional feature processing structure is used to divide local regions according to proximity and to perform cumulative calculation and averaging of features within the regions to obtain regional scale features.

[0034] The sample selection structure is used to jointly evaluate and select key samples based on the degree of deviation from physical constraints and the energy intensity of the flow field to update the sample set.

[0035] The encoding and decoding structure is a multi-layer encoding and decoding network structure built based on a self-attention mechanism.

[0036] The encoding and decoding structure introduces a nonlinear activation method that adapts to the changing characteristics of the flow during the feature modeling process.

[0037] The method for predicting complex turbulence, based on the aforementioned neural network, includes: Scale the flow field data to obtain spatiotemporal feature data; Frequency mapping is performed on spatiotemporal feature data to obtain high-dimensional feature representations; Region aggregation calculation is performed based on high-dimensional feature representation, and physical constraints are embedded to obtain constraint feature representation; Sample selection is performed based on constraint feature representation to obtain an optimized sample set; Encode and decode the feature representations corresponding to the optimized sample set to obtain decoded features; The decoded features are mapped to the flow field physical quantity space to obtain the flow field prediction value.

[0038] A computer storage medium for storing a computer program, which, when read by the computer, is executed by the computer using the method described thereon.

[0039] A computer, including a processor and a storage medium, executes the method when the processor reads a computer program stored in the storage medium.

[0040] A computer program product, which, as a computer program, implements the method when the computer program is executed.

[0041] Implementation Method Two: This implementation method is a further detailed description of the technical solution provided in Implementation Method One, specifically: When implementing the resampling region differential physical neural network and method for predicting complex turbulence, the overall process is to build an end-to-end model flow based on the mapping of spatiotemporal coordinates to flow field physical quantities. The process unfolds in sequence around high-frequency feature enhancement, regional physical constraint embedding, dynamic screening of key samples, and multi-constraint collaborative training, forming a complete technical path from data processing to model training and then to prediction output.

[0042] Obtain a flow field dataset containing spatial and temporal coordinates and corresponding velocity and pressure components. Perform a uniform scale transformation on the input coordinates and output physical quantities to eliminate dimensional differences. Divide the data into training, validation, and test data according to a preset ratio, and output a standardized dataset with a regular structure for subsequent model training.

[0043] The neural network structure is constructed based on a standardized dataset, and the parameters of each layer are initialized. The weights of the linear mapping layer are initialized in a uniform distribution manner, and the parameters of the normalization layer are set with fixed initial values. At the same time, a random frequency matrix is ​​generated and kept unchanged during training. The initial state configuration of the network is completed and the model to be trained is output.

[0044] The spatial and temporal coordinates in the training data are input into the feature mapping unit. The original coordinates are transformed by constructing multiple sets of periodic change functions with different frequencies. The transformed high-frequency features are then concatenated with the original coordinates and unified to a fixed dimension through linear mapping. This yields a feature tensor containing high-frequency information, which is then output to the physical constraint processing stage.

[0045] The high-frequency feature tensor is divided into multiple local regions according to spatial or temporal proximity. The flow field changes are accumulated in each region to obtain the overall trend of regional changes. The accumulated results are then averaged to obtain the regional scale change features. These features are then embedded into the loss calculation process as physical constraint information, and the output is a feature representation containing physical constraint information.

[0046] Initialize the sample set based on the current training state and set an upper limit for the sampling size. During the training process, periodically extract candidate samples from the dataset. For each sample, calculate its error with the actual value, the degree of violation of physical constraints, and the energy intensity of the corresponding flow field. Then, weight and fuse the two to form a comprehensive score. Based on the score results, select high-value samples to update the sampling set and remove redundant samples, and output the optimized training sample set.

[0047] The feature tensors, which are embedded with physical constraints and have been sampled, are input into the encoding and decoding structure. The features are encoded layer by layer through a multi-layer self-attention mechanism to extract global correlation information. The features are then reconstructed and refined by combining the encoding results with the decoding structure. At the same time, nonlinear transformation functions that adapt to the wave characteristics are introduced into each layer to enhance the ability to express the periodic changes of the fluid. The decoded high-level semantic features are then output.

[0048] The decoded features are input into the output mapping unit, and the features are converted into the corresponding velocity components and pressure prediction results through multi-layer linear transformation. Based on the scale parameters obtained from the training phase, the prediction results are reversed to restore the scale, so that the output results are consistent with the real physical quantities, and the predicted values ​​of the flow field physical quantities are obtained.

