Irregular grid engineering simulation method and system based on laplace neural operator

Abstract

The application discloses an irregular grid engineering simulation method and system based on a Laplace neural operator, and belongs to the technical field of engineering simulation, and solves the problems of traditional neural operators depending on regular grids, large interpolation error of irregular grid scenes, and low solving efficiency of multi-working-condition engineering simulation. The application constructs a graph Laplace matrix based on an irregular grid, obtains a geometric self-adaptive Laplace spectral basis through characteristic decomposition, replaces traditional Fourier bases to realize spectral domain transformation and global physical coupling modeling; further, geometric spectral bases and working-condition spectral bases are set to realize double-branch weighted fusion, a Laplace spectral basis neural operator network is constructed in combination with a local linear branch, and after offline training, the network can quickly output full-field physical quantities online. The application does not need regular grid interpolation, can retain geometric topology and boundary information, has excellent prediction accuracy and generalization, and can be widely applied to multi-physical field rapid simulation and design optimization of burners, chemical reactors, complex pipelines and other equipment.

Description

Technical Field

[0001] This invention belongs to the field of engineering simulation technology, specifically relating to an irregular mesh engineering simulation method and system based on the Laplace neural operator. Background Technology

[0002] Engineering simulation, as a tool for equipment development, process optimization, and performance analysis, is widely used in fields such as fluid mechanics, combustion dynamics, chemical mass and heat transfer, structural mechanics, and material microstructure response. The solutions are often governed by partial differential equations (PDEs), such as the Navier-Stokes equations for incompressible fluids, the Euler equations for transonic airfoil flow, the convection-diffusion-reaction equations for chemical reactors, the Darcy flow equation for porous media, heat conduction equations, and structural elastic / plastic constitutive equations. While traditional finite element methods, finite volume methods, finite difference methods, and classical spectral methods can obtain high-fidelity numerical solutions, in engineering scenarios such as multi-condition parameter scanning, structural geometry optimization, real-time digital twins, and uncertainty analysis, repeated iterative solutions are required for different geometries and operating conditions. This results in long computation cycles and high computational costs, making it difficult to meet the needs of rapid design and real-time analysis.

[0003] To improve the efficiency of engineering simulation solutions, deep learning-based proxy modeling techniques have rapidly developed. Among them, neural operator methods learn the mapping relationship from the input function space to the output function space. After training, only one forward inference is needed to approximate the output of a high-fidelity numerical solver, and it can output complete physical quantities across the entire field. Compared with traditional neural networks based on scalar prediction, it is more suitable for the proxy requirements of full-field simulation. Among various neural operator schemes, the Fourier Neural Operator (FNO) learns the frequency domain transformation matrix on truncated low-frequency modes by performing a Fast Fourier Transform (FFT) on a regular grid, and then maps it back to the physical space through an Inverse Fourier Transform. This can efficiently approximate PDE solvers with global coupling characteristics, achieving significant acceleration effects in regular grid scenarios such as the Burgers equation, Darcy flow, and Navier-Stokes equation.

[0004] However, the computational efficiency of standard FNO is highly dependent on regular rectangular tensor meshes and FFT operations. Real-world engineering simulation objects typically have complex geometric boundaries and non-uniform discrete meshes, such as burner cavities, internal components of chemical reactors, curved pipes, airfoil surfaces, perforated structural components, and heterogeneous material unit cells. Their discretization results are often triangular meshes, tetrahedral meshes, boundary layer refined meshes, curved surface meshes, or point cloud data. The node distribution and mesh topology are determined by both geometric boundaries and physical field gradients, and do not satisfy the constraints of regular meshes. Forcibly interpolating irregular mesh data into regular rectangular meshes before using FNO for solving introduces significant interpolation errors, resulting in the loss of boundary geometric details, amplified errors in local high-gradient regions, and difficulty in accurately characterizing the physical field properties of key areas such as walls, material interfaces, and reaction zones. This directly reduces the prediction accuracy and engineering usability of the surrogate model.

[0005] To address the adaptation problem of Graph Neural Networks (FNOs) in irregular geometric scenarios, existing technologies have proposed various improvement schemes. Among them, Geo-FNO, a representative approach, learns the coordinate deformation mapping from the physical domain to the regular latent domain and performs FFT operations in the latent domain, which alleviates the adaptation problem for complex geometries to some extent. However, this scheme requires the additional construction of coordinate deformation mapping relationships. For engineering scenarios with complex topologies, irregular boundary heights, and significant influence of mesh connectivity on the solution field, coordinate deformation itself becomes a new modeling challenge, and it is difficult to fully utilize the topological connectivity and boundary structure information of the original mesh. In addition, simulation proxy schemes based on graph neural networks mostly adopt local message passing mechanisms. Although they can directly adapt to irregular meshes, their modeling capability for long-range physical coupling is limited, making it difficult to efficiently approximate PDE operators with global integral properties. This results in accuracy bottlenecks in large-scale, strongly coupled engineering physical field predictions.

[0006] In summary, existing neural operator proxy solution techniques are difficult to directly adapt to the irregular meshes and complex geometries that are widely present in engineering. They are insufficient in terms of global physical coupling modeling capabilities, geometric information utilization efficiency, and complex topology generalization ability, and cannot fully meet the high-precision and high-efficiency full-field simulation requirements of objects such as burners, chemical reactors, complex pipelines, porous media, fluid machinery, and material components. Summary of the Invention

[0007] To address these issues, this invention provides an irregular mesh engineering simulation method and system based on the Laplacian neural operator. This method solves the problems of existing neural operators relying on regular meshes, resulting in large interpolation errors, loss of boundary geometric information, insufficient accuracy in global physical coupling modeling, and poor prediction accuracy, generalization stability, and engineering deployment applicability in complex geometric irregular mesh engineering simulation scenarios.

[0008] To achieve the above objectives, the present invention provides the following technical solution: an irregular mesh engineering simulation method based on the Laplace neural operator, used for the analysis of pressure field, velocity field, temperature field, concentration field, stress field and displacement field of burners, chemical reactors, complex pipelines, porous media, fluid machinery and material components, including an offline training stage and an online inference stage; The offline training phase includes the following steps: S1. Collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete grid data of the corresponding engineering object, working condition parameter fields, and real physical field obtained by a high-fidelity numerical solver. S2. Based on the node connection relationships, spatial distances and boundary features of the irregular discrete grid data, construct a corresponding graph structure, construct a graph Laplacian matrix according to the graph structure, and perform eigenvalue decomposition on the graph Laplacian matrix to obtain a set of Laplacian eigenvectors that are adapted to the irregular grid geometry, which serve as the basis for spectral domain transformation. S3. Construct a Laplace spectral basis neural operator network. After mapping the node-level input features to a high-dimensional feature space through the input encoding layer, input at least one Laplace spectral domain operator layer. The spectral domain operator layer performs spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis to realize global physical coupling modeling on irregular grids. After mapping through the output decoding layer, the predicted physical field is obtained. S4. Using the error between the predicted physical field and the real physical field as the training objective, iteratively optimize all learnable parameters of the Laplace spectral-based neural operator network to obtain the trained surrogate solution model. The online inference stage includes: receiving the target irregular grid data and target operating condition parameters to be predicted, constructing the graph structure and calculating the Laplace spectral basis, inputting it into the surrogate solution model to perform one forward inference, and outputting the prediction results of the full field physical quantities under the target operating condition.

[0009] As a preferred scheme for irregular mesh engineering simulation method based on Laplacian neural operator, in step S2, the specific process of constructing a graph Laplacian matrix according to the graph structure and performing eigenvalue decomposition on the graph Laplacian matrix is ​​as follows: The edge set between nodes is determined based on the cell connection relationship, proximity relationship, or physical parameter similarity of the irregular mesh; the edge weight is calculated based on the spatial distance between nodes, and an adjacency weight matrix is ​​constructed. The formula for calculating edge weight is: ; In the formula: For nodes With nodes Edge weights between them; , They are nodes ,node The coordinate vector; For distance scale parameters; The Euclidean norm of a vector; Based on the adjacency weight matrix Construct degree matrix , degree matrix The main diagonal elements are assigned the sum of the weights of the edges associated with the corresponding nodes, which is the sum of the row elements, while all other non-diagonal elements are set to zero. The Laplace matrix is ​​adopted in the following form: ; In the formula: The graph is a Laplace matrix; It is a degree matrix; This is the node adjacency weight matrix; The eigenvalue decomposition of the graph Laplacian matrix is ​​expressed as follows: ; In the formula: Let be the Laplacian eigenvector matrix, and its column vectors be... For the first Spectral basis vectors; It is an eigenvalue diagonal matrix. For the first The eigenvalues ​​corresponding to the order eigenvectors; This represents the total number of grid nodes. Before truncation The low-order eigenvectors constitute the truncated spectral basis. ,in To retain the number of spectral modes, satisfying .

