Neural operator network method based on grid parameterization

Abstract

The invention discloses a neural operator network method based on grid parameterization. The method comprises the following steps of: drawing grids of a physical domain by using a grid division tool or software; mapping grids of different physical domains to grids of a general domain, and sampling on the general domain; based on the geometric domain and the parameters, generating an input and output field data set, and sampling the input and output fields at the acquired sampling points; and based on the generated data set and the deep neural operator network, solving PDEs (partial differential equation). According to the method, operator problems on different domains are converted into solution on a general domain at the same time, so that PDEs solution across different domains is realized.

Description

Technical Field

[0001] The present invention belongs to the field of neural network solving partial differential equations, and in particular relates to a neural operator network method based on grid parameterization. Background Art

[0002] Partial differential equations (PDEs) are used to mathematically represent physical systems with varying geometric domains and initial / boundary conditions (parameters) in many scientific and engineering applications. The ubiquity and fundamental nature of PDEs make the development of efficient and robust methods for solving them crucial for scientific discovery and technological innovation. The diversity and complexity of real-world physical phenomena have given rise to a wide variety of PDEs, placing diverse demands on their solution methods. No single method is perfectly applicable to all types of PDE problems, driving researchers to continuously explore and develop new solution theories and techniques.

[0003] To date, different types of PDEs solvers have been developed, which can generally be divided into classical numerical solvers and neural network-based solvers.

[0004] Classical numerical methods, such as the finite element method and the finite difference method, solve PDEs by discretizing the space according to a specific mesh design. These methods are granularity-dependent, favoring fine-grained meshes for accuracy and coarse-grained meshes for efficiency. Therefore, when the target PDEs are complex, they inevitably face a trade-off between accuracy and efficiency. Solving PDEs with traditional numerical methods requires extensive computational recalculations, which is extremely time-consuming and resource-intensive even for minor changes in the domain and parameters.

[0005] With the rapid development of artificial intelligence and machine learning technologies, neural network-based methods for solving PDEs have gradually become a research hotspot in scientific computing and engineering applications. Recent advances in artificial intelligence (AI), including the application of deep learning techniques to solving PDEs, have demonstrated remarkable performance in mechanics, heat transfer, fluid dynamics, and climate modeling.

[0006] Grid-dependent finite-dimensional operators discretize the domain into a grid and then approximate the solution operator as a parameterized neural network in a finite Euclidean space. These solvers are grid-dependent, resulting in limited generalization. Neural finite element methods directly parameterize the solution to the equation as a neural network. While these solvers are grid-independent and accurate, they are limited to learning a single instance of PDEs, rather than an entire family. Therefore, similar to classical numerical solvers, these solvers require new network design and training every time the instance changes. Furthermore, most of these solvers are not applicable to situations where the underlying PDEs are unknown.

[0007] Neural operators learn grid-dependent infinite-dimensional solution operators through neural networks. The paper "Kovachki, N. et al. Neural operator: Learning maps between function spaces with applications to PDEs. J. Mach. Learn. 24, 1–97 (2023)" proposes viewing operators as mappings between parameter function spaces and solution function spaces, and directly learning physical laws from data on similar PDEs with different parameters. Because these solvers directly learn solution operators, they only need to be trained once for the target PDEs, which is computationally advantageous. Neural operators include the Deep Neural Operator (DeepONet), the Fourier Neural Operator (FNO), and the Koopman Neural Operator (KNO).

[0008] However, these neural operator frameworks have a significant limitation, that is, they are mainly designed for situations where parameters can change but the solution domain (i.e., the computational area of ​​the physical system) remains fixed. This means that once the domain undergoes even a slight change, such as adjustment of the shape or boundary, the model needs to be retrained. This retraining process is not only time-consuming, but also greatly increases computing resources and training costs, which brings great inconvenience to practical applications. At present, research on more general neural operator frameworks that can adapt to changing domains and handle parameter changes is still insufficient, and this field still faces many unresolved challenges. Despite this, some studies have attempted to address the problem of solving PDEs on different domains through various methods, such as transfer learning, Transformer architecture, and mapping to grid latent space, but these methods still have many problems that need to be solved. Specifically, while transfer learning can reuse existing models to a certain extent, it often struggles to effectively capture the geometric characteristics of the new domain when faced with domain changes, such as differences in boundary conditions or region shapes, resulting in a significant decrease in model performance in new scenarios. Expanding to larger domains attempts to address the problem by encompassing all possible shapes, but this significantly increases computational complexity, especially when dealing with highly irregular or complex geometric domains, where the results are often unsatisfactory. The Transformer architecture demonstrates strong capabilities in processing sequential data, but its adaptability to domain geometry remains insufficient in solving PDEs, making it difficult to flexibly address changing physical scenarios. Methods that map to a grid latent space attempt to simplify computation by unifying problems across different domains onto a standardized grid representation, but the accuracy and reversibility of this mapping are difficult to guarantee, especially when dealing with complex geometric domains, where it can easily introduce large approximation errors, affecting the accuracy of the solution. Furthermore, these methods also suffer from significant theoretical deficiencies, such as a lack of a solid mathematical foundation to guarantee model convergence and stability, which raises questions about their reliability and credibility in practical applications. At the same time, the generalization ability of neural operators when dealing with different domains is relatively limited. In particular, when the domain shapes vary greatly, the prediction accuracy of the model will drop significantly. This problem seriously restricts its widespread application in fields that require frequent adjustments to the domain shape (such as engineering design and biomedical modeling). Therefore, developing a general neural operator framework that can efficiently and accurately solve PDEs with varying domains and parameters has become an urgent need and an important direction of current research. Summary of the Invention

