Graph neural network ns partial differential equation solving method based on physical perception and cross-scale distillation

Abstract

This invention belongs to the field of partial differential equation solving, specifically disclosing a graph neural network method for solving NS partial differential equations based on physical perception and cross-scale distillation. The method includes the following steps: preparation and projection mapping of a multi-scale physical topology graph based on fluid dynamics; physical residual perception and cross-scale feature gating allocation; dual-track cross-distillation and dynamic evolution optimization; and outputting a prediction field of the joint solution of the partial differential equation after evolution iteration. This invention reduces extreme computational costs and grid overhead, overcoming the limitation of grid degree of freedom explosion faced by traditional numerical solutions when considering both macroscopic flow fields and microscopic high-frequency structures. It overcomes the "spectral bias" and slow global information transmission defects of traditional single-scale graph neural networks, avoiding the over-decoupling and accurate representation of high-frequency microscopic details (such as shock waves and boundary layer eddies) across frequency bands.

Description

Technical Field

[0001] This invention relates to the field of partial differential equation solving technology, specifically to a graph neural network method for solving NS partial differential equations based on physical perception and cross-scale distillation. Background Technology

[0002] In computational fluid dynamics and physical field analysis, the Navier-Stokes (NS) equations are often used to model the spatial distribution of fluid particles and their spatiotemporal evolution under complex boundary conditions to accurately describe them. However, the three-dimensional, compressible, and high-frequency unsteady characteristics of real fluids lead to significant nonlinear coupling and extremely high computational complexity. Therefore, physical assumptions are typically used to approximate the fluid state, thereby constructing numerically solvable incompressible fluid models. In specific applications (such as graph network benchmarks for flow around a cylinder, such as CYLINDERFLOW), these models are further reduced to two-dimensional Eulerian mesh models.

[0003] The incompressible approximation is a physical modeling method that treats fluid density as a constant throughout its spatiotemporal evolution. This method assumes that the fluid's Mach number is extremely low (typically less than 0.3) during motion, and that the rate of volume change of fluid particles under pressure is negligible. By eliminating the strong coupling interference of sound wave propagation to the flow field solution, this method significantly reduces the rigidity of the original dynamic model, enabling stable and efficient solutions to the spatial velocity and pressure distribution of the fluid while preserving key physical mechanisms such as viscous dissipation and eddy stripping.

[0004] Spatial discretization and dimensionality selection are the first steps in fluid modeling, requiring the determination of the computational grid's resolution and spatial dimensions based on the physical space's geometric topology. While high-resolution 3D grids can capture full-scale turbulence characteristics, their large node degrees of freedom result in an extremely heavy computational burden; conversely, 1D grids completely lose the nonlinear characteristics of fluid flow around and vortex street evolution. In AI fluid dynamics verification and practical engineering, fixed-topology 2D Eulerian grids are often chosen to capture the core sections of the flow field, achieving an optimal balance between computational power and physical complexity.

[0005] Traditional methods for solving the incompressible Navier-Stokes (NS) equations mainly include direct numerical simulation (DNS) based on fine grids and turbulence models based on macroscopic approximations (such as the Reynolds-averaged method (RANS) or large eddy simulation (LES)). The DNS method divides the fluid spatial domain into extremely fine grids (finite volume method or finite element method), and discretizes the nonlinear partial differential equations into a large system of algebraic equations on each grid cell, thus transforming the original continuum mechanics problem into a linear / nonlinear system that can be solved numerically iteratively. This method theoretically does not introduce any empirical assumptions and can accurately capture the details of the flow field at all scales, making it widely used in early fundamental fluid mechanics research. However, when dealing with high Reynolds numbers and complex geometric topologies, extremely small grid sizes are required to analyze the smallest-scale eddy dissipation, leading to an exponential expansion of the system's degrees of freedom and computational scale. This places enormous pressure on computational resources and memory storage, making it difficult to apply to rapid predictions at the engineering level.

[0006] Turbulence models based on macroscopic approximations (such as RANS) employ time averaging or spatial filtering to solve the mean flow field on a larger-scale grid. By introducing additional turbulent viscosity assumptions and semi-empirical formulas to close the equations, they can effectively reduce the nonlinear stiffness of partial differential equations and the number of solution variables, thus significantly reducing overall computational overhead. These methods improve computational efficiency while maintaining the accuracy of macroscopic engineering solutions and are widely applicable to steady-state macroscopic design of pipeline flows in aerospace and industrial applications. Their core problem lies in their heavy reliance on manually designed empirical parameters and the smoothing of high-frequency details within the grid. When faced with unsteady strong eddy stripping (such as the wake of a cylinder) or extreme boundary layer abrupt changes, reconstructing local microscopic physical information is extremely difficult, and the physical assumptions introduced by the model can easily lead to distorted flow field predictions and inaccurate energy dissipation estimates.

[0007] While traditional numerical fluid dynamics (CFD) methods have a well-established foundation in engineering practice, they still exhibit limitations in computational efficiency and reliance on experience when dealing with complex flow field problems involving high fidelity, multi-scale coupling, and strong nonlinearity. The increasing demands for high-precision real-time simulation, coupled with rising requirements for computational efficiency, the ability to distinguish local extrema, and the accuracy of spatiotemporal representation of flow fields, are driving the exploration of novel interdisciplinary solution methods. Deep learning technology, which integrates data-driven approaches with physical priors, is becoming a significant disruptive path for complex fluid dynamics modeling due to its powerful nonlinear mapping capabilities and adaptability to unstructured spaces.

[0008] In recent years, novel methods, represented by Physics-Informed Neural Networks (PINNs) and Graph Neural Networks (GNNs), have gained widespread attention in solving partial differential equations in fluid mechanics. These methods can not only handle complex irregular Eulerian / Lagrange grid data, but also guide model training by directly embedding conservation laws of the Navier-Stokes equations (such as divergence-free constraints and momentum balance residuals) into the network's loss function. This mechanism breaks the limitation of relying solely on massive amounts of high-fidelity flow field data, demonstrating considerable generalization potential in handling high-dimensional nonlinearity, complex boundary evolution, and sparse observation data. In the modeling of classic incompressible flow fields (such as steady-state and transient flow around CylinderFlow), existing physics-guided graph models have already achieved preliminary end-to-end velocity and pressure field predictions.