[0010] As a preferred scheme for irregular mesh engineering simulation methods based on the Laplace neural operator, the specific process of the spectral domain operator layer performing spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis is as follows: For the node feature matrix defined on irregular grid nodes By performing a forward spectral transform on the trunculated spectral basis, the spectral coefficient matrix of the nodal features under the spectral basis is obtained. The forward transform expression is: ; The corresponding inverse spectral transform expression is: ; In the formula: The input node feature matrix; The number of feature channels; This represents the spectral coefficient matrix of the node features under the Laplace spectral basis; For the truncated front The Laplacian eigenvector matrix of order 1; In the spectral domain, a learnable spectral weight matrix is ​​used. Performing a linear filtering transformation on the spectral coefficients, followed by an inverse transformation, yields the spectral domain convolution output. The complete truncated spectral domain filtering expression is as follows:

[0011] In the formula: Output the result of spectral domain convolution; For spectral filtering functions, This is a learnable spectral domain transformation weight matrix; This represents the diagonal matrix construction operator.

[0012] As a preferred scheme for the irregular mesh engineering simulation method based on the Laplacian neural operator, in step S2, two independent Laplacian spectral bases are constructed: the first is a geometric spectral base, which is constructed based on mesh node coordinates, cell topology connections and boundary partitioning identifiers, and is used to characterize the inherent geometric structure features of irregular meshes and complex boundaries.

[0013] The second category is the operating condition spectrum basis, which is constructed based on the node similarity of the material property field, fluid parameter field, source term distribution field or input physical field, and is used to characterize the spatial distribution characteristics of physical properties under different operating conditions.

[0014] As a preferred embodiment of the irregular mesh engineering simulation method based on the Laplacian neural operator, the spectral domain operator layer is a bispectral basis Laplacian neural operator layer, and the feature update process of the spectral domain operator layer specifically includes: For the Layer input hidden features Spectral domain transformation, spectral domain filtering, and inverse transformation are performed independently using geometric spectral basis and working condition spectral basis, respectively.

[0015] The output expression for the geometric spectrum branch is: ; The output expression for the load spectrum branch is: ; In the formula: For the first The hidden feature matrix is ​​input to the layer; This represents the number of grid nodes. For the first The number of feature channels in the layer; The eigenvector matrix of the geometric spectral basis; This is the diagonal matrix of Laplacian eigenvalues ​​corresponding to the geometric graph; For the first Learnable spectral domain transformation weight matrix of layer geometric spectral branch; The eigenvector matrix of the working condition spectrum basis; This is the diagonal matrix of Laplace eigenvalues ​​corresponding to the working condition diagram; For the first Learnable spectral domain transformation weight matrix of layer working condition spectrum branch; Output features for the geometric spectrum branch; Output features for the operating condition spectrum branch; The two-way fusion weights are obtained by normalizing the learnable parameters. Weighted fusion is then performed on the outputs of the two branches to obtain the total output of the spectral domain branches. The fusion expression is as follows: ; ; In the formula: For the first Total output of the spectral domain branch; , These are the fusion weighting coefficients for the geometric spectrum branch and the working condition spectrum branch, respectively. For the fusion weight vector; A learnable parameter vector; This indicates a Hadamard weighted operation based on elements or channels. The total output of the spectral domain branch is superimposed with the output of the local linear branch, and then processed by a nonlinear activation function to obtain the first... The hidden features of the layer and the layer update expression are: ; In the formula: For the first The layer outputs the hidden feature matrix; For the first The weight matrix of the local point-state linear mapping of the layer; For the corresponding bias vector; It is a non-linear activation function.

[0016] As a preferred embodiment of the irregular mesh engineering simulation method based on the Laplacian neural operator, the Laplacian spectral-based neural operator network includes the following from input to output: The engineering input layer is used to access node coordinates, boundary condition identifiers, material / fluid parameter fields, geometric partition markers, and source term data, and convert them into standardized node-level features. The input encoding layer maps low-dimensional node features to a high-dimensional hidden feature space through a multilayer perceptron or linear mapping layer. At least one cascaded layer of Laplacian spectral domain operators is used to alternately perform global spectral domain coupling modeling and local feature transformation; The output decoding layer maps high-dimensional hidden features into target physical quantities through a multilayer perceptron or linear mapping layer, and outputs the full-field prediction results of pressure field, velocity field, temperature field, displacement field, concentration field or stress field.

[0017] As a preferred scheme for irregular mesh engineering simulation methods based on the Laplacian neural operator, a graph Laplacian matrix is ​​constructed according to the graph structure. During the eigenvalue decomposition of the graph Laplacian matrix, for large-scale mesh scenes, a polynomial spectral filtering method is used instead of the spectral domain transformation of explicit eigenvalue decomposition. Specifically: structure The polynomial filtering function of order X represents the spectral domain convolution operation as a weighted sum of powers of the Laplacian matrix, expressed as: ; In the formula: for Polynomial spectral filtering operator of order 1; The order of the polynomial; For the first Learnable or preset filter coefficients of order; For the Laplace matrix of the graph Power; This is the feature matrix of the input nodes.

[0018] As a preferred scheme for irregular mesh engineering simulation methods based on the Laplacian neural operator, in step S4, the total loss function is used as the training optimization objective, and its expression is: ; In the formula: Total training loss; Loss due to data supervision; This is the boundary condition loss, used to constrain the physical quantities at the boundary nodes to meet the preset boundary conditions. The physical residual loss is used to constrain the predicted solution field to satisfy the control relationship of the corresponding partial differential equation. This is a smoothness regularization term used to suppress unreasonable oscillations in the solution field; , , These are the weighting coefficients for boundary condition loss, physical residual loss, and smoothness regularization term, respectively.

[0019] As a preferred scheme for irregular mesh engineering simulation methods based on the Laplacian neural operator, the data supervision loss adopts the relative L2 error between the predicted solution field and the true solution field, and the expression is: ; In the formula: The total number of training samples; It is the set of all learnable parameters of the model; For the Laplace spectral basis neural operator surrogate model; For the first Input of working conditions for each sample; For the first The irregular grid corresponding to each sample; For the first High-fidelity real solution field for each sample; Represents the L2 norm of a vector or matrix.

[0020] This invention also provides an irregular mesh engineering simulation system based on the Laplace neural operator, used to implement the above-mentioned irregular mesh engineering simulation method based on the Laplace neural operator, including an offline training module and an online inference module:

[0021] The offline training module includes: The training sample acquisition and construction submodule is used to collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete grid data of the corresponding engineering object, working condition parameter fields, and real physical field obtained by a high-fidelity numerical solver. The Laplace spectral basis calculation submodule is used to construct a corresponding graph structure based on the node connection relationship, spatial distance and boundary features of the irregular discrete grid data, construct a graph Laplace matrix according to the graph structure, and perform eigenvalue decomposition on the graph Laplace matrix to obtain a set of Laplace eigenvectors that are adapted to the irregular grid geometry, which serve as the basis for spectral domain transformation. The spectral domain neural operator inference submodule is used to build a Laplacian spectral basis neural operator network. After mapping the node-level input features to a high-dimensional feature space through the input encoding layer, it is input into at least one Laplacian spectral domain operator layer. The spectral domain operator layer performs spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis to realize global physical coupling modeling on irregular grids. After mapping through the output decoding layer, the predicted physical field is obtained. The surrogate model training and optimization submodule is used to iteratively optimize all learnable parameters of the Laplace spectral-based neural operator network with the error between the predicted physics field and the real physics field as the training objective, so as to obtain the trained surrogate solution model. The online inference module receives the target irregular grid data and target operating condition parameters to be predicted. After graph structure construction and Laplace spectral basis calculation, it is input into the surrogate solution model to perform one forward inference and outputs the prediction results of the full field physical quantities under the target operating condition.

[0022] The present invention has the following advantages: This invention uses Laplace eigenvectors instead of traditional Fourier basis as the basis for spectral domain transformation. It does not rely on regular rectangular grids and fast Fourier transform, and can directly adapt to irregular discrete forms such as triangular grids, tetrahedral grids, boundary layer refined grids, curved surface grids and point clouds commonly used in engineering. It avoids the problems of geometric information loss, boundary detail distortion and error amplification in high gradient regions during interpolation to regular grids, and significantly improves the prediction accuracy in complex geometric scenes.

