A multi-task partial differential equation solving method based on spectral-physical manifold learning and graph diffusion operator

Abstract

The application discloses a multi-task partial differential equation solving method based on spectral-physical manifold learning and a graph diffusion operator, obtains different geometric topologies of tasks (a calculation domain, a boundary condition and low-resolution sparse observation data); extracts spatial geometry, multi-scale dynamics and energy mode features, fuses the features into a hybrid spectral-physical embedding, constructs a task graph and derives a graph Laplacian matrix; selects a central prototype task based on a composite centrality; constructs a physical information neural network with parameter decoupling, divides parameters into a coding module of a shared invariant differential operator and an adaptive module adaptive to a geometric boundary; uses the graph diffusion operator to pass coding parameters of the prototype task to a target task along a task graph topology to initialize the network; after fine-tuning through a data fitting and a physical residual joint loss, high-resolution continuous physical field prediction is output; and the application is not dependent on a specific equation form and is suitable for fluid mechanics, heat and mass transfer, diffusion reaction and the like.

Description

Technical Field

[0001] This invention relates to the field of scientific machine learning and computational physics, specifically to a multi-task partial differential equation solution method based on spectral-physical manifold learning and graph diffusion operators. Background Technology

[0002] In recent years, Physical Information Neural Networks (PINNs) and neural operators have provided new technical approaches for constructing efficient physical surrogate models. However, in practical engineering applications, physical problems are often not single tasks, but rather a family of tasks governed by the same partial differential equations but with different complex geometries or boundary conditions. Although these tasks share invariant physical control laws at the underlying level, drastic changes in geometric topology can induce highly structured distortions in the mathematical solution space.

[0003] Existing physical information neural networks typically require retraining model parameters from scratch when faced with computational domains featuring new geometric topologies, failing to effectively utilize the potential physical structural connections between different tasks. On the other hand, while some deep transfer learning techniques or general neural operator methods proposed in recent years attempt to achieve physical knowledge sharing across different tasks, their cross-geometric generalization ability is severely limited because they fail to effectively decouple the "invariant differential operators" in the physical system from the "geometry-dependent local adaptive biases" at the mathematical level. Especially when traversing complex computational domains with drastically different topologies, forcibly mapping or transferring network parameters often triggers severe "negative transfer," causing predictions for the target task to violate physical laws or even leading to training failure.

[0004] Furthermore, in scenarios involving rapid iterative evaluation of complex industrial configurations, acquiring high-precision numerical simulations or high-resolution experimental data is extremely costly. Typically, only computational results based on low-resolution coarse meshes (e.g., only 20%-40% of the spatial degrees of freedom of high-fidelity fine meshes) or extremely sparse sensor sampling data can be obtained. Under such conditions of extreme data scarcity, relying entirely on sparse data from a single task to learn complex nonlinear multi-scale physical field distributions from scratch is mathematically a highly ill-posed inverse problem, making the model highly susceptible to getting trapped in local optima or producing severe numerical oscillations. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a multi-task partial differential equation solution method based on spectral-physical manifold learning and graph diffusion operators, which solves the problem that the same governing equations in complex fluid engineering systems need to be repeatedly subjected to expensive numerical calculations or model training under different geometric topological conditions.

[0006] Technical solution: The present invention provides a multi-task partial differential equation solution method based on spectral-physical manifold learning and graph diffusion operators, comprising the following steps:

[0007] Step 1: Use computational fluid dynamics to obtain multiple physical tasks with different geometric topologies. Each task includes its computational domain, boundary conditions, and sparse observation data.

[0008] Step 2: Extract the spatial geometric features, multi-scale dynamic features of the physical field, and dominant energy mode features for each task, and fuse the multi-source features to construct a hybrid spectrum-physical embedding representation for characterizing the intrinsic physical properties of the task;

[0009] Step 3: Based on the hybrid spectral-physical embedding representation, calculate the physical similarity between different tasks, and construct a spectral-physical manifold task graph that reflects the relationship structure between tasks based on the similarity, and then derive the corresponding graph Laplacian matrix.

[0010] Step 4: In the spectral-physical manifold task graph, considering both the local connectivity density and global connectivity importance of nodes, select the task node with the highest composite centrality metric as the central prototype task.

[0011] Step 5: Construct a physical information neural network with explicitly decoupled parameter space, wherein the physical information neural network parameters are divided into operator encoding module parameters for learning invariant differential operator features shared between different geometric topologies, and geometric adaptive module parameters for adapting to the geometric boundary conditions of a specific computational domain.

[0012] Step 6: Based on the graph Laplacian matrix, introduce a continuous graph diffusion operator. Starting from the network parameters of the central prototype task, along the topological path of the spectral-physical manifold task graph, transfer the parameters of the operator encoding module to the target task to be solved in a physically consistent manner, and complete the initialization of the target task network parameters.