[0009] In response to the deficiencies in the prior art, the present invention provides a neural operator network method based on grid parameterization, which solves the challenge of learning function mapping in different domains by mapping different physical domains to the same general domain and by converting the learning function mapping into the task of learning operators on the general domain.

[0010] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is as follows:

[0011] A neural operator network method based on grid parameterization includes the following steps:

[0012] Step 1: Discretize the computational domain of the physical problem in space and generate a standard OBJ file using a meshing tool. This file contains vertex and facet information describing the domain boundaries and internal nodes.

[0013] Step 2: To handle geometric domains of different shapes, the grids of each domain are parameterized and uniformly mapped to a standardized computational domain, and regular or irregular sampling is performed within this domain.

[0014] Step 3: Based on the given geometric constraints and physical parameters, programmatically construct data pairs of input and output fields, and extract the field values ​​at the previously generated sampling points to form a structured training data set.

[0015] Step 4: The dataset is used to drive supervised training of a deep neural operator network, and an independent test set is used to evaluate the prediction accuracy and effectiveness of the trained model.

[0016] The beneficial effects of the present invention are as follows: the present invention provides a neural operator network method based on grid parameterization, which solves the challenge of learning function mapping in different domains. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings used in the embodiments.

[0018] Figure 1 This is a core module diagram of the neural operator network method based on grid parameterization in an embodiment of the present invention;

[0019] Figure 2 is an architecture diagram of a neural operator network method based on grid parameterization in an embodiment of the present invention;

[0020] Figure 3 1 is a diagram showing the effect of solving the Darcy flow equation using a neural operator network method based on grid parameterization in an embodiment of the present invention;

[0021] Figure 4 1 and 2 are the initial mesh diagram (a) and the mapped mesh diagram (b) of the airfoil in an embodiment of the present invention. DETAILED DESCRIPTION

[0022] In order to make the technical solution of the present invention clearer, the present invention will be clearly and completely explained below with reference to the drawings in the embodiments of the invention. The described embodiments only use a part of the embodiments, rather than all the embodiments.

[0023] See also Figure 1 and Figure 2 As shown, the present invention is specifically implemented according to the following steps:

[0024] (1) Use meshing tools to mesh the physical domain. For example, for a pentagonal physical domain, first use MATLAB to randomly generate geometric shapes, and then use Abaqus to mesh. For the airfoil (the wing shape comes from the UIUC wing database), use Pointwise to generate the airfoil shape and mesh it.

[0025] (2) Map the grids of different physical domains to the grids of the general domain through the method of grid parameterization. For a simply connected domain, the present invention adopts a harmonic mapping. Usually, the harmonic mapping is only applicable to a simply connected domain. For a multiply connected domain, the present invention adopts a conformal mapping and a quasi-conformal mapping. The conformal mapping is the one with the smallest harmonic energy among all the harmonic mappings. Usually, the mapping that one hopes to find is subject to certain restrictions, and the conformal transformation that meets these restrictions may not exist. It is necessary to extend the conformal mapping, such as the quasi-conformal mapping. After the mapping is completed, regular sampling is performed on the general domain, and the sampled points are linearly interpolated to map the points of the general domain back to the physical domain. The geometric coordinates of the corresponding sampling points in the physical domain are saved.

[0026] (3) According to the geometric information of the physical domain and the parameters of PDEs, the numerical simulation method is used to obtain the solution of the PDEs system under different physical domains and parameters, generate the input and output field data sets of the neural operator network, and sample the input and output fields at the obtained sampling points. In the physical domain, the parameter function a(X pg ) and solution function u(X pg ) is sampled through the sampling points. Then, these sampled values ​​are mapped to the universal domain, so that in the universal domain X Ig The parameter function a is generated at the sampling points in I (X Ig ) and solution function u I (X Ig ). See Figure 2 .

[0027] (4) To learn the solution operator on a general domain, this method constructs a neural network model based on FNO. The training process of this model is data-driven and relies on a dataset of a given input field and the corresponding output field. After training, the model's prediction accuracy and generalization performance are systematically verified using a reserved test set.