[0009] However, when faced with complex fluid evolution characterized by strong spatiotemporal coupling and the coexistence of multi-scale features (such as the coexistence of global low-frequency background flow and local high-frequency vortex shear shocks), traditional single-scale deep learning models begin to show serious shortcomings. In terms of topology, conventional graph networks primarily rely on message passing within local neighborhoods, resulting in extremely slow propagation of global pressure boundary conditions and low-frequency macroscopic dependencies within the dense mesh, easily leading to the "oversmoothing" problem. At the optimization level, due to inherent "spectral bias," networks tend to quickly fit the low-frequency, smooth regions of the flow field, while failing to accurately capture local high-frequency features in highly nonlinear, high-gradient regions (such as object boundary layers or stripping zones). This significant scale difference in the flow field leads to extreme gradient imbalance during backpropagation, causing the training process to be frequently affected by local high-frequency oscillations, easily causing model divergence or getting trapped in local optima.

[0010] Some improved existing methods attempt to alleviate the aforementioned training bottlenecks by introducing dynamic weights (such as adaptive residual weighting) or attention mechanisms into the loss function. However, when faced with the greater spatial heterogeneity and cross-frequency feature coupling in real-world strongly nonlinear flow fields, such simple loss-level optimization remains fragile. The computational constraints of a single high-resolution grid mean that the network's expressive power is insufficient to simultaneously consider both "global macroscopic topological priors" and "local microscopic high-frequency details," and the model's historical states in complex iterations are easily erased by the next round of drastic gradients, resulting in extremely poor overall solution robustness. To address these core pain points, there is an urgent need to introduce a novel graph neural network framework that possesses multi-scale decoupling capabilities in the topological space, physical-aware adaptiveness in computational allocation, and the ability to effectively integrate historical stable states and macroscopic priors, in order to effectively alleviate the bottleneck of solving highly challenging partial differential equations. Summary of the Invention

[0011] To address the problem that single-scale networks struggle to simultaneously consider both global macroscopic physical laws and local high-frequency details in solving complex partial differential equations (PDEs), and that computational costs are high, this invention provides a graph neural network-based method for solving NS partial differential equations based on physical perception and cross-scale distillation, thus solving the problems mentioned in the background section.

[0012] To achieve the above objectives, the present invention provides the following technical solution: a method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks, comprising the following steps: S1. Preparation and projection mapping of multi-scale physical topology graphs based on fluid dynamics; S2, Physical residual perception and cross-scale feature gating assignment; S3, dual-track cross-distillation and dynamic evolution optimization; S4. After evolution and iteration, the output partial differential equation simultaneous solution predicts the field.

[0013] Preferably, step S1 specifically includes the following: S11. Fine-grained physical space discretization and initial graph construction: First, the mesh is discretized within the physical computation domain to obtain the initial finest-scale physical space graph. ,in For a fine-grained set of nodes, Let be the set of edges connecting nodes; when dealing with incompressible fluids, the underlying physical evolution strictly follows the continuity equation and momentum equation, i.e., the two-dimensional incompressible Navier-Stokes governing equations to be solved in the multi-scale graph network, expressed as follows: ; ; ; in, , The flow field nodes are respectively at and Velocity in the direction; For pressure; The density is constant. S12, Physical Interaction Degree Calculation and Mesoscale Graph Coarsening: Combining Purely Geometric Adjacency Matrix By calculating the velocity gradient or momentum exchange difference between nodes and weighting the flow field characteristics, the "physical interaction degree" feature is precisely defined and calculated for each node. :

[0014] in, For inclusion direction and The velocity vector of directional velocity, This represents the corresponding velocity gradient tensor; Using the Top-K sampling strategy, according to Sort all nodes in descending order of size, and select the top 50% of nodes as the coarse graph node set. Constructing a medium-scale map ; S13. Generation of the coarsest-scale map: from the medium-scale map. A second sampling is performed based on the physical connectivity using the Top-K strategy, retaining 20% ​​to 25% of the core nodes to generate a coarsest-scale map containing the global low-frequency energy distribution. ; S14. Calculate and solidify the upsampling projection matrix: For adjacent levels, calculate the feature distance from the fine map node to its spatial nearest coarse map node, and generate a sparse upsampling projection matrix. Matrix elements The value is generated by combining the Gaussian kernel function with spatial distance weights:

[0015] In the formula, Representing nodes in a fine-scale diagram spatial coordinate vector, Represents nodes in a coarse-scale graph spatial coordinate vector, Represents a node Within the local neighborhood of the coarse-scale map, the first The spatial coordinate vector of each node It is a neighborhood index variable; To control the hyperparameters of receptive field smoothness, It is the set of local neighborhoods of nodes in the fine graph in the coarse graph; this projection matrix is ​​fixed after initialization and used to construct cross-scale constraint channels for physical world priors.

[0016] Preferably, step S2 specifically includes the following: S21. Low-frequency prior extraction: Scale map during forward propagation. The computation is initiated first, converging to generate a "macro-elite teacher" network that captures the macroscopic topological structure of the entire graph. Its output low-frequency macroscopic global view is projected through a matrix. Precise interpolation mapping back to fine mesh Forming spatial teacher prediction values ; S22. Calculation of local physical residuals: The fine mesh nodes interact according to the precise partial differential equations. The derivatives of the model output with respect to spatial coordinates and time are extracted through automatic differentiation technology to calculate the local physical residuals of each node. S23. Generate physical residual confidence gating: Introduce a spatial attention module consisting of two fully connected layers and a ReLU activation function to extract local features of nodes. Its gradient features Dynamically output node-specific sensitivity coefficients Combined with physical residuals ,node Dynamic confidence gating weights Strictly defined as: ; When the node is in a high-frequency region such as a shock wave, the residual becomes extremely large, leading to Forced disconnection of distillation constraints for autonomous exploration; when nodes are in a flat region, the residual is minimal, leading to This facilitates the smooth inheritance of macroscopic expressions within the network.