[0013] Step 7: Fine-tune the initialized target task neural network using a joint loss function that includes data fitting terms and physical equation residual terms; output the high-resolution continuous physical field prediction results in the target task computational domain through the trained neural network.

[0014] Furthermore, in step 3, the construction of the hybrid spectrum-physical embedding representation is as follows: power spectral density analysis is performed on the geometric coordinates of the computational domain to extract geometric spectrum features; Fourier transform is performed on the physical field variables to extract multi-scale physical spectrum features; and intrinsic orthogonal decomposition is performed to extract dominant mode features reflecting the main energy of the flow field; the three features are spliced ​​or fused to form the hybrid spectrum-physical embedding representation.

[0015] Furthermore, in step 4, the construction of the spectral-physical solution manifold task graph is as follows: calculate the manifold distance between the mixed spectral-physical embedding vectors of different tasks, and convert the distance into similarity weights based on the kernel function, thereby constructing a graph structure with tasks as nodes and physical similarity as edge weights. The graph structure is used to represent the intrinsic physical correlation of different geometric configurations in the solution space of partial differential equations.

[0016] Furthermore, in step 4, the composite centrality metric is obtained by combining the degree centrality and eigenvector centrality of a node. The degree centrality is used to measure the representativeness of a node in its local neighborhood, while the eigenvector centrality is used to measure the connectivity importance of a node in the global manifold network. The central prototype task serves as a benchmark anchor point for the subsequent diffusion of physical knowledge to other tasks.

[0017] Furthermore, in step 5, in the decoupled physical information neural network, the operator encoding module contains network weights that are kept globally shared during cross-task migration to capture the operator features of the underlying partial differential equations that control the evolution of the physical system; the geometry adaptation module contains network weights that are learned independently for each specific task to adapt the shared operator features to the computational domain with specific geometric deformations to eliminate manifold distortion errors caused by geometric differences.

[0018] Furthermore, in step 6, the continuous graph diffusion operator realizes the propagation of parameter information by solving differential equations driven by the graph Laplacian matrix, so that the transmission strength of parameters between tasks is proportional to the spectral-physical similarity between tasks, thereby avoiding negative migration to tasks with mismatched physical characteristics.

[0019] Furthermore, during the fine-tuning training process, the sparse observation data used for supervised learning comes from the sampling results of a low-resolution coarse grid, which has fewer spatial degrees of freedom than the high-fidelity fine grid, thereby achieving high-fidelity reconstruction of the physical field under sparse data conditions.

[0020] The present invention discloses a multi-task partial differential equation solving system based on spectral-physical manifold learning and graph diffusion operators, comprising:

[0021] The task construction module is used to acquire and manage multiple physical tasks with different geometric topologies and their sparse observation data.

[0022] The hybrid embedding module is used to extract multi-scale geometric and physical spectral features for each task and construct a hybrid spectral-physical embedding representation of the task.

[0023] The manifold construction module is used to construct a spectral-physical understanding manifold task graph based on the similarity between task embedding representations and to compute its graph Laplacian matrix;

[0024] The central prototype selection module is used to select a central prototype task based on a composite centrality metric in the task graph.

[0025] The decoupled network building module is used to construct a physical information neural network that divides network parameters into a shared operator encoding module and an independent geometric adaptive module;

[0026] The graph diffusion propagation module is used to utilize the continuous graph diffusion operator to transfer physical knowledge and initialize network parameters along the task graph to the target task, starting from the central prototype task.

[0027] The physics constraint optimization module is used to fine-tune the initialized target task network based on the joint loss function and output high-fidelity continuous physics prediction results.

[0028] An electronic device according to the present invention includes a processor and a memory, wherein the memory stores a computer program, and when the processor executes the program, it implements any of the methods described above.

[0029] The present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements any of the methods described herein.

[0030] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0031] (1) This invention transforms multi-task learning into navigation on a spectral-physical manifold, and for the first time achieves effective decoupling between invariant differential operators and geometric dependence adaptation, which significantly improves the model’s generalization ability and small sample prediction accuracy when facing new geometric topologies.

[0032] (2) The graph diffusion operator is introduced to replace the traditional graph attention mechanism. The diffusion process driven by graph Laplace is used to realize the smooth transfer of parameters on the manifold, which effectively suppresses the numerical oscillation in the cross-geometric transfer process and ensures the consistency and stability of physical knowledge transfer.

[0033] (3) Through the hybrid spectrum-physical embedding representation, the mapping relationship between geometric distortion and multi-scale features of physical field can be accurately characterized, so that the model can achieve flow field reconstruction that is highly consistent with high-fidelity numerical simulation under the condition of using only low-resolution coarse grid data with a sampling rate of 20%-40%.