[0023] This invention directly constructs the spectral basis based on the node connection relationship, boundary structure and geometric scale of the mesh itself, without the need to learn the coordinate deformation mapping from the physical domain to the regular latent domain. Compared with coordinate deformation schemes, it reduces the modeling complexity and can more directly utilize the original mesh topology and boundary structure information. It is more suitable for engineering objects with complex topology, local mesh refinement and irregular boundary shape, and maintains higher solution stability in geometrically sensitive areas such as holes, airfoils, bends and internal components.

[0024] This invention implements a complete spectral domain forward transform, filtering, and inverse transform process on irregular grids, preserving the global coupling modeling characteristics of neural operators. Compared with graph neural network schemes that rely solely on local message passing, it can more efficiently express the long-range correlation and overall response characteristics of physical fields, and is more in line with the global integral nature of partial differential equations. It also has better global prediction accuracy in strongly coupled engineering scenarios such as fluid mechanics, heat transfer, and structural mechanics.

[0025] This invention, through a bi-branch design of geometric spectral basis and working condition spectral basis and learnable weighted fusion, can simultaneously take into account the spatial distribution characteristics of the inherent geometric structure of the mesh, working condition parameters, and material properties. This enables the model to have stronger generalization and adaptation capabilities in multiple scenarios with varying geometry, working conditions, and materials, and better support engineering tasks such as multi-working condition parameter scanning and structural optimization.

[0026] This invention can be used to analyze the contribution of each mode to the solution field, facilitating model diagnosis and engineering parameter tuning. At the same time, this method can directly interface with the mesh output format of existing CFD / CAE software without additional mesh preprocessing, making it easy to embed into existing engineering simulation platforms and significantly improving the simulation efficiency of multi-condition screening, design optimization, and digital twin scenarios. Attached Figure Description

[0027] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings in the following description are merely exemplary, and those skilled in the art can derive other embodiments based on the provided drawings without creative effort.

[0028] The structures, proportions, sizes, etc. illustrated in this specification are only for the purpose of assisting those skilled in the art in understanding and reading the content disclosed herein, and are not intended to limit the conditions under which the present invention can be implemented. Therefore, they have no substantial technical significance. Any modifications to the structure, changes in the proportions, or adjustments to the size, without affecting the effects and objectives that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0029] Figure 1 This is a schematic diagram of the irregular mesh engineering simulation method based on the Laplace neural operator provided in the embodiments of the present invention; Figure 2 This is a technical roadmap for the irregular mesh engineering simulation method based on the Laplace neural operator provided in this embodiment of the invention. Figure 3 This is a schematic diagram of the irregular mesh engineering simulation system architecture based on the Laplace neural operator provided in an embodiment of the present invention. Detailed Implementation

[0030] The following specific embodiments illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. 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.

[0031] Example 1 See Figure 1 and Figure 2 This invention provides an irregular mesh engineering simulation method based on the Laplace neural operator, used for pressure, velocity, temperature, concentration, stress, and displacement field analysis of burners, chemical reactors, complex pipelines, porous media, fluid machinery, and material components. The method includes an offline training phase and an online inference phase. Addressing the pain points of traditional numerical solvers in engineering simulations, such as high cost of repetitive calculations across multiple operating conditions and difficulty in adapting to complex geometric meshes, this invention adopts a two-stage architecture of offline centralized training and online rapid inference. This architecture broadly supports engineering tasks such as rapid generation of CAE / CFD simulation results, equipment structure optimization, and multi-condition parameter selection. In the offline phase, the neural operator model is trained using high-fidelity simulation data to learn the end-to-end mapping law between geometric boundaries, operating condition parameters, and physical fields. In the online phase, it replaces traditional finite element and finite volume solvers, achieving millisecond to second-level prediction of all field physical quantities under new operating conditions. This method is particularly suitable for engineering scenarios with prevalent irregular meshes, such as burner flow fields, heat and mass transfer in chemical reactors, flow in complex pipelines, seepage in porous media, and structural stress analysis.

[0032] The offline training phase includes the following steps: S1. Collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete mesh data of the corresponding engineering object, operating condition parameter fields, and a real physical solution field obtained by a high-fidelity numerical solver. Construct the input-output sample pairs required for supervised learning, which correspond to the training data foundation from the input function space to the output function space in operator learning theory. The irregular discrete mesh can cover various commonly used engineering forms such as triangular mesh, tetrahedral mesh, boundary layer refined mesh, and curved surface mesh, preserving the real geometric topology and boundary morphology of objects such as burner cavities, airfoil surfaces, internal components of chemical reactors, curved pipelines, and perforated structural components, avoiding geometric distortion caused by regular mesh interpolation. The operating condition parameter fields cover driving variables such as boundary conditions, material properties, source term strength, and geometric dimensions, covering the common parameter variation ranges in engineering. The high-fidelity real solution field is generated by a mature commercial CFD / CAE solver or a self-developed high-precision solver, serving as the true value label for model fitting.

[0033] S2. Based on the node connectivity, spatial distance, and boundary features of the irregular discrete mesh data, a corresponding graph structure is constructed. A graph Laplace matrix is ​​then constructed according to this graph structure, and eigenvalue decomposition is performed on the graph Laplace matrix to obtain a set of Laplace eigenvectors that fit the irregular mesh geometry, serving as the basis for spectral domain transformation. Traditional Fourier neural operators (FNO) rely on regular tensor meshes and fast Fourier transforms, and cannot natively adapt to non-uniform, non-rectangular engineering meshes. The graph Laplace matrix, however, is a mathematical approximation of the continuous Laplace differential operator on discrete meshes / manifolds. Its eigenvectors constitute a set of orthogonal bases that naturally fit the mesh geometry and topology, which can be understood as a non-Euclidean Fourier basis on irregular meshes. Taking a burner cavity, chemical reactor, or curved pipeline as an example, the node set can correspond to the combustion chamber wall, nozzle, flow channel, internal components, and internal fluid calculation nodes generated by CFD / CAE software, while the edge set corresponds to the finite element / finite volume element connection relationships and boundary adjacency relationships. The resulting spectral basis can accurately match the actual topology of the engineering mesh. From a modal perspective, low-order eigenvectors correspond to globally smooth geometric and physical modes, while high-order eigenvectors correspond to local details and rapid changes, which can fully characterize the spatial features of boundary layer refinement regions, high gradient regions, and material interface regions. The spectral basis obtained through eigenvalue decomposition avoids the problems of geometric information loss and error amplification caused by interpolation to regular meshes.

[0034] S3. Construct a Laplacian spectral-based neural operator network. After mapping node-level input features to a high-dimensional feature space via an input encoding layer, input at least one Laplacian spectral operator layer is applied. This spectral operator layer performs forward spectral transformation, linear spectral filtering, and inverse spectral transformation based on the spectral transformation basis, achieving global physical coupling modeling on irregular grids. After mapping by the output decoding layer, the predicted physical field is obtained. The Fourier layers dependent on regular grids are replaced with self-designed Laplacian spectral operator layers. Other structures such as input dimensionality increase, local branching, nonlinear activation, and output projection remain compatible with FNO, preserving the global modeling advantages of neural operators and achieving native support for irregular grids. The input encoding layer increases low-dimensional discrete features such as coordinates, boundary markers, and material parameters to a high-dimensional hidden space, giving nodes richer feature representations. The Laplacian spectral operator layer learns linear transformations in the spectral domain through a "forward transformation-spectral filtering-inverse transformation" process, equivalent to global integral convolution in physical space, directly capturing long-range coupling relationships in the physical field. For example, the influence of upstream and downstream fluid flow and the cross-regional transmission of structural stress align with the global integral nature of partial differential equation solvers; the output decoding layer maps high-dimensional hidden features back to the target physical quantity. From an engineering perspective, this transformation process can globally couple and extract features from nodal fields such as pressure, temperature, velocity, component concentration, displacement, or stress. The transformed spectral coefficients can directly drive the surrogate model to quickly predict the complete solution field under new operating conditions, serving practical engineering tasks such as equipment structure optimization, process parameter screening, operating condition comparison, and abnormal state prediction.