[0049] Multiple loss metrics are constructed based on the difference between the predicted results and the true labels. Differentiated weights are set for different physical quantities to strengthen the learning of key components. At the same time, the model output is physically consistent with the continuity constraints and momentum conservation constraints. The loss weights are adjusted in a way that gradually transitions from data fitting to physical constraint dominance, thus completing the stable training process of the model and outputting the trained model parameters.

[0050] The spatial and temporal coordinates in the test data are input into the trained model. After feature enhancement, regional physical constraint embedding, encoding and decoding processing, and output mapping, the corresponding velocity components and pressure prediction results are obtained, thereby achieving high-precision prediction of complex turbulent flow fields.

[0051] Implementation Method 3, in conjunction with Appendix Figure 1-7 This embodiment describes the technical solution provided above in further detail through specific examples. Specifically: Example 1: Construction of a Differential Physical Neural Network for Resampling Regions This embodiment constructs a Resampled Region Differential Physics Neural Network (RARP-Net) for complex turbulence prediction. The overall architecture is as follows: Figure 1As shown, this is an end-to-end sequence modeling framework for flow field prediction tasks based on the two-dimensional incompressible Navier-Stokes (NS) equations. It achieves accurate mapping from the input spatial-temporal coordinates (x, y, t) to the output velocity components u, v and pressure p. It includes a Fourier feature embedding layer 1, a regional differential physical constraint module 2, an Encoder-Decoder backbone network (including an Encoder layer 3 and a Decoder layer 4), a RE adaptive resampling module 5, a physical constraint-guided output layer 6, a channel weighting module 7, and a loss function calculation module 8.

[0052] 1.1 Fourier Feature Embedding Layer 1 The purpose of this layer is to address the shortcomings of traditional Transformers in representing low-dimensional spatial coordinates and to enhance the high-frequency representation capability of the input. Based on an input tensor x with dimensions of batch (B) × sequence length (S) × coordinate dimension (x, y, t), a random frequency matrix following a normal distribution and multiplied by a scale factor γ = 10.0 is constructed. =16 is the number of Fourier characteristic frequencies), obtained through the formula:

[0053] After completing the feature mapping, the mapped high-frequency features are concatenated with the original input, outputting a feature tensor with a dimension of 3 + 2 × 16 = 35. This tensor is then transformed into the embedding dimension required by the Transformer backbone network through a linear layer. .

[0054] 1.2 Domain Differential Physical Constraint Module 2 This module is the core module that integrates physical laws and alleviates noise interference in traditional point differentiation. It adopts a "first integrate, then average" regional differentiation calculation strategy. The specific process is as follows: 1) Divide the flow field sequence data into several local regions according to the adjacency principle. The number of adjacent points in each region is adaptively adjusted according to the flow field sequence length and turbulence scale characteristics to ensure coverage of continuous flow structure.

[0055] 2) For each local region, calculate the integral value by combining the physical meaning of the Navier-Stokes equations, accumulate the uvp changes of all data points in the region, characterize the cumulative change characteristics of the flow field in the region, and filter out noise interference from individual data points.

[0056] 3) The integral results of each region are averaged to obtain the average integral value as the region's differential feature, which is then embedded into the loss function calculation module 8 to provide a physical supervision basis for network training that fits the actual flow field changes.

[0057] 1.3 Encoder-Decoder Backbone Network This network is built on the Transformer architecture and embeds the WaveAct wave activation function to adapt to the wave and periodic characteristics of fluid dynamics. The expression of the WaveAct function is as follows:

[0058] in, , As learnable parameters, they are deeply embedded into the FeedForward layer, Encoder layer 3, Decoder layer 4, and output layer 6.

[0059] The Encoder section consists of four stacked EncoderLayers. Each EncoderLayer follows the process of "layer normalization + WaveAct activation → self-attention module → residual connection → layer normalization + WaveAct activation → feedforward network → residual connection". The queries, keys, and values ​​in the self-attention module are all inputs to the Encoder itself. The feedforward network consists of three linear layers, with the intermediate layers having a dimensionality of [missing information]. 512, to achieve deep encoding of input features.

[0060] The Decoder section consists of four stacked DecoderLayers. The attention module's query is the Decoder's own input, with the key and value being the output features of the Encoder. The rest of the structure is consistent with the Encoder, enabling accurate decoding of encoded features and uncovering the mapping relationship between input features and flow field output.