[0034] (4) In design space optimization scenarios such as complex industrial membrane modules, this invention can achieve efficient solutions across nearly a thousand heterogeneous geometric tasks. Compared with the traditional method of training from scratch for each task, it significantly shortens the computation convergence time and greatly reduces the computational overhead in engineering design, thus having extremely broad industrial application value.

[0035] (5) It has good versatility: The method of the present invention does not depend on the specific form of partial differential equations and can be applied to small sample modeling and prediction problems of various physical systems such as fluid mechanics, heat and mass transfer, and diffusion reaction. Attached Figure Description

[0036] Figure 1 This is a flowchart of the present invention;

[0037] Figure 2 This is a schematic diagram of the multi-geometric-domain partial differential equation task of the present invention;

[0038] Figure 3 This is a flowchart of the spectral-physical feature embedding construction process of the present invention;

[0039] Figure 4 This is the spectral-physical understanding manifold and task relationship diagram of the present invention;

[0040] Figure 5 This is the graph neural network diffusion transfer graph of the present invention;

[0041] Figure 6 This is a schematic diagram of the decomposed physical information neural network structure of the present invention. Detailed Implementation

[0042] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0043] like Figure 1 As shown, this embodiment of the invention provides a joint cognitive method for spectrum perception and prediction based on reciprocal learning, including the following steps:

[0044] Step 1: CFD computation constructs a task set containing complex heterogeneous geometries and sparse observation data. For computational domains with different topologies, sparse sampling data based on low-resolution coarse grids (e.g., spatial degrees of freedom only 20%-40% of fine grids) is acquired as the raw input for multi-task learning.

[0045] Step 2: Construct a hybrid spectrum-physical feature embedding representation of the task. By extracting the geometric spectrum, the dynamic spectrum of the physical field, and the dominant energy modes, the high-dimensional heterogeneous task is mapped into a low-dimensional feature vector in the latent space, thereby achieving a fine characterization of the physical task structure.

[0046] Step 3: Establish the spectral-physical solution manifold and construct the task association graph. Based on the similarity metric of embedded vectors, construct the manifold topology in the solution space to characterize the physical associations between different geometric configurations, and derive the corresponding graph Laplacian matrix.

[0047] Step 4: Construct a physical information neural network architecture based on mathematical decoupling. Design a decoupled network consisting of an "invariant differential operator encoding module" and a "geometric adaptive module" to achieve effective separation of physical law sharing and geometric deviation compensation from the bottom layer.

[0048] Step 5: Implement cross-task knowledge transfer based on graph diffusion operators. Utilize the continuous diffusion mechanism driven by graph Laplace to guide the physical knowledge of the central prototype task to smoothly diffuse along the solution manifold to the task to be solved, thus completing the efficient initialization of network parameters.

[0049] Step 6: Construct a joint physics constraint loss function and fine-tune the model. Combining graph diffusion priors and partial differential equation residual constraints, the neural network is driven to converge quickly under extremely sparse observation conditions, outputting high-fidelity continuous physics field prediction results.

[0050] Step 1: Figure 2 As shown, a set of multi-task partial differential equations containing heterogeneous geometric topologies and observation data are constructed.

[0051] This embodiment takes the design of the feed water baffle channel in a seawater desalination reverse osmosis membrane module as an example. To capture the influence of different geometric structures on the physical field distribution, a family of partial differential equations containing various three-dimensional topologies is first constructed, represented as:

[0052]

[0053] Where: T represents the complete set of the constructed task family; N represents the total number of tasks included. To ensure the learnability and learning stability of the geometric manifold, N is generally large (e.g., 970 cases in this example). k This represents the k-th specific physics prediction task.

[0054] For each physical task T in the set k They are governed by the same physical conservation laws within a specific geometric computational domain, and their mathematical description is as follows:

[0055]

[0056] And it conforms to specific computational domain boundary constraints:

[0057]

[0058] in: The input is a dimensionless three-dimensional spatial coordinate system, containing coordinates in three directions: (x, y, z). The three-dimensional geometric computation domain corresponds to the specific mesh topology of the k-th task; This represents the boundary of the computational domain; In this embodiment, the physical field variables that the neural network model needs to predict specifically include the velocity components and pressure fields in three dimensions. ;

[0059] In this embodiment, the differential operator governing the physical system corresponds to the incompressible Navier-Stokes equations. is the boundary condition operator, representing the non-slip solid wall boundary and the inlet / outlet conditions; and These are the source term and the known boundary term of the equation, respectively.

[0060] All subsequent operations are performed under dimensionless normalization conditions.