[0035] S4. Using the error between the predicted physical solution field and the actual physical solution field as the training objective, iteratively optimize all learnable parameters of the Laplace spectral-based neural operator network to obtain the trained surrogate solution model. The training process drives the network to fit the solution operator mapping of the target partial differential equation through the error backpropagation algorithm. Using the error of the overall solution as the optimization objective, the model's output at all grid nodes is constrained to closely approximate the high-fidelity solution, ensuring the overall accuracy of the physical field spatial distribution. During the iteration process, all learnable variables, such as spectral domain filter weights, encoding / decoding layer parameters, and dual-branch fusion weights, are simultaneously optimized, enabling the model to gradually master the evolution of the physical field under different operating conditions. According to engineering requirements, the training objective can also be expanded to include superimposed boundary condition constraints, physical equation residuals, smoothness regularization, and other losses to further improve the model's physical consistency and generalization ability. After training, the model only requires matrix multiplication-level forward computation to output the solution field, without iteratively solving the partial differential equation, improving the computation speed by several orders of magnitude compared to traditional numerical solvers.

[0036] The online inference stage includes: receiving the target irregular mesh data and target operating condition parameters to be predicted; constructing a graph structure and calculating the Laplace spectral basis; inputting the data into the surrogate solution model to perform one forward inference; and outputting the predicted results of the full-field physical quantities under the target operating condition. In actual engineering scenarios, engineers can submit new burner structural parameters, reactor internal component layout, pipeline geometry, material properties, inlet flow rate, temperature, or pressure boundary conditions in the simulation platform. The system automatically reads or generates the corresponding irregular mesh, and only needs to complete one neural network forward propagation to obtain the full-field physical response and key engineering indicators such as pressure drop, maximum temperature, and maximum stress under the target operating condition, without having to perform time-consuming high-fidelity numerical solutions. The single-sample inference time is typically in the millisecond to second range, which can efficiently support scenarios with a large number of repetitive solutions in engineering, such as multi-operating condition parameter screening, design space traversal, real-time performance evaluation, and uncertainty analysis, significantly shortening the equipment development cycle and reducing computing resource consumption.

[0037] In one possible embodiment, step S2, constructing a graph Laplacian matrix based on the graph structure and performing eigenvalue decomposition on the graph Laplacian matrix, specifically involves the following process:

[0038] The edge set between nodes is determined based on the cell connection relationship, proximity relationship, or physical parameter similarity of the irregular mesh; the edge weight is calculated based on the spatial distance between nodes, and an adjacency weight matrix is ​​constructed. The formula for calculating edge weights is: ; In the formula: For nodes With nodes Edge weights between them; , They are nodes ,node The coordinate vector; For distance scale parameters; The Euclidean norm of a vector is denoted by .

[0039] Specifically, the closer the spatial distance between nodes, the larger the edge weight, indicating a stronger geometric connection and physical coupling between the nodes; scale parameter Used to control the scope of the neighborhood. The smaller the value, the faster the edge weights decay with distance, and the spectral basis emphasizes local features more. The larger the value, the wider the scope of influence, and the more the spectral basis emphasizes global features. Besides distance-weighted methods, in engineering, the native connectivity of finite element / finite volume elements can be directly reused to construct edges. In this case, the edge weight can be set to 1, which strictly preserves the topological connectivity of the original mesh. For boundary layer refinement regions, high-gradient regions near reaction zones, and material interface regions, edge weights can simultaneously characterize local geometric relationships and physical correlations, thus providing an input structure completely consistent with the actual mesh for subsequent temperature, velocity, pressure, or concentration field predictions.

[0040] Among them, based on the adjacency weight matrix Construct degree matrix , degree matrix The main diagonal elements are assigned the sum of the weights of the edges associated with the corresponding nodes, which is the sum of the row elements, while all other non-diagonal elements are set to zero.

[0041] The Laplace matrix is ​​adopted in the following form: ; In the formula: The graph is a Laplace matrix; It is a degree matrix; This is the node adjacency weight matrix; Specifically, the graph Laplace matrix is ​​a mathematical approximation of the continuous Laplace operator on a discrete graph, corresponding to the differential operators of processes such as diffusion, wave motion, and heat conduction in physics, and serves as the carrier for spectral domain modeling. Symmetrically normalized Laplace matrices, through scale normalization using the degree matrix, can eliminate numerical inhomogeneities caused by differences in node degree. Its eigenvalue range is constrained to the [0,2] interval, facilitating spectral truncation and filtering design, and exhibiting superior numerical stability. For irregular meshes with significant differences in node degree, such as boundary layer refinement meshes and locally refined meshes commonly used in engineering, the symmetrically normalized form typically performs better, and can be flexibly selected based on specific mesh characteristics and task requirements.

[0042] The eigenvalue decomposition of the graph Laplacian matrix is ​​expressed as follows: ; In the formula: Let be the Laplacian eigenvector matrix, and its column vectors be... For the first Spectral basis vectors; It is an eigenvalue diagonal matrix. For the first The eigenvalues ​​corresponding to the order eigenvectors; This represents the total number of grid nodes. Before truncation The low-order eigenvectors constitute the truncated spectral basis. ,in To retain the number of spectral modes, satisfying .

[0043] Specifically, eigenvalue decomposition diagonalizes the graph Laplacian matrix, resulting in orthogonal eigenvectors that form a complete spatial basis. For regular Euclidean meshes, sine / cosine Fourier bases can be used to describe periodic structures. For irregular meshes, these eigenvectors are determined by mesh connectivity, boundary shape, and geometric scale. From a physical perspective, eigenvalues... Corresponding frequencies in the spectral domain The smaller the value, the smoother the spatial variation of the corresponding feature vector, representing a global low-frequency pattern. The larger the value, the more drastic the spatial variation, representing a local high-frequency mode. In engineering, physical fields such as velocity, temperature, and stress fields are typically dominated by low-frequency components, while high-frequency components are mostly local details or numerical noise; therefore, only the first few values ​​need to be retained. Using a low-order eigenvector as a truncated spectral basis can significantly reduce computational load and model parameter count while preserving the main physical information of the solution field.

[0044] In one possible embodiment, the spectral domain operator layer performs forward spectral domain transformation, linear spectral domain filtering, and inverse spectral domain transformation based on the spectral domain transformation basis as follows: For the node feature matrix defined on irregular grid nodes By performing a forward spectral transform on the trunculated spectral basis, the spectral coefficient matrix of the nodal features under the spectral basis is obtained. The forward transform expression is: ; The corresponding inverse spectral transform expression is: ; In the formula: The input node feature matrix; Number of feature channels; This represents the spectral coefficient matrix of the node features under the Laplace spectral basis; For the truncated front The Laplace eigenvector matrix is ​​a matrix of eigenvalues. The essence of the forward spectral transform is to project the nodal features of the physical space onto an orthogonal Laplace basis, obtaining the coefficients corresponding to each spectral mode, corresponding to the process of transforming the physical space to the spectral domain in the Fourier transform. The inverse transform reconstructs the physical space by weighted superposition of the spectral coefficients according to the basis vectors, corresponding to the inverse Fourier transform. Since the Laplace eigenvectors are naturally adapted to the mesh geometry, this projection and reconstruction process does not introduce geometric interpolation errors, preserving complex geometric information such as irregular boundaries, holes, bends, and internal components. In engineering simulation scenarios, this transform can globally couple and extract features from nodal fields such as pressure, temperature, velocity, component concentration, displacement, or stress. The spectral coefficients obtained from the transform can directly drive the surrogate model to quickly predict the complete solution field under new working conditions, serving practical tasks such as equipment structure optimization and process parameter selection.

[0045] In the spectral domain, a learnable spectral weight matrix is ​​used. Performing a linear filtering transformation on the spectral coefficients, followed by an inverse transformation, yields the spectral domain convolution output. The complete truncated spectral domain filtering expression is as follows: ; In the formula: Output the result of spectral domain convolution; For spectral filtering functions, This is a learnable spectral domain transformation weight matrix; This represents the construction operator for the diagonal matrix. Global convolution on the graph is equivalent to element-wise multiplication in the spectral domain. Compared to ordinary graph neural networks, which can only capture local message passing within a k-order neighborhood, this spectral domain convolution possesses a global receptive field, enabling direct modeling of long-range coupling and overall response of the physical field, and better aligning with the global integral characteristics of partial differential equation solution operators. The learnable spectral weights automatically fit the contributions of different frequency components to the solution field during training, thereby learning the operator mapping rules of the target PDE.

[0046] In one possible embodiment, in step S2, two independent Laplace spectral bases are constructed: the first is a geometric spectral base, which is constructed based on grid node coordinates, cell topology connections and boundary partitioning identifiers, and is used to characterize the inherent geometric structure features of irregular grids and complex boundaries; The second category is the operating condition spectrum basis, which is constructed based on the node similarity of the material property field, fluid parameter field, source term distribution field or input physical field, and is used to characterize the spatial distribution characteristics of physical properties under different operating conditions.