[0017] Preferably, step S3 specifically includes the following: S31. Constructing a "dual-track" elite renewal strategy and EMA evolution and renewal; S32. Node-aware dynamic cross-attention feature fusion and weight allocation; S33. Construction of the composite loss function; S34, Global Network Training.

[0018] Preferably, step S31 specifically includes: introducing a "dual-track" elite update strategy, which not only includes a "spatial projection teacher" that provides a stable spatial low-frequency overall view, but also specifically introduces an exponential moving average (EMA) mechanism to construct a "time trajectory teacher" in the time domain update; and using a cross-attention mechanism to dynamically weight and fuse the historically optimal EMA parameter state with the current network feature output in each round of backpropagation iteration.

[0019] Preferably, in the "spatial projection teacher", the teacher's characteristics Derived from the forward calculation of the upper-level multi-scale coarse map, and through the previously fixed projection matrix Mapped to the current detailed image node; In the "time trajectory teacher," this teacher does not participate in independent backpropagation training, but is used to provide temporal smoothing features. Its network parameters These are the current parameters based on the fine-scale student network. The result is dynamically smoothed and updated using the exponential moving average (EMA) mechanism.

[0020] Preferably, in step S32, a fine-grained feature-level cross-attention mechanism is introduced to calculate the fusion weights, and the specific calculation includes the following: First, the current detailed student network is mapped to nodes. The hidden features are represented as the basis of the query vector Query. At the same time, the two teachers will focus on the nodes. The output features are concatenated to construct a dimension of . Context matrix ; Subsequently, through a learnable weight matrix Perform a linear mapping to obtain the corresponding query vector. Key matrix Sum matrix : ; Next, the correlation score between the student node state and the dual-track teacher features is calculated using scaled dot product attention, and then normalized using the Softmax activation function to output the three-dimensional probability fusion weights for the two teachers:

[0021] in, The feature scaling factor is used to prevent the softmax function from entering the vanishing gradient region due to excessively large inner product values; the output... and These represent the confidence levels of the current node's demand for spatial and temporal teachers, respectively, and strictly satisfy the normalization condition. ; In step S32, the weight allocation specifically includes the following: when a node experiences a drastic physical residual change in the current Epoch, the characteristics of the student network... This will deviate from the stable manifold, at which point the attention mechanism will adaptively increase. The weighting of the time-stability state is forcibly increased. The dependence on the flow field helps to smooth out oscillations; conversely, if the nodes deviate from the conservation laws of the overall flow field, it will increase the risk of oscillations. The weight depends on the macroscopic constraints of space. To bring it back to the correct physical evolutionary trajectory.

[0022] Preferably, step S33 specifically includes the following: For fine-scale images any node on First, based on the extracted gating weights and attention fusion weights Constructing a dual-track distillation loss :

[0023] pass Identify the physical frequency band of the node, high frequency region low frequency region ; activate distillation constraints in the low-frequency region, and adaptively adjust the "spatial teacher prediction error" within the distillation process using cross-attention weights. Compared to "time teacher prediction error" Distribute the error correction computing power among them; Subsequently, boundary condition constraint loss to ensure physical objectivity is introduced. The penalty coefficient is calculated using the mean square error between the predicted node values ​​and the actual boundary values, and is fixedly amplified to 10 to 100 times that of other loss terms to impose a hard constraint; at the same time, a partial differential equation residual loss is introduced. Calculated The square mean; Finally, the three sets of mutually constraining loss terms are transformed into dynamic hyperparameters based on Epoch evolution. By fusing the data, a global dynamic composite total loss function is constructed. : ; In this composite loss function, the boundary loss Provides an absolute spatial reference; and It acts like a valve that evolves over time, dynamically adjusting the network during training to address the residual loss of partial differential equations. Whether to conduct a strong-constraint exploration or to investigate the losses in dual-track distillation Perform strong dependency inheritance.

[0024] Preferably, step S34 specifically includes the following: within 20,000 training cycles, the model uses the AdamW optimizer with weight decay characteristics for network optimization and scheduling, and combines cosine annealing with a hot restart mechanism to smoothly decay the learning rate from 1e-3 to 1e-6; during this process, the dynamic fusion coefficient α follows a smooth decay evolution trajectory, which is fixed at 0.95 in the first 10% of training cycles, at which time the network is in a "re-distillation, light physics" state; it decays exponentially smoothly from 10% to 60%, and stabilizes below 0.05 in the last 40% stage, at which time the network enters a "light distillation, heavy physics" state.

[0025] Preferably, step S4 specifically includes the following: In the specific iterative process, the network first calculates the dynamic gating weights through forward propagation in each Epoch. With cross attention weights Subsequently, the model is based on the constructed composite total loss. The back gradient is calculated and the backbone network parameters are updated. At the same time, the smoothing parameters of the time trajectory teacher are updated synchronously using the exponential moving average mechanism. The evolution process continues to iterate until the set maximum convergence cycle is reached, and finally the predicted field of the partial differential equation simultaneous solution with high robustness is output.

[0026] The beneficial effects of this invention are: 1) This invention introduces several key improvements to the graph neural network topology and physical guidance strategy, significantly enhancing the model's physical field representation and solution accuracy in extreme nonlinear, multi-scale coupling, and high-frequency feature mutation scenarios (such as complex fluid flow and shock wave evolution). By introducing a multi-resolution graph structure hierarchy based on Top-K sampling, the model effectively overcomes the limitations of traditional single-scale graph neural networks in terms of limited local receptive fields and "spectral bias," enabling rapid capture of global low-frequency macroscopic views with extremely low computational power consumption. Simultaneously, it creatively introduces an adaptive confidence gating mechanism based on physical residuals, allowing the network to dynamically adjust the cross-scale knowledge transfer path according to the magnitude of the residuals in local partial differential equations. It smoothly inherits macroscopic priors in the gentle low-frequency region and autonomously cuts off distillation and focuses on exploring microscopic details in the high-frequency region with dramatic gradient changes, thereby achieving efficient adaptive extraction of cross-frequency physical features and effectively alleviating the grid degree-of-freedom explosion and computational power bottleneck faced by traditional numerical solutions when balancing macroscopic flow fields and microscopic high-frequency details.