[0061] Step 2: Construct a hybrid spectral-physical feature embedding representation for the task, such as Figure 3 As shown, in order to accurately measure the physical similarity between different heterogeneous geometric tasks in the solution space of partial differential equations, this embodiment abandons the traditional distance metric that relies solely on geometric shapes or parameters, and innovatively constructs a hybrid spectrum-physical feature embedding representation that can simultaneously reflect spatial geometric complexity, multi-scale dynamic behavior, and flow field energy distribution. For any task T in the task set... i The specific construction process of its embedded representation includes the following four sub-steps:

[0062] (21) Geometric spectral feature extraction based on power spectral density: For the three-dimensional geometric spatial coordinate sequence of the i-th task computation domain, the discrete Fourier transform is applied to calculate its power spectral density in order to extract spatial geometric frequency features with translation invariance. The calculation formula is as follows:

[0063]

[0064]

[0065] Where: M is the total number of discrete sampling points in the geometric space; x i For the m-th discrete geometric coordinate point in the i-th task

[0066] k is the discrete spatial frequency index X i [k] represents the amplitude of the discrete Fourier transform of the coordinate sequence; PSD i [k] represents the calculated power spectral density.

[0067] In engineering implementation, 20 dominant frequency components are extracted from three directions, resulting in a dimension of R. (60) The geometric spectral feature vector of the task is obtained by extracting the first few dominant frequency components of the power spectral density. It is used to characterize the spatial structural features of flow channels or meshes.

[0068] (21) Physical dynamic spectrum feature extraction based on fast Fourier transform: In order to capture the multi-scale evolution structure of fluid under different geometric structures, fast Fourier transform is performed on the physical field variable sequence (such as velocity component and pressure) of the i-th task, and amplitude normalization is performed:

[0069]

[0070]

[0071] Where: f i [n] represents the observed value of the physical field variable at the nth sampling point; F i [k] is the complex frequency domain representation of the physical field;

[0072] N freq The number of effective frequencies to be retained; This represents the normalized relative energy spectrum amplitude. This normalization operation completely eliminates the differences in absolute physical amplitudes between different tasks, focusing on the structural characteristics of the relative energy distribution. In this experiment, the first 20 frequency components of each of the four physical variables are concatenated to obtain a total dimension of R. (80) Multiscale physical dynamic eigenvectors .

[0073] (22) Dominant energy mode feature extraction based on intrinsic orthogonal decomposition. In order to capture the data-driven coherent flow field structure that transcends fixed basis functions, the dominant energy mode feature extraction is performed on task T. i Flow field snapshot matrix A i Perform singular value decomposition, i.e. Calculate the energy percentage of each mode:

[0074]

[0075] Among them: A i s represents the physical field snapshot matrix constructed from observational data; ik S is a singular value matrix i The k-th singular value in;

[0076] K total E represents the rank or total number of singular values ​​of the snapshot matrix. i [k] represents the proportion of total energy accounted for by the k-th mode. The energy mode feature vector is constructed by extracting the top few dominant modes that cover the vast majority of the system's energy (in engineering implementation, the modal cutoff order K=10, which generally covers more than 90% of the system's total energy). .

[0077] (24) Construction of hybrid embedding vectors. The feature vectors extracted from different dimensions are directly concatenated and fused in the latent space to generate the final high-dimensional hybrid feature embedding vector:

[0078]

[0079] Where: z i Let be the final hybrid spectrum-physical embedding vector for the i-th task. This embedding vector mathematically provides a continuous and consistent coordinate system. Through this feature combination, physical tasks that differ greatly in geometric appearance but are similar in underlying flow transport mechanisms (such as boundary layer separation and large-scale vortices) will be close to each other in this implicit embedding space, thus laying the data foundation for the next step of accurately constructing the manifold topology in the solution space.

[0080] Step 3: As Figure 4 As shown, a spectral-physical manifold is established, and a task association graph and a central prototype anchor point are constructed. After completing the high-dimensional hybrid spectral-physical embedding of all tasks, this step aims to accurately measure the physical structure correlation between different heterogeneous geometric tasks in the latent space and discretize it into a graph structure topology, which serves as a discrete mathematical expression for the solution manifold of partial differential equations.

[0081] First, cosine similarity is calculated between embedding vectors from different tasks. This metric is mathematically equivalent to energy normalization of the feature vectors, effectively eliminating the influence of absolute energy scales between different tasks and focusing on the alignment of spectral-physical structure feature directions.

[0082]

[0083] in: For task T i With Task T j Spectral-physical similarity score between them; Z i With Z j The task T extracted in the previous step is respectively i With T j The mixed spectrum-physical embedding vector; ||·|| denotes the L2 norm of the vector.

[0084] Based on the calculated similarity, a task relationship graph representing the topology of the solution manifold is constructed. Where each node v in the vertex set v. i Corresponding to a specific physical task T i edge set This represents the connection relationships between tasks. The weight coefficients A in the adjacency matrix A of the graph. ij By analyzing the above similarity This is achieved by applying a Gaussian kernel function or a threshold truncation mapping, ensuring that strong connection weights exist only between tasks with high spectral-physical consistency. This graph topology enables tasks that differ greatly in geometric appearance but are similar in underlying fluid transport mechanisms to establish effective physical connections.