[0047] Specifically, the geometric spectral basis is determined solely by the mesh spatial structure. Different operating conditions of the same geometric object can share this spectral basis, which is specifically responsible for capturing the constraint effect of boundary shape and mesh topology on the solution field. The operating condition spectral basis is dynamically constructed based on input material properties, source term distribution, boundary conditions, and other parameters, reflecting the spatial distribution differences of physical properties under different operating conditions and characterizing the driving influence of material properties on the physical field. For example, in airfoil external flow simulation, the geometric spectral basis can highlight the structural features of the airfoil, trailing edge, and boundary layer mesh, while the operating condition spectral basis can reflect changes in incoming flow velocity, angle of attack, pressure, and viscous parameters. In chemical reactor or burner simulation, the geometric spectral basis can characterize the connection relationships of nozzles, baffles, channels, and internal components, while the operating condition spectral basis can reflect differences in inlet flow rate, temperature, component concentration, heat source intensity, and reaction conditions. The two spectral bases are complementary, and compared to a single geometric spectral basis, they can better adapt to engineering simulation scenarios with varying operating conditions and materials.

[0048] In one possible embodiment, the spectral domain operator layer is a bispectral basis Laplacian neural operator layer, and the feature update process of the spectral domain operator layer specifically includes: For the Layer input hidden features Spectral domain transformation, spectral domain filtering, and inverse transformation are performed independently using geometric spectral basis and working condition spectral basis, respectively.

[0049] The output expression for the geometric spectrum branch is: ; The output expression for the load spectrum branch is: ; In the formula: For the first The hidden feature matrix is ​​input to the layer; This represents the number of grid nodes. For the first The number of feature channels in the layer; The eigenvector matrix of the geometric spectral basis; This is the diagonal matrix of Laplacian eigenvalues ​​corresponding to the geometric graph; For the first Learnable spectral domain transformation weight matrix of layer geometric spectral branch; The eigenvector matrix of the working condition spectrum basis; This is the diagonal matrix of Laplace eigenvalues ​​corresponding to the working condition diagram; For the first Learnable spectral domain transformation weight matrix of layer working condition spectrum branch; Output features for geometric spectrum branches; The output features are provided for the operating condition spectrum branch. Dual-branch parallel computation is a specific implementation of dual-spectral-base fusion. The two branches have independent learnable spectral weights, allowing them to separately fit the influence mechanisms of geometric structure and operating condition parameters on the solution field, avoiding the bottleneck of a single spectral base being unable to simultaneously capture both types of information. The geometric spectrum branch performs spectral domain transformation based on a fixed geometric structure spectral base, learning the physical field evolution laws under geometric boundary constraints. The operating condition spectrum branch performs spectral domain transformation based on a dynamically changing operating condition spectral base, learning the field change patterns driven by physical property parameters. Taking an airfoil flow scenario as an example, the geometric branch can stably capture the flow field structural constraints brought by the airfoil shape, while the operating condition branch can flexibly adapt to flow field changes under different inflow parameters. Taking burner simulation as an example, the geometric branch can characterize the fixed structural influence of the cavity and nozzle, while the operating condition branch can respond to changes in different fuel compositions and inlet flow rates. The two branches compute independently without interference, and are ultimately combined through a fusion mechanism to achieve feature complementarity between geometric and operating condition information.

[0050] The two-way fusion weights are obtained by normalizing the learnable parameters, and the weighted fusion of the two branch outputs is performed to obtain the total output of the spectral domain branch. The fusion expression is as follows: ; ; In the formula: For the first Total output of the spectral domain branch; , These are the fusion weighting coefficients for the geometric spectrum branch and the working condition spectrum branch, respectively. For the fusion weight vector; A learnable parameter vector; This represents Hadamard weighted operations by element or by channel. Adaptive weighted fusion allows the model to automatically learn the importance of two branches based on data distribution, without requiring manual weight fixing. The function guarantees that the sum of the weights of the two paths is 1, achieving a normalized weighted average. The learnable parameters are automatically adjusted during training. For geometry-driven scenarios, such as complex boundary flows and stress analysis of irregular structures, the model automatically increases the weight of the geometry spectrum branch; for property-driven scenarios, such as variable material heat transfer and multi-component diffusion reactions, the model automatically increases the weight of the working condition spectrum branch. Hadamard operations support weighting by channel or by element, enabling finer-grained feature fusion. This allows the model to flexibly adapt to different types of engineering simulation tasks, achieving optimal feature combination results in various scenarios.

[0051] The total output of the spectral domain branch is superimposed with the output of the local linear branch, and then processed by a nonlinear activation function to obtain the first... The hidden features of the layer and the layer update expression are: ; In the formula: For the first The layer outputs the hidden feature matrix; For the first The weight matrix of the local point-state linear mapping of the layer; For the corresponding bias vector; The activation function is nonlinear. The global spectral branch is responsible for capturing long-range physical coupling and the overall field distribution, corresponding to global effects such as diffusion and integral terms in partial differential equations. The local linear branch (point-state 1×1 convolution) is responsible for capturing node-level nonlinear transformations and local feature interactions, corresponding to convection, source, and nonlinear effects in partial differential equations. The superposition of these two branches, through the introduction of nonlinear expressive power via the nonlinear activation function, enables the network to fit complex nonlinear partial differential equation solvers. This structure retains the global coupling expressive power of neural operators while adapting to irregular mesh geometry and physical distributions under different operating conditions, achieving a good balance between accuracy and efficiency.

[0052] One possible embodiment of the Laplace spectral-based neural operator network includes the following from input to output: The engineering input layer is used to access node coordinates, boundary condition identifiers, material / fluid parameter fields, geometric partition markers, and source term data, and convert them into standardized node-level features. The input encoding layer maps low-dimensional node features to a high-dimensional hidden feature space through a multilayer perceptron or linear mapping layer. At least one cascaded layer of Laplacian spectral domain operators is used to alternately perform global spectral domain coupling modeling and local feature transformation; The output decoding layer maps high-dimensional hidden features into target physical quantities through a multilayer perceptron or linear mapping layer, and outputs the full-field prediction results of pressure field, velocity field, temperature field, displacement field, concentration field or stress field.

[0053] Specifically, from the perspective of engineering software integration, the engineering input layer can directly interface with mesh file parsing, boundary condition reading, and working condition parameter reading, converting multi-source heterogeneous engineering data into node features in a unified format; the input encoding layer completes feature upscaling, mapping simple input parameters into high-dimensional implicit features, providing a rich feature foundation for subsequent operator layers; the multi-layered cascaded spectral domain operator layer gradually extracts and evolves physical field features through layer-by-layer global and local spectral domain transformations, realizing complex operator mapping from input to output; the output decoding layer can interface with cloud map generation, key engineering index calculation, and simulation result write-back, mapping high-dimensional hidden features into physical quantities such as pressure, velocity, temperature, and displacement that engineers can directly use.

[0054] One possible embodiment involves constructing a graph Laplacian matrix based on the graph structure, and during the eigenvalue decomposition process on the graph Laplacian matrix, for large-scale mesh scenes, using polynomial spectral filtering instead of the spectral domain transformation of explicit eigenvalue decomposition, specifically: structure The polynomial filtering function of order X represents the spectral domain convolution operation as a weighted sum of powers of the Laplacian matrix, expressed as: ; In the formula: for Polynomial spectral filtering operator of order 1; The order of the polynomial; For the first Learnable or preset filter coefficients of order; For the Laplace matrix of the graph Power; This is the feature matrix of the input nodes.

[0055] Specifically, this solution is an efficient and equivalent alternative implementation for large-scale industrial grids. When the number of nodes in the engineering grid is very large, the computational complexity of performing full eigenvalue decomposition on the Laplacian matrix is ​​extremely high, resulting in significant memory and time overhead, making it difficult to directly apply to large-scale 3D simulation scenarios. Polynomial spectral filtering, based on the spectral mapping theorem, directly achieves spectral domain filtering through power operations on the Laplacian matrix, eliminating the need to explicitly calculate all eigenvectors and significantly reducing computational and storage costs. The polynomial order corresponds to the equivalent receptive field; the larger the order, the wider the captured neighborhood, which can be flexibly adjusted according to the grid size and task requirements. This method is mathematically equivalent to spectral domain convolution in explicit eigenvalue decomposition and is an optimization technique for large-scale industrial grid engineering deployments, effectively expanding the applicable scenarios of this method.