[0027] 2) Regarding model optimization and training evolution strategies, the method of this invention effectively solves the model oscillation and divergence collapse problems caused by drastic gradient fluctuations in high-frequency nonlinear regions during the solution of complex partial differential equations. During training, an innovative dual-track elite update strategy based on cross-attention is introduced. While utilizing the upper-level network to provide spatial low-frequency regularization, an exponential moving average (EMA) mechanism is combined to construct a "temporal trajectory teacher," smoothing the dynamic trajectory of network parameters over time and effectively mitigating gradient oscillations caused by multi-scale spatial coupling. Furthermore, a dynamic alpha fusion mechanism that evolves with the epoch is introduced into the total loss function, enabling a smooth transition from "re-distillation, light physics" in the early training stage to quickly overcome local minima, and then to "light distillation, heavy physics" in the later stage to accurately lock onto physical extrema. This comprehensive and collaborative design endows the network with strong anti-interference capabilities and adaptive optimization capabilities, significantly enhancing the model's generalization performance and convergence robustness under complex physical field distributions. Attached Figure Description

[0028] Figure 1 This is a schematic diagram of the flowchart for solving the NS partial differential equations using a graph neural network based on physical perception and cross-scale distillation. Figure 2This is a schematic diagram of a graph neural network architecture based on physical perception and cross-scale distillation. Detailed Implementation

[0029] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0030] In scientific computing, Physics-Informed Neural Networks (PINNs) embed the physical governing laws of partial differential equations (PDEs), such as mass conservation and momentum conservation, into the loss function of the neural network, guiding the model to strictly adhere to physical laws while approximating observed data. However, traditional PINNs often employ multilayer perceptrons (MLPs), which have limitations when dealing with engineering problems with complex geometric boundaries and unstructured meshes. To handle complex and irregular physical geometric domains, Graph Neural Networks (GNNs) have been introduced into solving PDEs. GNNs abstract the discretized physical space as a graph structure G=(V,E), where nodes V represent spatial coordinates or physical observation points, and edges E represent spatial adjacency relationships or physical interactions. When solving PINNs, GNNs utilize message passing mechanisms to update the feature representation of the central node by aggregating information from local neighboring nodes, thus naturally adapting to mesh data with arbitrary topologies and efficiently extracting the geometric and topological features of the physical space during network forward propagation.

[0031] Knowledge distillation (KD) is a classic deep learning model optimization and knowledge transfer technique. Its core idea is to construct a "teacher-student" learning paradigm: using soft labels or intermediate hidden features output by a "teacher model" with a large number of parameters, strong expressive power, or that has already been trained, to guide the training of a "student model" with fewer parameters or that is still being optimized. Compared to simply relying on hard labels, the soft labels provided by the teacher model contain the relative probability distribution between different categories or predicted values, providing the student model with richer hidden knowledge and regularization signals, thereby improving the student model's generalization ability and prediction accuracy.

[0032] In complex PDE solving tasks, the optimization space of the loss function often exhibits extreme nonconvexity due to the highly nonlinear physical residual term, making the model prone to getting trapped in local optima or oscillations. To address this, researchers introduced the "self-distillation" technique into the PINN solution process. Self-distillation refers to the network acting as both teacher and student without the need for an external pre-trained teacher model. In PINN applications, the model's favorable states in its training history (such as parameter snapshots from previous rounds) are typically used to generate supervisory signals to guide the network update in the current round. This self-distillation of historical trajectories provides a smooth anchor point for the current gradient optimization, effectively constraining the network's blind exploration in the complex physical gradient space, accelerating the model's convergence process, and improving the stability of numerical predictions.

[0033] This invention provides a technical solution: a graph neural network method for solving NS partial differential equations based on physical perception and cross-scale distillation, the overall process of which is as follows: Figure 1 As shown, the specific steps of the method include the following: S1. Preparation and Projection Mapping of Multi-Scale Physical Topology Graphs Based on Fluid Dynamics. This step aims to overcome the limitation of traditional pure geometric graph coarsening, which easily loses key physical evolution features. It deeply integrates physical priors to achieve frequency decoupling and spatial dimensionality reduction, specifically including the following: S11. Fine-grained physical space discretization and initial graph construction: First, the mesh is discretized within the physical computation domain to obtain the initial finest-scale physical space graph. ,in For a fine-grained set of nodes, This is the set of edges connecting nodes.

[0034] Based on the fluid continuum hypothesis, a set of generalized Navier-Stokes equations can be derived from the law of conservation of mass and Newton's second law. Assume the fluid velocity vector is... Then the general equations for unsteady fluid diffusion and convection can be expressed as: ; The first equation is the continuity equation, and the second equation is the momentum conservation equation.

[0035] To further simplify the above equations, relevant fluid parameters are defined to uniformly represent each physical quantity: The spatial velocity vector of the fluid; The pressure at each node in space; For fluid density; External volume forces (such as gravity) acting on fluid micro-elements; Let be the viscous stress tensor, which, for Newtonian fluids, is linearly related to the velocity gradient. The left-hand side of the momentum equation... The term represents nonlinear convective acceleration, which is the core factor inducing the complex multi-scale evolution of the flow field.

[0036] Although the 3D compressible NS model has extremely high physical fidelity, its computational and memory costs are extremely high in tasks such as multi-scale graph neural network training and nonlinear optimization. Therefore, this embodiment uses a 2D incompressible model to further simplify the computational domain, projecting the 3D spatial field onto... Two-dimensional plane. In two-dimensional incompressible models, the following physical approximation is often introduced to further simplify the solution: It is assumed that the fluid density is constant, i.e. Therefore, the mass conservation equation degenerates into the divergence-free condition of the velocity field ( Assuming the fluid is a Newtonian fluid with a constant viscosity, i.e., dynamic viscosity... The constant is assumed, and external volume forces are ignored. The dominant influence on the local cross-sectional flow field. Under the above assumptions, the velocity vector... Decomposed into directional velocity components and directional velocity components .