[0085] In multi-task neural network transfer learning, directly passing parameters in an unordered manner is highly susceptible to interference from the network's internal "parameter permutation symmetry," which can lead to negative transfer. To establish a physically consistent reference benchmark, a "central prototype task" needs to be selected in the task relationship graph.

[0086] This embodiment comprehensively considers the representativeness of a node in its local manifold neighborhood and the connectivity of the global manifold, and constructs a composite centrality metric for each task node:

[0087]

[0088] Wherein: C center (i) represents the composite centrality score of the i-th task node; C eig (i) represents the eigenvector centrality, used to capture the connectivity importance of nodes in the global manifold network, and it satisfies the eigenvalue equation. , Let C be the largest eigenvalue of the adjacency matrix A. eig (i) is the i-th component of the corresponding principal feature vector x; C deg (i) Degree centrality, used to reflect the task density of the node in its local neighborhood, is calculated as the sum of normalized adjacency weights: , where |v| is the total number of graph nodes; α is a weight adjustment coefficient used to balance local and global importance, with a value range of [0,1], which can be determined through few-sample Bayesian optimization in specific engineering implementations. By traversing and calculating the composite score of all nodes, the node with the highest score is selected as the central prototype task anchor point for the entire task family. Selected central prototype task i * This will serve as the absolute reference frame for the physically invariant operator encoder throughout the parameter transfer process. All subsequent graph diffusion processes will proceed along the solution manifold starting from this anchor point, ensuring from the algorithm's underlying layers that cross-geometric topology knowledge transfer unfolds along a path completely consistent with physical laws.

[0089] Step 4: Construct a multi-task physical information neural network architecture based on a mathematical decoupling mechanism. In traditional multi-task transfer in physical information neural networks, the network is usually treated as a single black box for overall parameter transfer or random initialization. This approach is highly susceptible to interference from the "parameter permutation symmetry" within deep neural networks. That is, functionally equivalent but misaligned network layers in the parameter space can cause destructive interference during fusion, leading to severe "negative transfer" phenomena.

[0090] To completely solve this problem, this embodiment constructs a physical information neural network architecture based on a mathematical decoupling mechanism. Specifically, the overall parameter space θ of the physical information neural network to be trained is strictly divided into two independent functional modules: .

[0091] Where: θ is the set of all weights and bias parameters of the neural network surrogate model; θ enc For the parameter set of the operator encoding module; θ adapt This is the parameter set for the geometry adaptive module.

[0092] These two modules perform completely different representation tasks in terms of physical mechanisms:

[0093] Operator encoding module θ enc (Physical anchor point):

[0094] This module corresponds to the deep feature extraction network in a neural network. Its core function is to learn and represent invariant differential operators shared throughout the entire multi-task family, which are independent of the geometric topology of the computational domain (i.e., corresponding to the fundamental partial differential equations governing fluid motion in step 1). This module eliminates the misalignment problem caused by parameter permutation, ensuring that the underlying model is always driven by consistent physical conservation laws regardless of how the macroscopic geometry is distorted.

[0095] Geometric Adaptive Module θ adapt (Deformation compensation):

[0096] This module corresponds to the shallow / output mapping network of a neural network. Its core function is to receive the latent physical features output by the operator encoding module and accurately map and adapt them to the geometric topology and boundary responses specific to the target task (i.e., corresponding to the variable boundary constraints in step 1). Since the distortion of the flow field solution space caused by changes in geometry is absorbed directionally by this module, it is designed to be highly tunable in cross-task migration.

[0097] Through the decoupled network architecture design described above, this invention perfectly maps the two major components of a partial differential equation system (i.e., invariant governing equations and variable boundary conditions) at the mathematical level. This structure not only provides a universal operator expression with a consistent physical benchmark for all heterogeneous tasks, but also lays the structural foundation for the next step of using "graph diffusion" for precise parameter allocation and knowledge transfer.

[0098] Step 5: As Figure 5 As shown, cross-task decoupling knowledge transfer and parameter initialization based on graph diffusion operators are implemented.

[0099] After constructing the decoupled physical information neural network and the spectral-physical understanding manifold, this step aims to progressively diffuse knowledge outward along the manifold topology, starting from the central prototype task, thereby providing physically consistent parameter initialization for all unseen heterogeneous geometric tasks. The specific diffusion process includes the following sub-steps:

[0100] (51) Central Prototype Anchor Point Initiation. First, for the central prototype task i selected in step 3... * It trains independently using its own sparse observation data without relying on any prior parameters until the physical loss function converges. The optimal network parameters after training are denoted as... Add this task to the set of trained tasks. In, that is It serves as the sole starting point for the diffusion of knowledge about the entire manifold.