[0056] In one possible embodiment, in step S4, the total loss function is used as the training optimization objective, and its expression is: ; In the formula: Total training loss; Loss due to data supervision; This is the boundary condition loss, used to constrain the physical quantities at the boundary nodes to meet the preset boundary conditions. The physical residual loss is used to constrain the predicted solution field to satisfy the control relationship of the corresponding partial differential equation. This is a smoothness regularization term used to suppress unreasonable oscillations in the solution field; , , These are the weighting coefficients for boundary condition loss, physical residual loss, and smoothness regularization term, respectively.

[0057] Specifically, the training objective of combining multiple loss terms constrains the model from the dimensions of data fitting, physical constraints, and regularization, thereby improving prediction accuracy and physical consistency. Pure data loss can only guarantee fitting of training samples, and is prone to overfitting or physically unreasonable prediction results. After superimposing multi-dimensional constraints, the model's generalization ability and physical reliability are significantly improved. In specific engineering applications, boundary condition loss can be used to constrain burner wall temperature, inlet / outlet pressure, or velocity boundaries, improving the prediction accuracy of critical boundary regions; physical residual loss can be used to constrain the consistency of the governing equations of flow, heat and mass transfer, or structural stress processes, enhancing the model's extrapolation ability; smoothness regularization terms can suppress numerical oscillations, making the predicted field more consistent with the smooth characteristics of the real physical field. For scenarios without explicit partial differential equations and using only simulation samples for supervised training, only data and regularization losses can be retained to adapt to different engineering data conditions.

[0058] In one possible implementation, the data supervision loss uses the relative L2 error between the predicted solution and the true solution, expressed as: ; In the formula: The total number of training samples; It is the set of all learnable parameters of the model; For the Laplace spectral basis neural operator surrogate model; For the first Input of working conditions for each sample; For the first The irregular grid corresponding to each sample; For the first High-fidelity real solution field for each sample; Represents the L2 norm of a vector or matrix.

[0059] Specifically, relative L2 error is used instead of absolute L2 error to eliminate the impact of differences in the magnitude of physical quantities on training, while maintaining consistency with common evaluation standards in the field of engineering simulation. Different physical quantities (such as pressure, temperature, and velocity) have different numerical magnitudes, and the overall amplitude of the solution field may also vary greatly under different working conditions. Absolute error will cause the model to bias towards fitting large numerical regions, ignoring small but critical regions. Relative error, using the L2 norm of the true solution field as the denominator, measures the proportion of prediction error to the overall amplitude of the solution field, which is more in line with the engineering evaluation habits of relative accuracy, allowing the model to maintain a balanced fitting accuracy across the entire region and all magnitudes. This error metric is also fully aligned with the evaluation system of existing neural operator benchmark methods such as Geo-FNO, facilitating horizontal comparisons and performance verification.

[0060] To verify the correspondence between this invention and the problem of surrogate solution in irregular mesh engineering simulation, a comparative experimental table was compiled with reference to the settings of the complex geometry PDE dataset in the Geo-FNO literature (Li Z, Huang DZ, Liu B, et al. Fourier neural operator with learned deformations for pdes on general geometries[J]. Journal of Machine Learning Research, 2023, 24(388): 1-26.). This section is used to illustrate the application objects and verification paths to be solved by this invention, and does not limit this invention to the following datasets.

[0061] The Elasticity Mechanics dataset is used to evaluate the ability of a model to predict the stress field of a structure under irregular point cloud input. The computational domain of this problem is a hyperelastic material unit cell with a central void, the input is geometric / material information on a point cloud or irregular mesh, and the output is the stress field. Since the void boundary and local stress concentration regions have a significant impact on the solution field, this dataset is suitable for validating whether the model can utilize irregular geometric information.

[0062] The airfoil flow dataset is used to evaluate the model's surrogate prediction capability on complex aerodynamic geometries. This problem describes transonic airfoil flow based on the Euler equations, with inputs being airfoil geometry and mesh node information, and outputs being the Mach number or the associated flow field. The pipe flow dataset is based on the incompressible Navier-Stokes equations, with inputs being curved pipe geometry and a structured deformable mesh, and outputs being the velocity field. The plastic forming dataset is used to evaluate the model's surrogate prediction capability for time-dependent structural deformation and displacement evolution. These scenarios are all typical problems in engineering design that require extensive and repeated simulations.

[0063] Public experiments with Geo-FNO typically use relative L2 error to evaluate training and testing errors, and report model size and training time. The results show that directly interpolating irregular geometry onto a regular grid before using FNO or UNet often leads to increased testing errors due to interpolation errors and loss of boundary information. This invention aims to model the spectral domain directly on irregular grids using Laplacian spectral bases, avoiding or reducing information loss caused by interpolation on regular grids.

[0064] Table 1 below presents the baseline experimental statistics for the elasticity dataset in the Geo-FNO literature. The input is a point cloud, and the output is the structural stress field. The last row of the table shows the experimental results of the method of this invention.

[0065] Table 1. Results of the benchmark experiment for inputting point clouds in elasticity (relative to L2 error)

[0066] As shown in Table 1, in the elasticity point cloud input scenario, although the FNO / UNet method, which directly interpolates to a regular mesh, has a shorter training time, the test error is still affected by the loss of interpolation and boundary information. Geo-FNO reduces the impact of irregular geometry through geometric deformation. The difference between the method in this invention and Geo-FNO is that it does not rely on mapping the physical domain to a regular latent domain before performing FFT, but directly uses the Laplacian eigenvectors of the irregular mesh as the spectral basis. Therefore, it is more suitable for further processing of irregular connectivity, local mesh refinement, and complex boundary topology.

[0067] Table 2 below compares the results of three engineering scenarios—airfoil flow, pipe flow, and plastic forming—from the Geo-FNO literature. Airfoil and pipe problems belong to fluid mechanics scenarios, while plastic forming problems belong to time-dependent structural mechanics scenarios. The last row of the table shows the experimental results of the method described in this invention.

[0068] Table 2. Results of baseline experiments on airfoils, duct flow, and plastic forming (relative to L2 error)

[0069] As shown in Table 2, in airfoil and pipe fluid simulations, forcibly interpolating complex geometric fields into regular rectangular regions leads to a significant increase in test errors, indicating that geometric boundaries and mesh distribution have a significant impact on flow field prediction. This invention directly expresses the geometric patterns and physical field distributions on irregular meshes using Laplace spectral bases, enabling the model to maintain better geometric consistency near complex boundaries. For plastic forming problems, the output includes not only physical quantities but also the mesh position and displacement field evolving over time; the Laplace spectral base can also serve as the geometric representation basis for the spatiotemporal mesh response.

[0070] This invention addresses the rapid prediction and simulation acceleration of irregular mesh physical fields for complex geometric engineering equipment. It can directly adapt to various irregular discretization forms such as triangular meshes, tetrahedral meshes, boundary layer refined meshes, curved surface meshes, and point clouds. It is widely used in simulation acceleration, design optimization, and performance extrapolation in fields such as energy and power, petrochemicals, fluid machinery, advanced materials, and structural engineering. Typical application scenarios are as follows:

[0071] Rapid simulation of flow fields in burners and power equipment: For power equipment such as gas burners, aero-engine combustion chambers, and industrial furnaces, the internal cavities, nozzles, swirl blades, and flame stabilization structures have complex geometries. The simulation meshes are mostly unstructured tetrahedral meshes with local refinement on the walls and reaction zones. Traditional CFD solvers can take hours to days to complete a single-condition calculation. This method can directly adapt to the original irregular mesh without interpolation to reconstruct a regular mesh. After offline training, it can quickly output the full-field velocity field, temperature field, and component concentration field under different inlet flow rates, fuel compositions, equivalence ratios, and temperature boundaries within seconds. This supports burner nozzle structure optimization, combustion efficiency assessment, NOx emission prediction, and multi-condition parameter selection, significantly shortening the R&D iteration cycle of combustion equipment.

[0072] Simulation of heat and mass transfer and reaction processes in chemical reactors: For chemical process equipment such as fixed-bed reactors, stirred reactors, packed towers, and microchannel reactors, the internal baffles, catalyst particles, internal components, and complex flow channels result in complex geometric topologies, highly irregular simulation meshes, and computationally intensive multiphysics coupling solutions involving convection, diffusion, and reaction. This method natively adapts to the unstructured meshes of reactors. Through dual-path fusion of geometric and operating condition spectral bases, it simultaneously characterizes the influence of reactor internal structure and operating parameters such as inlet flow rate, temperature, component concentration, and reaction intensity. It rapidly predicts the concentration field, temperature field, reaction rate distribution, and pressure drop characteristics within the reactor, supporting reactor structure optimization, process parameter tuning, and catalyst arrangement design, thereby improving the mass transfer efficiency and reaction conversion rate of chemical processes.