[0028] The present invention provides two embodiments, one of which is Darcy flow. Darcy's law describes the pressure of a fluid flowing in a porous medium with a specific permeability. In this embodiment, the steady state of the Darcy flow equation is considered, and the PDE is a second-order linear elliptic equation with a Dirichlet boundary. The parametric geometric representation of the Darcy flow problem consists of the coordinates of five points forming a polygon. Another embodiment is the airfoil flow problem in fluid dynamics. For the airfoil flow embodiment, the airfoil shape is from the UIUC airfoil database.

[0029] Example 1: Darcy Flow Equation. The two-dimensional Darcy flow equation has many applications in science and engineering, including modeling the pressure of underground flows, the deformation of linear elastic materials, and the electric potential in conductive materials. The Darcy flow equation is as follows:

[0030] -▽(a(x,y)▽(u(x,y)))=F(x,y) (x,y)∈Ω (1)

[0031]

[0032] Where a(x,y) is the diffusion coefficient (parametric function), u(x,y) is the solution function, and F(x,y) is the force function.

[0033] For Example 1, the pentagonal physical domain is mapped to the square general domain by the grid parameterization method. The solution effect is shown in Figure 3 . Figure 3 Shown are the true solution to the Darcy flow equation, the predicted solution from the neural network, and the error between the two.

[0034] Example 2: Fluid Dynamics Equations. Fluid dynamics analysis is important across a variety of engineering disciplines, helping engineers and scientists design and optimize various systems in aerospace, automotive, energy, and environmental infrastructure. Most fluids can be modeled using the Navier-Stokes (NS) equations, which are nonlinear equations that describe the interaction of velocity and pressure fields within a fluid domain. These equations are expressed as follows:

[0035]

[0036] ▽·v=0 (4)

[0037] in, μ is the viscosity, ρ is the fluid density, f is the body force, and the velocity field is the solution function.

[0038] In the specific airfoil flow problem, the PDE equation is described by the Reynolds-averaged Navier-Stokes (RANS) equation, which is derived from the more basic NS equation by introducing additional terms in the turbulence effect model. In the embodiment, the square domain containing the airfoil is mapped to a ring domain with a unit outer radius by quasi-conformal mapping. The initial mesh of the airfoil and the mesh after mapping are shown in FIG. Figure 4 .

[0039] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may be modified and varied in various ways.

Claims

1. A neural operator network method based on grid parameterization, characterized in that: Specifically include: Discretize the computational domain of the physical problem in space and generate a standard OBJ file using a meshing tool. This file contains vertex and facet information describing the domain boundaries and internal nodes. In order to handle geometric domains of different shapes, the grids of each domain are parameterized and mapped uniformly to a standardized computational domain, and regular or irregular sampling is performed within this domain. Based on given geometric constraints and physical parameters, programmatically construct data pairs of input and output fields, and extract field values ​​at previously generated sampling points to form a structured training dataset; The dataset is used to drive the supervised training of a deep neural operator network, and an independent test set is used to evaluate the prediction accuracy and effectiveness of the trained model.

2. A neural operator network method based on grid parameterization according to claim 1, characterized in that: The spatial discretization can be performed by using different meshing tools for different problems, including: The first category targets procedurally generated geometry, using MATLAB scripts to batch create a series of convex polygonal domains and calling Abaqus software to perform automated meshing. The second category targets complex engineering shapes, selecting standard airfoils from the UIUC wing database and using the professional mesh generation software Pointwise for refined mesh drawing. All final meshes are processed into a unified OBJ data format, which fully records the vertex and facet information required to define the physical domain.

3. The neural operator network method based on grid parameterization according to claim 1, characterized in that: The grid parameterization method specifically includes: First, based on topological analysis, the topological connectivity of the physical domain mesh is judged; if the physical domain mesh is a simply connected domain mesh, harmonic mapping is used to complete the parameterization; if the physical domain mesh is a multi-connected domain mesh with complex topology, conformal mapping or quasi-conformal mapping is used.

4. The neural operator network method based on grid parameterization according to claim 1, characterized in that: The generating of input and output field data sets specifically includes: To ensure the reliability and accuracy of neural operator model training, a rigorous "true value" data curation process was designed. With the help of mature numerical simulation solvers, each problem instance was solved with high precision to generate authoritative solution data, which constitute the "true value" portion of this dataset. By evaluating these paired (input, output) functions at discrete sampling points, this high-dimensional function space mapping problem was converted into finite-dimensional vector pairs that can be learned by neural networks.

5. The neural operator network method based on grid parameterization according to claim 1, characterized in that: The training and testing phases of the operator network specifically include: This operator network can reveal and grasp the physical laws behind diverse PDEs data. Its essence is to learn a universal solver that can directly map "problems" (parameter functions) to "answers" (solution functions). To achieve this goal, the network uses an efficient Fourier neural operator (FNO) for training, and further integrates the geometric features of the physical domain and the abstract domain to make the network's understanding and prediction of complex problems more profound and accurate.

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