[0101] (52) Progressive target task selection. To ensure the smoothness of knowledge diffusion and avoid crossing gaps with excessive physical differences, a progressive propagation strategy is adopted. In each round of diffusion, a target task is selected from the untrained task set that is compatible with the currently trained set. The task node i that is closest (most similar) in the spectral-physical latent space is taken as the current target task.

[0102] (53) Hard transfer of invariant operator encoder. Parameters of the operator encoding module for target task i. Since this module is responsible for representing the invariant differential operators shared by the entire task family, to ensure that the operator features do not produce semantic misalignment during transmission (parameter permutation symmetry problem), it performs an absolute copy (hard migration) from the nearest neighbor node:

[0103]

[0104] in: Let k be the initial parameters of the operator encoding module for target task i before fine-tuning training; k is the trained set. The nearest neighbor task node with the highest physical similarity to task i in the spectrum; The parameters are the optimal operator encoding module parameters for task k that have converged.

[0105] This operation forcibly anchors the target task i to a known functional basis that is consistent with physical laws.

[0106] Graph-guided soft diffusion of the geometry-adaptive module. Parameters of the geometry-adaptive module for target task i. Since this module needs to adapt to the unique geometric topology and boundary perturbations of the target task, this embodiment uses a graph attention-based diffusion operator to weightedly fuse the knowledge within its local neighborhood:

[0107]

[0108] Diffusion weight The calculation is based on the deep node embeddings updated by the graph neural network, and its calculation formula is as follows:

[0109]

[0110] in: Let N(i) be the initial parameters of the geometric adaptive module for target task i; N(i) is the set of neighbor nodes of task i in the task graph G. represents the optimal geometric adaptive module parameters that have converged for neighbor node j; m represents the index of trained nodes in the local neighborhood that participate in the weight normalization summation. These are the high-level solution manifold feature embeddings extracted from task nodes i, j, and m after graph Laplacian message passing. In the engineering implementation, the hybrid spectral-physical embedding vector extracted in step 2 is used as the initial input (i.e., The neighbor feature aggregation and non-linear update are performed layer by layer using the following formula: , where l is the current network layer number. It is a non-linear activation function. For learnable weight matrix, Let || denote the graph attention weights between node i and its neighbor node k (measuring the relative importance of neighbor features), || denotes vector concatenation, and finally outputs the features of the Lth layer. Used to calculate the diffusion weight; Indicates the inner product operation; A temperature scalar hyperparameter is used to control the diffusion smoothness. This soft diffusion mechanism is mathematically equivalent to linear interpolation in the local tangent space of the solution manifold, thus providing the target task with an extremely accurate geometric deformation prior.

[0111] After the combined allocation of hard migration and soft diffusion described above, target task i obtained complete initial network parameters. Then, it undergoes fine-tuning training with physical constraints (proceed to step 6), and after convergence, it is merged into the already trained set. In the middle. By continuously executing sub-steps (2) to (4), knowledge will spread steadily from the central anchor point along the connection path of the task relationship graph like ripples, until it covers and completes the parameter initialization of all massive heterogeneous geometric tasks.

[0112] Step 6: As Figure 6 As shown, a joint physical constraint loss function is constructed and fine-tuned using small samples. Well-optimized parameters for target task i are obtained through the graph diffusion operator. Next, this step aims to fine-tune the neural network using extremely sparse observation data combined with the physical constraints of partial differential equations, driving the model to converge to a physically consistent optimal solution. The following joint loss function is constructed:

[0113]

[0114] in: Let i be the total joint loss function for objective task i; This is the current set of decoupling parameters for the network;

[0115] These are adaptive weighting coefficients used to balance the data fitting term and the physical residual term. Data loss term. The formula used to quantify the error between the neural network prediction results and the low-resolution coarse-grid observation data obtained in step 1 is as follows:

[0116]

[0117] Where: M is the number of coarse grid sampling points; X i,j Let be the spatial coordinates of the j-th sparse observation point; This represents the physical field prediction value of the neural network at this coordinate. These correspond to the actual sparse observations. Physical loss term. This is used to force the neural network to satisfy the underlying physical control laws governing fluid motion throughout the entire computational domain, and it is calculated based on the residuals of partial differential equations at collocation points:

[0118]

[0119] Where: N f To calculate the number of unlabeled physical coordinates densely sampled within the computational domain of target task i; For differential operators, the partial differential equations representing the evolution of the control system are used. is the source term of the equation.

[0120] During the model parameter optimization phase, the results were diffused graphically. Starting from this point, a hybrid optimization strategy combining the Adam optimizer and the L-BFGS quasi-Newton method is used to iteratively minimize the loss function $\mathcal{L}(\theta_i)$.