[0037] When dealing with incompressible fluids, the underlying physical evolution strictly follows the continuity equation and momentum equation, namely the two-dimensional incompressible Navier-Stokes governing equations to be solved in a multi-scale graph network, expressed as follows: ; ; ; Node features are defined by the aforementioned two-dimensional spatial coordinates. With initial physical state quantities (velocity) and pressure The structure consists of ( ). To ensure effective gradient propagation and avoid over-smoothing, the K-Nearest Neighbors (KNN) rule is used to construct edges, with the number of edges connected to a single node being ( ). The preferred setting is 20. , The flow field nodes are respectively at and Velocity in the direction; For pressure; The density is constant.

[0038] The three sets of partial differential equations described above together constitute the two-dimensional incompressible Navier-Stokes governing equations to be solved in the multi-scale graph network of this invention. In discretized graph structures such as Cylinderflow, the unknown physical quantities to be solved include those on all fluid mesh nodes. Shaft speed , Shaft speed and pressure .

[0039] S12, Physical Interaction Degree Calculation and Mesoscale Graph Coarsening: Combining Purely Geometric Adjacency Matrix By calculating the velocity gradient or momentum exchange difference between nodes and weighting the flow field characteristics, the "Physical Interaction Degree" feature is precisely defined and calculated for each node. :

[0040] in, For inclusion direction and The velocity vector of directional velocity, This represents the corresponding velocity gradient tensor; Using the Top-K sampling strategy, according to Sort all nodes in descending order of size, and select the top 50% of nodes as the coarse graph node set. Constructing a medium-scale map This measure ensures that critical and sensitive nodes located in high-frequency physical interaction regions such as eddy current stripping and boundary layer abrupt changes are prioritized for preservation.

[0041] S13. Generation of the coarsest-scale map: from the medium-scale map. A second sampling is performed based on the physical connectivity using the Top-K strategy, retaining 20% ​​to 25% of the core nodes to generate a coarsest-scale map containing the global low-frequency energy distribution. ; S14. Calculate and solidify the upsampling projection matrix: For adjacent levels, calculate the feature distance from the fine map node to its spatial nearest coarse map node, and generate a sparse upsampling projection matrix. Matrix elements The value is generated by combining the Gaussian kernel function with spatial distance weights:

[0042] In the formula, Representing nodes in a fine-scale diagram spatial coordinate vector, Represents nodes in a coarse-scale graph spatial coordinate vector, Represents a node Within the local neighborhood of the coarse-scale map, the first The spatial coordinate vector of each node It is a neighborhood index variable; To control the hyperparameters of receptive field smoothness, It is the set of local neighborhoods of nodes in the fine graph in the coarse graph; this projection matrix is ​​fixed after initialization and used to construct cross-scale constraint channels for physical world priors.

[0043] S2. Physical Residual Awareness and Cross-Scale Feature Gating Assignment. This step aims to address the "spectral bias" problem that is easily encountered in modeling complex physical behaviors, giving the model the ability to intelligently identify and decouple local physical characteristics. Specifically, it includes the following: S21. Low-frequency prior extraction: Scale map during forward propagation. The computation is initiated first, converging to generate a "macro-elite teacher" network that captures the macroscopic topological structure of the entire graph. Its output low-frequency macroscopic global view is projected through a matrix. Precise interpolation mapping back to fine mesh Forming spatial teacher prediction values ; S22. Calculation of local physical residuals: The fine mesh nodes interact according to the precise partial differential equations. The derivatives of the model output with respect to spatial coordinates and time are extracted through automatic differentiation technology to calculate the local physical residuals of each node. S23. Generate physical residual confidence gating: Introduce a spatial attention module consisting of two fully connected layers and a ReLU activation function to extract local features of nodes. Its gradient features Dynamically output node-specific sensitivity coefficients Combined with physical residuals ,node Dynamic confidence gating weights Strictly defined as: ; When the node is in a high-frequency region such as a shock wave, the residual becomes extremely large, leading to Forced disconnection of distillation constraints for autonomous exploration; when nodes are in a flat region, the residual is minimal, leading to This facilitates the smooth inheritance of macroscopic expressions within the network.

[0044] S3. Dual-track cross-distillation and dynamic evolution optimization. This step aims to smooth out network parameter oscillations caused by polar gradients in high-frequency physical regions (such as shock waves and turbulent edges), achieving deep interplay and adaptive adjustment between spatial macroscopic topological constraints and temporal historical evolutionary steady states. This invention provides a graph neural network architecture based on physical perception and cross-scale distillation, the network diagram of which is attached. Figure 2 As shown, a multi-resolution spatial topology hierarchy (e.g., coarsest, medium, and finest scales) is constructed by introducing graph coarsening techniques. Each scale focuses on expressing the macroscopic flow field or microscopic local physical behavior in different frequency bands. Based on this, an adaptive confidence gating mechanism based on physical residuals is creatively introduced. This gating module dynamically generates confidence weights based on the magnitude of the local PDE physical residuals calculated on the fine-grained grid, guiding the allocation of macroscopic prior knowledge transmitted from the upper-level coarse grid among the fine-graph nodes. This achieves adaptive matching and accurate decoupling modeling of different physical frequency band characteristics in complex spaces (e.g., the smooth low-frequency region and the shock-wave abrupt high-frequency region).

[0045] At the model evolution and parameter update level, considering that excessive reliance on local high-frequency physical gradients in the early stages of cross-scale joint training can easily lead to solution oscillations and network divergence, severely affecting the convergence process and prediction stability, this method further introduces a "dual-track" elite update strategy. This mechanism not only includes a "spatial projection teacher" that provides a stable low-frequency global view, but also specifically introduces an exponential moving average (EMA) mechanism to construct a "temporal trajectory teacher" in the temporal update. By using a cross-attention mechanism to dynamically fuse the historically optimal EMA parameter state with the current network feature output in each round of backpropagation iteration, the drastic gradient change trajectory caused by high-frequency mutations is greatly smoothed, ensuring the robustness of the entire multi-scale network under extreme physical conditions.