[0061] 1.4 RE Adaptive Resampling Module 5 This module optimizes training resource allocation through a dynamic sampling mechanism of "residual-energy joint scoring," the process of which is as follows: Figure 2 As shown, the core hyperparameters are configured as follows: sampling budget `colloc_budget` = 2000, initial sample count `init_colloc` = 400, dynamic update interval `rar_interval` = 10, candidate pool size `rar_cand` = 3000, and encryption count `rar_topk` = 400. The specific steps are as follows: 1) Sampling initialization: 400 sequences are randomly selected from the training set as initial physical constraint samples.

[0062] 2) Candidate pool construction: Every 10 epochs, 3000 sequences are randomly selected from the training set as candidate samples.

[0063] 3) Residual-Energy Joint Scoring: Calculate the physical residual intensity res and flow field energy E for each candidate sample, using the scoring formula:

[0064] The sample score is obtained, where β=1.0 is the energy weight. It is the minimum value; Physical residual strength (res): measures the degree to which sequence samples violate the Navier-Stokes equation constraints, and is calculated as follows:

[0065] in It is the residual of the Navier-Stokes equations and momentum equations in the x / y directions, obtained by differentiating the velocity field output by the model. λc is the incompressible continuity equation residual, λc is the weighting coefficient of the continuity residual (set to 1.0 in this paper), B is the batch size, and S is the sequence length.

[0066] Flow field energy (E): Measures the kinetic energy density of the flow field corresponding to the sequence samples, reflecting the complexity of the flow field. It is calculated as follows:

[0067] , It is the physical domain velocity field after inverse normalization, used to avoid the impact of normalization on energy calculation.

[0068] 4) Sample update: Select the 400 highest-scoring sequences, merge them into the existing sample set, truncate them to the upper limit of the sampling budget of 2000 after deduplication, and rebuild the physical constraint sample loader.

[0069] 1.5 Physically Constrained Guided Output Layer 6 This layer maps the features output from Decoder layer 4 to the target physical quantity dimension. The specific process is as follows: The decoder's output features first pass through an output head consisting of three linear layers. The intermediate layers use the WaveAct activation function, ultimately mapping to a 3-dimensional feature tensor corresponding to the three physical quantities u, v, and p, yielding the flow field prediction in the normalized domain. Then, an inverse normalization function converts the predictions to the true values ​​in the physical domain, completing the mapping from the feature space to the physical space. The inverse normalization formula is:

[0070] in , The predicted values ​​are obtained from the training set statistics and converted into the true values ​​in the physical domain, thus completing the mapping from the feature space to the physical space.

[0071] 1.6 Weighting Module 7 and Loss Function Calculation Module 8 Channel weighting module 7 assigns differentiated supervisory weights to different output dimensions and sets... =1.0、 =2.0、 =1.0, enhancing the learning of lateral velocity v (secondary flow component); Loss function calculation module 8 designs a multi-constraint weighted composite loss function, mathematically in the form of:

[0072] in, Weighted monitoring of channel losses, For the residual loss of the continuity equation, For momentum conservation residual loss, , These are the weighting coefficients.

[0073] Example 2: A Complex Turbulence Prediction Method Based on RARP-Net (1) Step 1 Dataset Preprocessing 1) Dataset Acquisition: The 2D chamber flooding dataset was generated by STAR-CCM + fluid simulation software, containing spatial and temporal coordinates (x, y, t) at different times during the chamber flooding process, along with corresponding u, v, and p labels. Classic fluid datasets include 1D Durgers equations, 2D incompressible Navier-Stokes equations with external forces, and a cylinder flow dataset.

[0074] 2) Normalization: Normalize the input coordinates (x,y,t) and output physical quantities (u,v,p) of all datasets to eliminate the impact of dimensional differences on model training; 3) Dataset partitioning: All datasets are partitioned according to the ratio of training set: validation set: test set = 6:2:2, and the proportion of supervised sampling samples is set to 0.7.

[0075] (2) Step 2 Network Initialization The core configuration parameters are shown in Table 1:

[0076] Parameter initialization: Linear layers are initialized using Xavier uniform initialization, and the weights of the normalized layers are initialized to 1.0 and the biases are initialized to 0.0; the frequency matrix of the Fourier feature mapping module is initialized using a random normal distribution and remains fixed during training. Initialize the RE sampling module: Set the sampling budget to 2000, the initial number of samples to 400, the update interval to 10, the candidate pool size to 3000, and the number of encryptions to 400.