[0121] Thanks to the "mathematical decoupling" design in step 4 and the "graph diffusion propagation" in step 5, the parameters of the operator encoding module in the neural network are now... The low-frequency dominant physical modes have been precisely aligned. Therefore, fine-tuning the calculated gradients... At this time, the optimization rigidity is significantly weakened, and parameter updates are mainly concentrated in the geometric adaptation module. The above is used to fit high-frequency boundary perturbations. This mechanism perfectly matches the "spectral bias" characteristic of deep neural networks, enabling the model to effectively suppress non-physical high-frequency numerical oscillations and avoid getting trapped in local optima, even under extremely small sample conditions with only 20% of the observed data.

[0122] Continue iterating until the gradient converges. Finally, the optimal neural network surrogate model is obtained after training. By inputting continuous spatial coordinates into this model, high-fidelity, multi-scale continuous physical field (such as velocity field, pressure field, concentration field, etc.) prediction results (far exceeding the resolution of the input data) can be output instantaneously under the target geometric configuration, thereby achieving a one to two-order-of-magnitude improvement in computational efficiency in the evaluation of massive configurations of the design space.

[0123] In industrial scenarios such as desalination membrane modules and heat exchanger channels, even minor changes in geometric configuration can lead to drastic distortions in the flow field distribution, causing traditional physical information neural networks to face severe generalization failures and negative transfer risks when transferring data across geometric topologies. This invention transforms multi-task learning into a navigation process on a spectral-physical manifold by constructing a hybrid spectral-physical embedding representation of the task. It effectively decouples invariant differential operators from geometrically dependent adaptive biases using mathematical methods and combines a graph diffusion mechanism to achieve stable transfer of physical knowledge across geometric topologies. This invention can significantly improve the reconstruction accuracy and training efficiency of models in heterogeneous geometric spaces under extremely sparse or low-resolution coarse-grid data conditions, providing efficient physical proxy model support for the rapid iterative design of complex industrial systems.