[0046] To enhance the model's adaptability to spatial heterogeneity and physical law exploration at different training stages, this method introduces a dynamic Alpha fusion control mechanism based on Epoch evolution in the construction of the total loss function. By pre-setting a fusion coefficient that evolves smoothly with the training process, the dependence of the network on teacher prior knowledge and micro-level autonomous exploration is quantified. Based on this dynamic parameter, the weights of the dual-track cross-distillation loss term and the PDE physics residual term in the total loss are adaptively adjusted. This mechanism enables the model to smoothly transition from the initial "re-distillation, light physics" (quickly traversing local minima) to the later "light distillation, heavy physics" (precisely capturing physical extrema), significantly improving the model's sensitivity to high-gradient nonlinear regions and the overall efficiency of solving the physics field.

[0047] This step specifically includes the following: S31. Construct a "dual-track" elite update strategy and EMA evolution update; In the backpropagation and parameter optimization path of the fine-scale student network, in order to avoid the "short-sighted" optimization and gradient collapse caused by relying solely on the current batch of data, the network is simultaneously controlled by two "teacher" models that provide elite prior knowledge.

[0048] Specifically, this includes: introducing a "dual-track" elite update strategy, which not only includes a "spatial projection teacher" that provides a stable, low-frequency overall view of space, but also introduces an exponential moving average (EMA) mechanism to construct a "time trajectory teacher" in the time domain update; and using a cross-attention mechanism to dynamically weight and fuse the historically optimal EMA parameter state with the current network's feature output in each round of backpropagation iteration.

[0049] The characteristics of the teacher in the "Spatial Teacher" concept. Derived from the forward calculation of the upper-level multi-scale coarse map, and through the previously fixed projection matrix Mapped to the current fine graph node; its core physical significance lies in providing a global spatial overview and low-frequency structural regularization for fine graph exploration.

[0050] In the aforementioned "Temporal Teacher," the teacher does not participate in independent backpropagation training but is instead used to provide temporal smoothing features. Its network parameters These are the current parameters based on the fine-scale student network. The result is dynamically smoothed and updated using the Exponential Moving Average (EMA) mechanism. Let... For the current training epoch, the EMA parameter update formula is as follows: ; in, The momentum decay rate is preferably set to [value missing] in this embodiment. Between. Through extremely high momentum hysteresis, the time trajectory teacher effectively filters out local gradient noise caused by high-frequency nonlinear regions, representing the most stable historical prior solution at this spatial scale.

[0051] S32. Node-aware dynamic cross-attention feature fusion and weight allocation; To enable the network to intelligently assess whether it relies more on "spatial big picture" or "historical stability" based on the transient physical environment (such as an oscillation amplification region or a stable convergence region) of different nodes, this embodiment introduces a fine-grained feature-level cross-attention mechanism to calculate the fusion weights. The specific calculation includes the following: First, the current detailed student network is mapped to nodes. The hidden features are represented as the basis of the query vector Query. At the same time, the two teachers will focus on the nodes. The output features are concatenated to construct a dimension of . Context matrix ; Subsequently, through a learnable weight matrix Perform a linear mapping to obtain the corresponding query vector. Key matrix Sum matrix : ; Next, Scaled Dot-Product Attention is used to calculate the correlation score between the student node state and the dual-track teacher features, and the probability is normalized using the Softmax activation function to output the three-dimensional probability fusion weights for the two teachers:

[0052] in, The feature scaling factor is used to prevent the softmax function from entering the vanishing gradient region due to excessively large inner product values; the output... and These represent the confidence levels of the current node's demand for spatial and temporal teachers, respectively, and strictly satisfy the normalization condition. .

[0053] The weight allocation specifically includes the following: when a node experiences a drastic physical residual mutation in the current Epoch, the characteristics of the student network... This will deviate from the stable manifold, at which point the attention mechanism will adaptively increase. The weighting of the time-stability state is forcibly increased. The dependence on the flow field helps to smooth out oscillations; conversely, if the nodes deviate from the conservation laws of the overall flow field, it will increase the risk of oscillations. The weight depends on the macroscopic constraints of space. This brings it back to the correct physical evolution trajectory. This node-level dynamic game enables adaptive optimization in the joint solution process of complex nonlinear flow fields.

[0054] S33. Construction of the composite loss function: This step provides a unified mathematical expression for the network's micro-level autonomous exploration, spatial global prior constraints, and temporal stability constraints. Specifically, it includes the following: For fine-scale images any node on First, based on the extracted gating weights and attention fusion weights Constructing a dual-track distillation loss :

[0055] pass Identify the physical frequency band of the node, high frequency region low frequency region ; activate distillation constraints in the low-frequency region, and adaptively adjust the "spatial teacher prediction error" within the distillation process using cross-attention weights. Compared to "time teacher prediction error" Distribute the error correction computing power among them; Subsequently, boundary condition constraint loss to ensure physical objectivity is introduced. The penalty coefficient is calculated using the mean square error between the predicted node values ​​and the actual boundary values, and is fixedly amplified to 10 to 100 times that of other loss terms to impose a hard constraint; at the same time, a partial differential equation residual loss is introduced. Calculated The square mean; Finally, the three sets of mutually constraining loss terms are transformed into dynamic hyperparameters based on Epoch evolution. By fusing the data, a global dynamic composite total loss function is constructed. : ; In this composite loss function, the boundary loss Provides an absolute spatial reference; and It acts like a valve that evolves over time, dynamically adjusting the network during training to address the residual loss of partial differential equations. Whether to conduct a strong-constraint exploration or to investigate the losses in dual-track distillation Perform strong dependency inheritance.