[0077] (3) Step 3: Fourier feature enhancement The spatiotemporal coordinates (x, y, t) of the training set are input into the Fourier feature embedding layer. The low-dimensional to high-dimensional mapping is completed according to the feature mapping formula in Example 1. After splicing the high-frequency features with the original input, the data is converted into a 64-dimensional feature tensor through a linear layer and output to the regional differential physical constraint module.

[0078] (4) Step 4: Embedding of regional differential physical constraints The 64-dimensional feature tensor is input into the regional differential physical constraint module, and the regional differential features are calculated according to the process of "integration first and then averaging". This feature is then embedded into the loss function calculation module to provide physical supervision constraints for network training.

[0079] (5) Step 5 RE adaptive resampling 400 sequences are randomly selected from the training set as initial physical constraint samples for calculating the physical loss during the first training. Every 10 epochs, the following operations are performed: candidate pool construction, residual-energy joint scoring, high-scoring sample selection, sampling set update, and data loader reconstruction. The specific process is as described in Example 1.4, which realizes dynamic selection of key samples and simplification of redundant samples.

[0080] (6) Step 6 Feature Encoding and Decoding The 64-dimensional feature tensor with embedded regional differential physical constraints is input into the Encoder-Decoder backbone network. It is then deep encoded by 4 Encoder layers to extract global and local features of the flow field. The encoded features are then accurately decoded by 4 Decoder layers, and the decoded feature tensor is output to the physical constraint-guided output layer.

[0081] (7) Step 7 Multi-constraint weighted loss training A composite loss function is constructed using the loss function calculation module, and the model is trained by combining the channel weighting module and the course learning strategy. The specific calculation formula is as follows: Channel weighted monitoring loss:

[0082] Physical residual loss: Calculating the residual loss of the continuity equation and momentum conservation residual loss

[0083]

[0084] Course learning strategy: In the early stage (Epoch<50), only optimize channel-weighted supervised loss. = The model first learns the basic spatiotemporal distribution characteristics of the flow field. In the intermediate stage (50 ≤ Epoch < 150), a continuity equation residual loss is introduced. = +0.1 Strengthen the mass conservation constraint. In the later stages (epoch ≥ 150), add momentum conservation residual losses. = +0.1 +0.01 It fully satisfies the physical constraints of the Navier-Stokes equations.

[0085] (8) Step 8 Flow field prediction The spatiotemporal coordinates (x, y, t) of the test set are input into the trained RARP-Net. After passing through Fourier feature enhancement, regional differential physical constraint embedding, and Encoder-Decoder encoding and decoding, the output layer guided by physical constraints is denormalized to obtain the physical domain prediction values ​​of the flow field velocity components u, v and pressure p.

[0086] Example 3: Model Performance Validation 3.1 Benchmark Experiment On a two-dimensional chamber flooding dataset, RARP-Net was compared with baseline models such as DeepONet, FNO, KAN, PINN, PINNformer, and TNO. The results are shown in Table 2. RARP-Net showed significantly lower L2 errors than other baseline models, with a 0.35% error in the u direction and a 0.72% error in the p direction. Only its error in the v direction (3.86%) was slightly higher than TNO, indicating the best overall performance. A visualization of the prediction at t=0.0 is also provided. Figure 3 The results show that the model's predicted values ​​are highly consistent with the actual values, with extremely small absolute errors; the comparison visualization at t=1.25 ( Figure 4 The results show that RARP-Net has significantly better prediction accuracy than other models in high gradient regions and secondary flow regions, and can accurately capture the detailed features of the flow field.

[0087]

[0088] 3.2 General applicability experiment Cross-scenario tests were conducted on 1D Burgers equations, 2D incompressible Navier-Stokes equations with external forces, flow around a cylinder, and a flooded compartment dataset. The results are shown in Table 3. RARP-Net's relative L2 error does not exceed 5% in all scenarios, with a maximum error of only 0.354% in the u-direction of the flooded compartment dataset, verifying the model's good cross-scenario generalization ability.