Claims

1. A multi-task partial differential equation solution method based on spectral-physical manifold learning and graph diffusion operators, characterized in that, Includes the following steps: Step 1: Obtain multiple physical tasks with different geometric topologies. Each task includes its computational domain, boundary conditions, and sparse observation data. The data was obtained through low-resolution numerical simulation calculations using a computational fluid dynamics solver. Step 2: Extract the spatial geometric features, multi-scale dynamic features of the physical field, and dominant energy mode features for each task, and fuse the multi-source features to construct a hybrid spectrum-physical embedding representation for characterizing the intrinsic physical properties of the task; Step 3: Based on the hybrid spectrum-physics embedding representation, calculate the physical similarity between different tasks, and construct a spectrum-physics manifold task graph reflecting the relationship structure between tasks based on the similarity, and then derive the corresponding graph Laplacian matrix; the construction of the hybrid spectrum-physics embedding representation is as follows: extract geometric spectrum features by performing power spectral density analysis on the geometric coordinates of the computational domain; extract multi-scale physical spectrum features by performing Fourier transform on the physical field variables; and extract the dominant mode features reflecting the main energy of the flow field by performing intrinsic orthogonal decomposition. The three features are spliced ​​or fused to form a hybrid spectrum-physical embedding representation; Step 4: In the spectral-physical manifold task graph, considering both the local connectivity density and global connectivity importance of nodes, select the task node with the highest composite centrality metric as the central prototype task. The construction of the spectral-physical manifold task graph is as follows: calculate the manifold distance between the mixed spectral-physical embedding vectors of different tasks, and convert the distance into similarity weights based on the kernel function, thereby constructing a graph structure with tasks as nodes and physical similarity as edge weights. The graph structure is used to represent the intrinsic physical connections of different geometric configurations in the solution space of partial differential equations. The composite centrality metric is obtained by combining the degree centrality and eigenvector centrality of nodes. The degree centrality is used to measure the representativeness of a node in its local neighborhood, and the eigenvector centrality is used to measure the connectivity importance of a node in the global manifold network. The central prototype task serves as a benchmark anchor point for the subsequent diffusion of physical knowledge to other tasks. Step 5: Construct a physical information neural network with explicitly decoupled parameter spaces. The parameters of the physical information neural network include operator encoding module parameters and geometric adaptation module parameters. The operator encoding module parameters are used to learn invariant differential operator features shared across different geometric topologies. The geometric adaptation module parameters are used to adapt to the geometric boundary conditions of the computational domain. In the decoupled physical information neural network, the operator encoding module contains network weights that are globally shared during cross-task transfer, used to capture the operator features of the fundamental partial differential equations that control the evolution of the physical system. The geometric adaptation module contains network weights that are learned independently for each task, used to adapt the shared operator features to the computational domain with geometric deformation to eliminate manifold distortion errors caused by geometric differences. Step 6: Introducing a continuous graph diffusion operator based on the graph Laplacian matrix, starting with the network parameters of the central prototype task, the parameters of the operator encoding module are transferred to the target task in a physically consistent manner along the topological path of the spectral-physical manifold task graph, thus initializing the network parameters of the target task. The continuous graph diffusion operator achieves parameter information propagation by solving differential equations driven by the graph Laplacian matrix, making the transmission strength of parameters between tasks proportional to the spectral-physical similarity between tasks, thereby avoiding negative migration to tasks with mismatched physical characteristics. The diffusion process is as follows: (61) Central Prototype Anchor Point Initiation; First, for the selected central prototype task i * It trains independently using its own sparse observation data without relying on any prior parameters until the physical loss function converges; the optimal network parameters after training are denoted as... Add the task to the set of trained tasks. In, that is , serving as the sole starting point for the diffusion of knowledge across the entire manifold; (62) Progressive target task selection; adopting a progressive propagation strategy; in each round of propagation, selecting from the untrained task set and the currently trained task set. The task node i that is closest in the spectral-physical latent space is taken as the current target task. (63) Hard transfer of invariant operator encoder; Operator encoding module parameters for target task i Perform an absolute hard migration from the nearest neighbor node. ; in, Let k be the initial parameters of the operator encoding module for target task i before fine-tuning training; k is the trained set. The nearest neighbor task node with the highest physical similarity to task i in the spectrum; The parameters of the optimal operator encoding module for task k have converged; Graph-guided soft diffusion of the geometry-adaptive module; Parameters of the geometry-adaptive module for target task i. A graph attention-based diffusion operator is used to weightedly fuse knowledge within its local neighborhood: ； Diffusion weight The calculation is based on the deep node embeddings updated by the graph neural network, and its calculation formula is as follows: ； in, Let N(i) be the initial parameters of the geometric adaptive module for target task i; N(i) is the set of neighbor nodes of task i in the task graph G. represents the optimal geometric adaptive module parameters that have converged for neighbor node j; m represents the index of trained nodes in the local neighborhood that participate in the weight normalization summation. These are the high-level solution manifold feature embeddings extracted from task nodes i, j, and m after graph Laplacian message passing, respectively. The extracted mixed spectral-physical embedding vector is used as the initial input. The neighbor feature aggregation and non-linear update are performed layer by layer using the following formula: , where l is the current network layer number. It is a non-linear activation function. For learnable weight matrix, Let || denote the graph attention weights between node i and its neighbor node k, and || denote vector concatenation. The final output is the feature of the Lth layer. Used to calculate diffusion weights; Indicates the inner product operation; Temperature scalar hyperparameter for controlling diffusion smoothness; After a combination of hard migration and soft diffusion, target task i obtained complete initial network parameters. ; continuously execute sub-steps (62) to (63), steadily spreading from the central anchor point along the connection path of the task relationship graph until covering and completing the parameter initialization of all massive heterogeneous geometric tasks; Step 7: Fine-tune the initialized target task neural network using a joint loss function that includes data fitting terms and physical equation residual terms; output the high-resolution continuous physical field prediction results in the target task computational domain through the trained neural network.

2. The multi-task partial differential equation solution method based on spectral-physical manifold learning and graph diffusion operator according to claim 1, characterized in that, During the fine-tuning training process, the sparse observation data used for supervised learning comes from the sampling results of a low-resolution coarse grid, which has fewer spatial degrees of freedom than a high-fidelity fine grid, thereby achieving high-fidelity reconstruction of the physical field under sparse data conditions.

3. A multi-task partial differential equation solving system based on spectral-physical manifold learning and graph diffusion operators, implemented using the method described in any one of claims 1-2, characterized in that, include: The task construction module is used to acquire and manage multiple physical tasks with different geometric topologies and their sparse observation data. The hybrid embedding module is used to extract multi-scale geometric and physical spectral features for each task and construct a hybrid spectral-physical embedding representation of the task. The manifold construction module is used to construct a spectral-physical understanding manifold task graph based on the similarity between task embedding representations and to compute its graph Laplacian matrix; The central prototype selection module is used to select a central prototype task based on a composite centrality metric in the task graph. The decoupled network building module is used to construct a physical information neural network that divides network parameters into a shared operator encoding module and an independent geometric adaptive module; The graph diffusion propagation module is used to utilize the continuous graph diffusion operator to transfer physical knowledge and initialize network parameters along the task graph to the target task, starting from the central prototype task. The physics constraint optimization module is used to fine-tune the initialized target task network based on the joint loss function and output high-fidelity continuous physics prediction results.

4. An electronic device comprising a processor and a memory, characterized in that, The memory stores a computer program, which, when executed by the processor, implements the method as described in any one of claims 1-2.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the method as described in any one of claims 1-2.

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