[0056] S34. Global Network Training. Specifically, this includes the following: Within a 20,000 training epochs, the model uses the AdamW optimizer with weight decay characteristics (1e-4) for network optimization and scheduling, combined with cosine annealing and a warm restart mechanism to smoothly decay the learning rate from 1e-3 to 1e-6. The warm restart mechanism allows the network to "jump out" of traps like minima or saddle points by instantly increasing the learning rate, continuing to explore better global solutions. During this process, the dynamic fusion coefficient α follows a smooth decay evolution trajectory, fixed at 0.95 in the first 10% of training epochs, where the network is in a "redistillation, light physics" state; it decays exponentially from 10% to 60%, and stabilizes below 0.05 in the last 40%, at which point the network transitions to a "light distillation, heavy physics" state. This fully releases computational power to the fine mesh for autonomous exploration of heavy physics. The above proportions and values ​​are for illustrative purposes only and can be adaptively adjusted according to the specific flow field Reynolds number and convergence situation.

[0057] S4. After evolution and iteration, the network outputs the prediction field of the simultaneous solution of the partial differential equations. Specifically, this includes the following: In each epoch, the network first calculates the dynamic gating weights through forward propagation. With cross attention weights Subsequently, the model is based on the constructed composite total loss. The back gradient is calculated and the backbone network parameters are updated. At the same time, the smoothing parameters of the time trajectory teacher are updated synchronously using the exponential moving average mechanism. The evolution process continues to iterate until the set maximum convergence cycle is reached, and finally the predicted field of the partial differential equation simultaneous solution with high robustness is output.

[0058] This invention effectively solves the model oscillation and divergence collapse problems caused by drastic gradient fluctuations in high-frequency nonlinear regions during the solution of complex partial differential equations. During training, it innovatively introduces a dual-track elite update strategy based on cross-attention. While utilizing the upper-level network to provide spatial low-frequency regularization, it combines an exponential moving average (EMA) mechanism to construct a "temporal trajectory teacher," smoothing the dynamic trajectory of network parameters over time and effectively mitigating gradient oscillations caused by multi-scale spatial coupling. Furthermore, a dynamic alpha fusion mechanism that evolves with the epoch is introduced into the total loss function, enabling a smooth transition from "re-distillation, light physics" in the early training stage to quickly overcome local minima, and then to "light distillation, heavy physics" in the later stage to accurately lock onto physical extrema. This comprehensive and collaborative design endows the network with strong anti-interference capabilities and adaptive optimization abilities, significantly enhancing the model's generalization performance and convergence robustness under complex physical field distributions.

[0059] It should be noted that, in this document, 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. Unless otherwise specified, 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.

[0060] The terminology used in the embodiments of this invention is for the purpose of describing particular embodiments only and is not intended to limit the invention. The singular forms “a,” “the,” and “the” as used in the embodiments of this invention and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise.

[0061] It should be understood that the term "and / or" used in this article is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, and B existing alone. Additionally, the character " / " in this article generally indicates that the preceding and following related objects have an "or" relationship.

[0062] Depending on the context, the word "if" as used here can be interpreted as "when," "when," "in response to determination," or "in response to detection." Similarly, depending on the context, the phrase "if determination" or "if detection (of the stated condition or event)" can be interpreted as "when determination," "in response to determination," "when detection (of the stated condition or event)," or "in response to detection (of the stated condition or event)."

[0063] Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A graph neural network NS partial differential equation solving method based on physical perception and cross-scale distillation, characterized in that, Includes the following steps: S1. Preparation and projection mapping of multi-scale physical topology graphs based on fluid dynamics; S2, Physical residual perception and cross-scale feature gating assignment; S3, dual-track cross-distillation and dynamic evolution optimization; S4. After evolution and iteration, the output partial differential equation simultaneous solution predicts the field.

2. The physical perception and cross-scale distillation based graph neural network NS partial differential equation solving method according to claim 1, characterized in that: Step S1 specifically includes the following: S11. Fine-grained physical space discretization and initial graph construction: First, the mesh is discretized within the physical computation domain to obtain the initial finest-scale physical space graph. ,in For a fine-grained set of nodes, Let be the set of edges connecting nodes; when dealing with incompressible fluids, the underlying physical evolution strictly follows the continuity equation and momentum equation, i.e., the two-dimensional incompressible Navier-Stokes governing equations to be solved in the multi-scale graph network, expressed as follows: ； ； ； in, , The flow field nodes are respectively at and Velocity in the direction; For pressure; The density is constant. S12, Physical Interaction Degree Calculation and Mesoscale Graph Coarsening: Combining Purely Geometric Adjacency Matrix By calculating the velocity gradient or momentum exchange difference between nodes and weighting the flow field characteristics, the "physical interaction degree" feature is precisely defined and calculated for each node. : ； in, For inclusion direction and The velocity vector of the direction of velocity. This represents the corresponding velocity gradient tensor; Using the Top-K sampling strategy, according to Sort all nodes in descending order of size, and select the top 50% of nodes as the coarse graph node set. Constructing a medium-scale map ; S13. Generation of the coarsest-scale map: from the medium-scale map. A second sampling is performed based on the physical connectivity using the Top-K strategy, retaining 20% ​​to 25% of the core nodes to generate a coarsest-scale map containing the global low-frequency energy distribution. ; S14. Calculate and solidify the upsampling projection matrix: For adjacent levels, calculate the feature distance from the fine map node to its spatial nearest coarse map node, and generate a sparse upsampling projection matrix. Matrix elements The value is generated by combining the Gaussian kernel function with spatial distance weights: ； In the formula, Representing nodes in a fine-scale diagram spatial coordinate vector, Represents nodes in a coarse-scale graph spatial coordinate vector, Represents a node Within the local neighborhood of the coarse-scale map, the first The spatial coordinate vector of each node For neighborhood index variables, To control the hyperparameters of receptive field smoothness, It is the set of local neighborhoods of nodes in the fine graph in the coarse graph; this projection matrix is ​​fixed after initialization and used to construct cross-scale constraint channels for physical world priors.

3. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 1, characterized in that: Step S2 specifically includes the following: S21. Low-frequency prior extraction: Scale map during forward propagation. The computation is initiated first, converging to generate a "macro-elite teacher" network that captures the macroscopic topological structure of the entire graph. Its output low-frequency macroscopic global view is projected through a matrix. Precise interpolation mapping back to fine mesh Forming spatial teacher prediction values ; S22. Calculation of local physical residuals: The fine mesh nodes interact according to the precise partial differential equations. The derivatives of the model output with respect to spatial coordinates and time are extracted through automatic differentiation technology to calculate the local physical residuals of each node. S23. Generate physical residual confidence gating: Introduce a spatial attention module consisting of two fully connected layers and a ReLU activation function to extract local features of nodes. Its gradient features Dynamically output node-specific sensitivity coefficients Combined with physical residuals ,node Dynamic confidence gating weights Strictly defined as: ; When the node is in a high-frequency region such as a shock wave, the residual becomes extremely large, leading to Forced disconnection of distillation constraints for autonomous exploration; when nodes are in a flat region, the residual is minimal, leading to This facilitates the smooth inheritance of macroscopic expressions within the network.

4. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 1, characterized in that: Step S3 specifically includes the following: S31. Constructing a "dual-track" elite renewal strategy and EMA evolution and renewal; S32. Node-aware dynamic cross-attention feature fusion and weight allocation; S33. Construction of the composite loss function; S34, Global Network Training.

5. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 4, characterized in that: In step S31, the specific steps include: introducing a "dual-track" elite update strategy, which not only includes a "spatial projection teacher" that provides a stable spatial low-frequency overall view, but also introduces an exponential moving average (EMA) mechanism to construct a "time trajectory teacher" in the time domain update; and using a cross-attention mechanism to dynamically fuse the historically optimal EMA parameter state with the current network feature output in each round of backpropagation iteration.

6. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 5, characterized in that: The characteristics of the teacher in the "spatial projection teacher" Derived from the forward calculation of the upper-level multi-scale coarse map, and through the previously fixed projection matrix Mapped to the current detailed image node; the "time trajectory teacher" does not participate in independent backpropagation training, but is used to provide temporal smoothing features. Its network parameters These are the current parameters based on the fine-scale student network. The result is dynamically smoothed and updated using the exponential moving average (EMA) mechanism.

7. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 4, characterized in that: In step S32, a fine-grained feature-level cross-attention mechanism is introduced to calculate the fusion weights. The specific calculation includes the following: First, the current detailed student network is mapped to nodes. The hidden features are represented as the basis of the query vector Query. At the same time, the two teachers will focus on the nodes. The output features are concatenated to construct a dimension of . Context matrix ; Subsequently, through a learnable weight matrix Perform a linear mapping to obtain the corresponding query vector. Key matrix Sum matrix : ； Next, the correlation score between the student node state and the dual-track teacher features is calculated using scaled dot product attention, and then normalized using the Softmax activation function to output the three-dimensional probability fusion weights for the two teachers: ； in, The feature scaling factor is used to prevent the softmax function from entering the vanishing gradient region due to excessively large inner product values; the output... and These represent the confidence levels of the current node's demand for spatial and temporal teachers, respectively, and strictly satisfy the normalization condition. ; In step S32, the weight allocation specifically includes the following: when a node experiences a drastic physical residual change in the current Epoch, the characteristics of the student network... This will deviate from the stable manifold, at which point the attention mechanism will adaptively increase. The weighting of the time-stability state is forcibly increased. The dependence on the flow field helps to smooth out oscillations; conversely, if the nodes deviate from the conservation laws of the overall flow field, it will increase the risk of oscillations. The weight depends on the macroscopic constraints of space. To bring it back to the correct physical evolutionary trajectory.

8. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 4, characterized in that: Step S33 specifically includes the following: For fine-scale images any node on First, based on the extracted gating weights and attention fusion weights Constructing a dual-track distillation loss : ； pass Identify the physical frequency band of the node, high frequency region low frequency region ; activate distillation constraints in the low-frequency region, and adaptively adjust the "spatial teacher prediction error" within the distillation process using cross-attention weights. Compared with "time teacher prediction error" Distribute the error correction computing power among them; Subsequently, boundary condition constraint loss to ensure physical objectivity is introduced. The penalty coefficient is calculated using the mean square error between the predicted node values ​​and the actual boundary values, and is fixedly amplified to 10 to 100 times that of other loss terms to impose a hard constraint; at the same time, a partial differential equation residual loss is introduced. Calculated The square mean; Finally, the three sets of mutually constraining loss terms are transformed into dynamic hyperparameters based on Epoch evolution. By fusing the data, a global dynamic composite total loss function is constructed. : ； In this composite loss function, the boundary loss Provides an absolute spatial reference; and It acts like a valve that evolves over time, dynamically adjusting the network during training to address the residual loss of partial differential equations. Whether to conduct a strong-constraint exploration or to investigate the losses in dual-track distillation Perform strong dependency inheritance.

9. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 4, characterized in that: In step S34, the specific steps include the following: Within a training cycle of 20,000 rounds, the model uses the AdamW optimizer with weight decay characteristics for network optimization and scheduling, and combines cosine annealing with a hot restart mechanism to smoothly decay the learning rate from 1e-3 to 1e-6; during this process, the dynamic fusion coefficient α follows a smooth decay evolution trajectory, which is fixed at 0.95 in the first 10% of training rounds, at which point the network is in a "re-distillation, light physics" state; it decays exponentially smoothly from 10% to 60%, and stabilizes below 0.05 in the last 40% of the training rounds, at which point the network transitions to a "light distillation, heavy physics" state.

10. The method for solving NS partial differential equations based on physical perception and cross-scale distillation using graph neural networks according to claim 1, characterized in that: In step S4, the specific steps include the following: In the specific iterative process, the network first calculates the dynamic gating weights through forward propagation in each Epoch. With cross attention weights ; Subsequently, the model is based on the constructed composite total loss. The back gradient is calculated and the backbone network parameters are updated. At the same time, the smoothing parameters of the time trajectory teacher are updated synchronously using the exponential moving average mechanism. The evolution process continues to iterate until the set maximum convergence cycle is reached, and finally the predicted field of the partial differential equation simultaneous solution with high robustness is output.

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