[0089]

[0090] 3.3 Ablation Experiment Ablation experiments were conducted on core modules / parameters such as region differentiation, RE sampling, Fourier feature mapping, and supervised sampling ratio. The results are as follows: Figure 5As shown: Replacing region differentiation with point differentiation significantly increases the model prediction error, with the largest increase in error in the v-direction. Replacing RE sampling with RAR sampling results in a slight increase in error. Decreasing the supervised sampling ratio from 0.7 to 0.3 / 0.5 leads to a gradual increase in error. Disabling the Fourier feature mapping module causes a significant increase in error, verifying the module's enhancement effect on high-frequency features.

[0091] Experiments have shown that each core module plays an irreplaceable role in the model's performance and exhibits a synergistic optimization effect.

[0092] 3.4 Robustness Experiment Global migration noise of levels 1 to 5 was applied to the spatiotemporal coordinates of the test set to test the model's robustness to interference. The results are as follows: Figure 6 , Figure 7 As shown, the model's relative L2 error increases slowly with increasing noise level. At the highest level, the error increases by only 0.35% in the u direction, 29.44% in the v direction, and 0.78% in the p direction, while still maintaining a low error. Visualization of flow field prediction at different noise levels ( Figure 7 The results show that the predicted flow field is highly consistent with the baseline flow field, with no obvious distortion, which verifies the good robustness of the model.

[0093] The above description of several specific embodiments further details the technical solution provided by the present invention in order to highlight the advantages and benefits of the technical solution provided by the present invention. However, the above-described specific embodiments are not intended to limit the present invention. Any reasonable modifications and improvements to the present invention, combinations of embodiments, and equivalent substitutions based on the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A resampling region differential physics neural network for predicting complex turbulence, characterized in that, include: It includes a feature mapping structure, a region feature processing structure, an encoding / decoding structure, and an output mapping structure connected in sequence, and a sample filtering structure interactively connected to the encoding / decoding structure; The feature mapping structure is used to perform frequency mapping on the input spatial-temporal coordinates to obtain a high-dimensional feature representation; The region feature processing structure is used to perform region aggregation calculations based on the high-dimensional feature representation to obtain region-scale features and embed physical constraints to form a constraint feature representation. The sample selection structure is used to evaluate and select samples based on the constraint feature representation to obtain an optimized sample set; The encoding / decoding structure is used to perform feature modeling on the feature representation corresponding to the optimized sample set to obtain decoded features; The output mapping structure is used to map the decoded features to the flow field physical quantity space to obtain the flow field prediction value.

2. The resampling region differential physics neural network for complex turbulence prediction according to claim 1, characterized in that, The feature mapping structure is used to transform the spatial-temporal coordinates through multi-frequency periodic mapping, concatenate them with the original input, and then perform a linear transformation to obtain a high-dimensional feature representation.

3. The resampling region differential physics neural network for predicting complex turbulence according to claim 1, characterized in that, The regional feature processing structure is used to divide local regions according to proximity and to perform cumulative calculation and averaging of features within the regions to obtain regional scale features.

4. The resampling region differential physics neural network for predicting complex turbulence according to claim 1, characterized in that, The sample selection structure is used to jointly evaluate and select key samples based on the degree of deviation from physical constraints and the energy intensity of the flow field to update the sample set.

5. The resampling region differential physics neural network for predicting complex turbulence according to claim 1, characterized in that, The encoding and decoding structure is a multi-layer encoding and decoding network structure built based on a self-attention mechanism.

6. The resampling region differential physics neural network for predicting complex turbulence according to claim 1, characterized in that, The encoding and decoding structure introduces a nonlinear activation method that adapts to the changing characteristics of the flow during the feature modeling process.

7. A method for predicting complex turbulence, characterized in that, The neural network implementation based on claim 1 includes: Scale the flow field data to obtain spatiotemporal feature data; Frequency mapping is performed on spatiotemporal feature data to obtain high-dimensional feature representations; Region aggregation calculations are performed based on high-dimensional feature representations, and physical constraints are embedded to obtain constraint feature representations. Sample selection is performed based on constraint feature representation to obtain an optimized sample set; The feature representations corresponding to the optimized sample set are encoded and decoded to obtain decoded features; The decoded features are mapped to the flow field physical quantity space to obtain the flow field prediction value.

8. A computer storage medium for storing computer programs, characterized in that, When the computer program is read by the computer, the computer executes the method of claim 7.

9. A computer, comprising a processor and a storage medium, characterized in that, When the processor reads the computer program stored in the storage medium, the computer executes the method of claim 7.

10. A computer program product, as a computer program, is characterized by: When the computer program is executed, it implements the method of claim 7.

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