[0073] Simulation of complex pipeline and fluid mechanical properties: For fluid machinery and piping systems such as curved pipelines, irregularly shaped flow channels, valves, pump bodies, impellers, and turbines, the flow channel geometry is often a spatial curved surface, and the simulation mesh is mostly an unstructured boundary layer mesh. This results in a large amount of repetitive computation in simulating hydraulic / aerodynamic performance under different flow rates, speeds, and pressure ratios. This method can directly connect to the original irregular mesh of the pipeline and impeller, quickly predict key performance indicators such as pressure field, velocity field, vorticity distribution, overall pressure drop, head, and efficiency, evaluate flow losses under different pipeline layouts, optimize impeller profiles and valve flow channel structures, and rapidly complete multi-condition performance traversal and scheme comparison, replacing a large amount of repetitive high-fidelity CFD calculations.

[0074] Simulation of energy transport characteristics in porous media and materials: For porous media in oil and gas reservoirs, fuel cell electrodes, catalyst materials, and soil media, the microstructure pore morphology is highly irregular, and after discretization, it is mostly non-uniform mesh or point cloud data. The coupled simulation of seepage, mass transfer, and heat transfer processes requires a large mesh size and high computational cost. This method can directly adapt to the irregular discretization form of porous media microstructures, quickly predict the permeability field, concentration field, temperature field, and equivalent transport parameters under different pore structures and physical property parameters, analyze the influence of pore morphology and material properties on macroscopic transport performance, and support the microstructure design of energy materials, seepage assessment of oil and gas reservoirs, and performance optimization of catalyst materials.

[0075] Rapid simulation of mechanical response of irregularly shaped structural components: For mechanical structural components, composite material components, and complex 3D-printed parts with holes, gaps, or irregular cross-sections, stress concentration areas and material interfaces are typically addressed using locally refined unstructured finite element meshes. Strength verification and deformation analysis under multiple loads and material parameters require repeated calls to the finite element solver. This method natively adapts to irregular finite element meshes in structural components, rapidly predicting displacement, stress, and strain fields under different loads, boundary constraints, and material properties. It accurately identifies stress concentration areas, supporting lightweight structural design, strength safety margin assessment, and multi-load condition traversal, significantly improving the analytical efficiency of structural mechanics simulations.

[0076] Example 2 See Figure 3 Embodiment 2 of the present invention also provides an irregular mesh engineering simulation system based on the Laplacian neural operator, used to implement the irregular mesh engineering simulation method based on the Laplacian neural operator in the above embodiments, including an offline training module 100 and an online inference module 200: The offline training module 100 includes: The training sample acquisition and construction submodule 101 is used to collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete grid data of the corresponding engineering object, working condition parameter fields, and real physical field obtained by a high-fidelity numerical solver. The Laplace spectral basis calculation submodule 102 is used to construct a corresponding graph structure based on the node connection relationship, spatial distance and boundary features of the irregular discrete grid data, construct a graph Laplace matrix according to the graph structure, and perform eigenvalue decomposition on the graph Laplace matrix to obtain a set of Laplace eigenvectors that are adapted to the irregular grid geometry, which serve as the basis for spectral domain transformation. The spectral domain neural operator inference submodule 103 is used to build a Laplacian spectral basis neural operator network. After mapping the node-level input features to a high-dimensional feature space through the input encoding layer, it is input into at least one Laplacian spectral domain operator layer. The spectral domain operator layer performs spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis to realize global physical coupling modeling on irregular grids. After mapping through the output decoding layer, the predicted physical field is obtained. The surrogate model training and optimization submodule 104 is used to iteratively optimize all learnable parameters of the Laplace spectral basis neural operator network with the error between the predicted physics field and the real physics field as the training objective, so as to obtain the trained surrogate solution model. The online inference module 200 is used to receive the target irregular grid data and target operating condition parameters to be predicted. After graph structure construction and Laplace spectrum basis calculation, it is input into the surrogate solution model to perform one forward inference and output the prediction results of the full field physical quantities under the target operating condition.

[0077] It should be noted that the information interaction and execution process between the modules of the above system are based on the same concept as the method embodiment in Embodiment 1 of this application, and the resulting technical effects are the same as those in the method embodiment of this application. For details, please refer to the description in the method embodiment shown above in this application, and it will not be repeated here.

[0078] Example 3 Embodiment 3 of the present invention provides a non-transitory computer-readable storage medium storing program code for an irregular mesh engineering simulation method based on the Laplacian neural operator. The program code includes instructions for executing the irregular mesh engineering simulation method based on the Laplacian neural operator of Embodiment 1 or any possible implementation thereof.

[0079] Computer-readable storage media can be any available medium that a computer can access, or a data storage device such as a server or data center that integrates one or more available media. The available medium can be magnetic media (e.g., floppy disks, hard disks, magnetic tapes), optical media (e.g., DVDs), or semiconductor media (e.g., solid-state drives (SSDs)).

[0080] Example 4 Embodiment 4 of the present invention provides an electronic device, including: a memory and a processor; The processor and the memory communicate with each other via a bus; the memory stores program instructions that can be executed by the processor, and the processor can execute the irregular mesh engineering simulation method based on the Laplacian neural operator according to Embodiment 1 or any possible implementation thereof by calling the program instructions.

[0081] Specifically, a processor can be implemented in hardware or software. When implemented in hardware, the processor can be a logic circuit, an integrated circuit, etc. When implemented in software, the processor can be a general-purpose processor that reads software code stored in memory. This memory can be integrated into the processor or located outside the processor and exist independently.

[0082] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of the present invention are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means.

[0083] It is obvious to those skilled in the art that the modules or steps of the present invention described above can be implemented using general-purpose computing devices. They can be centralized on a single computing device or distributed across a network of multiple computing devices. Optionally, they can be implemented using computer-executable program code, thereby storing them in a storage device for execution by a computing device. In some cases, the steps shown or described can be performed in a different order than those presented herein, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. Thus, the present invention is not limited to any particular combination of hardware and software.

[0084] Although the present invention has been described in detail above with general descriptions and specific embodiments, modifications or improvements can be made to it, which will be obvious to those skilled in the art. Therefore, all such modifications or improvements made without departing from the spirit of the present invention fall within the scope of protection claimed by the present invention.

Claims

1. An irregular mesh engineering simulation method based on the Laplace neural operator, used for the analysis of pressure, velocity, temperature, concentration, stress, and displacement fields in burners, chemical reactors, complex pipelines, porous media, fluid machinery, and material components, characterized by: Includes an offline training phase and an online inference phase; The offline training phase includes the following steps: S1. Collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete grid data of the corresponding engineering object, working condition parameter fields, and real physical field obtained by a high-fidelity numerical solver. S2. Based on the node connection relationships, spatial distances and boundary features of the irregular discrete grid data, construct a corresponding graph structure, construct a graph Laplacian matrix according to the graph structure, and perform eigenvalue decomposition on the graph Laplacian matrix to obtain a set of Laplacian eigenvectors that are adapted to the irregular grid geometry, which serve as the basis for spectral domain transformation. S3. Construct a Laplace spectral basis neural operator network. After mapping the node-level input features to a high-dimensional feature space through the input encoding layer, input at least one Laplace spectral domain operator layer. The spectral domain operator layer performs spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis to realize global physical coupling modeling on irregular grids. After mapping through the output decoding layer, the predicted physical field is obtained. S4. Using the error between the predicted physical field and the real physical field as the training objective, iteratively optimize all learnable parameters of the Laplace spectral-based neural operator network to obtain the trained surrogate solution model. The online inference stage includes: receiving the target irregular grid data and target operating condition parameters to be predicted, constructing the graph structure and calculating the Laplace spectral basis, inputting it into the surrogate solution model to perform one forward inference, and outputting the prediction results of the full field physical quantities under the target operating condition.

2. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 1, characterized in that, In step S2, the graph Laplacian matrix is ​​constructed based on the graph structure, and the eigenvalue decomposition of the graph Laplacian matrix is ​​performed as follows: The edge set between nodes is determined based on the cell connection relationship, proximity relationship, or physical parameter similarity of the irregular mesh; the edge weight is calculated based on the spatial distance between nodes, and an adjacency weight matrix is ​​constructed. The formula for calculating edge weights is: ； In the formula: For nodes With nodes Edge weights between them; , They are nodes ,node The coordinate vector; For distance scale parameters; The Euclidean norm of a vector; Based on the adjacency weight matrix Construct degree matrix , degree matrix The main diagonal elements are assigned the sum of the weights of the edges associated with the corresponding nodes, which is the sum of the row elements, while all other non-diagonal elements are set to zero. The Laplace matrix is ​​adopted in the following form: ； In the formula: The graph is a Laplace matrix; It is a degree matrix; This is the node adjacency weight matrix; The eigenvalue decomposition of the graph Laplacian matrix is ​​expressed as follows: ； In the formula: Let be the Laplacian eigenvector matrix, and its column vectors be... For the first Spectral basis vectors; It is an eigenvalue diagonal matrix. For the first The eigenvalues ​​corresponding to the order eigenvectors; This represents the total number of grid nodes. Before truncation The low-order eigenvectors constitute the truncated spectral basis. ,in To retain the number of spectral modes, satisfying .

3. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 2, characterized in that, The specific process of the spectral domain operator layer performing spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis is as follows: For the node feature matrix defined on irregular grid nodes By performing a forward spectral transform on the trunculated spectral basis, the spectral coefficient matrix of the nodal features under the spectral basis is obtained. The forward transform expression is: ； The corresponding inverse spectral transform expression is: ； In the formula: The input node feature matrix; Number of feature channels; This represents the spectral coefficient matrix of the node features under the Laplace spectral basis; For the truncated front The Laplacian eigenvector matrix of order 1; In the spectral domain, a learnable spectral weight matrix is ​​used. Performing a linear filtering transformation on the spectral coefficients, followed by an inverse transformation, yields the spectral domain convolution output. The complete truncated spectral domain filtering expression is as follows: ； In the formula: Output the result of spectral domain convolution; For spectral filtering functions, This is a learnable spectral domain transformation weight matrix; This represents the diagonal matrix construction operator.

4. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 1, characterized in that, In step S2, two independent Laplace spectral bases are constructed: the first is a geometric spectral base, which is constructed based on grid node coordinates, cell topology connections and boundary partitioning identifiers, and is used to characterize the inherent geometric structure features of irregular grids and complex boundaries; The second category is the operating condition spectrum basis, which is constructed based on the node similarity of the material property field, fluid parameter field, source term distribution field or input physical field, and is used to characterize the spatial distribution characteristics of physical properties under different operating conditions.

5. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 4, characterized in that, The spectral domain operator layer is a bispectral basis Laplacian neural operator layer, and the feature update process of the spectral domain operator layer specifically includes: For the Layer input hidden features Spectral domain transformation, spectral domain filtering, and inverse transformation are performed independently using geometric spectral basis and working condition spectral basis, respectively. The output expression for the geometric spectrum branch is: ； The output expression for the load spectrum branch is: ； In the formula: For the first The hidden feature matrix is ​​input to the layer; This represents the number of grid nodes. For the first The number of feature channels in the layer; The eigenvector matrix of the geometric spectral basis; This is the diagonal matrix of Laplacian eigenvalues ​​corresponding to the geometric graph; For the first Learnable spectral domain transformation weight matrix of layer geometric spectral branch; The eigenvector matrix of the working condition spectrum basis; This is the diagonal matrix of Laplace eigenvalues ​​corresponding to the working condition diagram; For the first Learnable spectral domain transformation weight matrix of layer working condition spectrum branch; Output features for geometric spectrum branches; Output features for the operating condition spectrum branch; The two-way fusion weights are obtained by normalizing the learnable parameters. Weighted fusion is then performed on the outputs of the two branches to obtain the total output of the spectral domain branches. The fusion expression is as follows: ； ； In the formula: For the first Total output of the spectral domain branch; , These are the fusion weighting coefficients for the geometric spectrum branch and the working condition spectrum branch, respectively. For the fusion weight vector; A learnable parameter vector; This indicates a Hadamard weighted operation based on elements or channels. The total output of the spectral domain branch is superimposed with the output of the local linear branch, and then processed by a nonlinear activation function to obtain the first... The hidden features of the layer and the layer update expression are: ； In the formula: For the first The layer outputs the hidden feature matrix; For the first The weight matrix of the local point-state linear mapping of the layer; For the corresponding bias vector; It is a non-linear activation function.

6. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 1, characterized in that, The Laplace spectral-based neural operator network includes the following from input to output: The engineering input layer is used to access node coordinates, boundary condition identifiers, material / fluid parameter fields, geometric partition markers, and source term data, and convert them into standardized node-level features. The input encoding layer maps low-dimensional node features to a high-dimensional hidden feature space through a multilayer perceptron or linear mapping layer. At least one cascaded layer of Laplacian spectral domain operators is used to alternately perform global spectral domain coupling modeling and local feature transformation; The output decoding layer maps high-dimensional hidden features into target physical quantities through a multilayer perceptron or linear mapping layer, and outputs the full-field prediction results of pressure field, velocity field, temperature field, displacement field, concentration field or stress field.

7. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 1, characterized in that, Based on the graph structure, a graph Laplacian matrix is ​​constructed. During the eigenvalue decomposition process of the graph Laplacian matrix, for large-scale mesh scenes, a polynomial spectral filtering method is used instead of the spectral domain transformation of explicit eigenvalue decomposition. Specifically: structure The polynomial filtering function of order X represents the spectral domain convolution operation as a weighted sum of powers of the Laplacian matrix, expressed as: ； In the formula: for Polynomial spectral filtering operator of order 1; The order of the polynomial; For the first Learnable or preset filter coefficients of order; For the Laplace matrix of the graph Power; This is the feature matrix of the input nodes.

8. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 1, characterized in that, In step S4, the total loss function is used as the training optimization objective, and its expression is: ； In the formula: Total training loss; Loss due to data supervision; This is the boundary condition loss, used to constrain the physical quantities at the boundary nodes to meet the preset boundary conditions. The physical residual loss is used to constrain the predicted solution field to satisfy the control relationship of the corresponding partial differential equation. This is a smoothness regularization term used to suppress unreasonable oscillations in the solution field; , , These are the weighting coefficients for boundary condition loss, physical residual loss, and smoothness regularization term, respectively.

9. The irregular mesh engineering simulation method based on the Laplacian neural operator according to claim 8, characterized in that, The data supervision loss uses the relative L2 error between the predicted solution and the true solution, and is expressed as: ； In the formula: The total number of training samples; It is the set of all learnable parameters of the model; For the Laplace spectral basis neural operator surrogate model; For the first Input of working conditions for each sample; For the first The irregular grid corresponding to each sample; For the first High-fidelity real solution field for each sample; Represents the L2 norm of a vector or matrix.

10. An irregular mesh engineering simulation system based on the Laplacian neural operator, used to implement the irregular mesh engineering simulation method based on the Laplacian neural operator as described in any one of claims 1 to 9, characterized in that, Includes offline training modules and online inference modules: The offline training module includes: The training sample acquisition and construction submodule is used to collect several sets of engineering simulation training samples. Each set of training samples includes irregular discrete grid data of the corresponding engineering object, working condition parameter fields, and real physical field obtained by a high-fidelity numerical solver. The Laplace spectral basis calculation submodule is used to construct a corresponding graph structure based on the node connection relationship, spatial distance and boundary features of the irregular discrete grid data, construct a graph Laplace matrix according to the graph structure, and perform eigenvalue decomposition on the graph Laplace matrix to obtain a set of Laplace eigenvectors that are adapted to the irregular grid geometry, which serve as the basis for spectral domain transformation. The spectral domain neural operator inference submodule is used to build a Laplacian spectral basis neural operator network. After mapping the node-level input features to a high-dimensional feature space through the input encoding layer, it is input into at least one Laplacian spectral domain operator layer. The spectral domain operator layer performs spectral domain forward transformation, spectral domain linear filtering, and spectral domain inverse transformation based on the spectral domain transformation basis to realize global physical coupling modeling on irregular grids. After mapping through the output decoding layer, the predicted physical field is obtained. The surrogate model training and optimization submodule is used to iteratively optimize all learnable parameters of the Laplace spectral-based neural operator network with the error between the predicted physics field and the real physics field as the training objective, so as to obtain the trained surrogate solution model. The online inference module receives the target irregular grid data and target operating condition parameters to be predicted. After graph structure construction and Laplace spectral basis calculation, it is input into the surrogate solution model to perform one forward inference and outputs the prediction results of the full field physical quantities under the target operating condition.

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