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Neural Network Element Method for Partial Differential Equations

Yifan Wang, Zhongshuo Lin, Hehu Xie

TL;DR

The paper addresses solving second-order elliptic PDEs on complex geometries by merging a finite element mesh with neural networks to form a neural network element space $V_{\rm NNh}(\mathcal{T}_h,\theta)$. It constructs this space by coupling FEM envelope functions with local NN components, then solves a Galerkin problem and adaptively updates NN parameters via a Ritz-type loss, yielding a two-stage training process that separates linear solves from neural optimization. A partition-of-unity-based error analysis shows that the global NN element approximation inherits the local NN approximation properties, and the authors demonstrate the approach on a 2D Laplace problem with a structured NN architecture, achieving high accuracy and efficiency. This method strategically leverages FEM's geometry handling and NN's expressive power, enabling accurate PDE solvers for engineering applications and paving the way for extensions to 3D domains and varied boundary conditions.

Abstract

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The application of finite element mesh makes the neural network element space satisfy the boundary value conditions directly on the complex geometric domains. The use of neural networks allows the accuracy of the approximate solution to reach the high level of neural network approximation even for the problems with singularities. We also provide the error analysis of the proposed method for the understanding. The proposed numerical method in this paper provides the way to enable neural network-based machine learning algorithms to solve a broader range of problems arising from engineering applications.

Neural Network Element Method for Partial Differential Equations

TL;DR

The paper addresses solving second-order elliptic PDEs on complex geometries by merging a finite element mesh with neural networks to form a neural network element space . It constructs this space by coupling FEM envelope functions with local NN components, then solves a Galerkin problem and adaptively updates NN parameters via a Ritz-type loss, yielding a two-stage training process that separates linear solves from neural optimization. A partition-of-unity-based error analysis shows that the global NN element approximation inherits the local NN approximation properties, and the authors demonstrate the approach on a 2D Laplace problem with a structured NN architecture, achieving high accuracy and efficiency. This method strategically leverages FEM's geometry handling and NN's expressive power, enabling accurate PDE solvers for engineering applications and paving the way for extensions to 3D domains and varied boundary conditions.

Abstract

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The application of finite element mesh makes the neural network element space satisfy the boundary value conditions directly on the complex geometric domains. The use of neural networks allows the accuracy of the approximate solution to reach the high level of neural network approximation even for the problems with singularities. We also provide the error analysis of the proposed method for the understanding. The proposed numerical method in this paper provides the way to enable neural network-based machine learning algorithms to solve a broader range of problems arising from engineering applications.

Paper Structure

This paper contains 8 sections, 2 theorems, 39 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural network element space and its basis
  3. Envelop function from finite element mesh
  4. Neural network element basis
  5. Error estimates of NN element approximation
  6. NN element based machine learning method
  7. Numerical examples
  8. Concluding remarks

Key Result

Lemma 2.1

The cover $\{\Omega_i\}_{i=1}^N$ of $\Omega$ satisfies the following pointwise overlap condition Then $\left\{\psi_i\right\}_{i=1}^N$ is a Lipschitz partition of unity subordinate to the cover $\mathcal{T}_h$ satisfying where ${\rm diam}(\Omega_i)$ denotes the diameter of $\Omega_i$, $C_{\infty}$ and $C_G$ are two constants independent of the mesh size.

Figures (6)

  • Figure 1: The finite element mesh $\mathcal{T}_h$
  • Figure 2: The patch $\omega_Z$ for one node $Z$
  • Figure 3: The patch $\omega_E$ for one edge $E$ and the corresponding basis $\lambda_2\lambda_3$
  • Figure 4: The triangle $K$ and the corresponding basis $\lambda_1\lambda_2\lambda_3$
  • Figure 5: The mesh and corresponding vertices, edges and triangles for the L shape domain
  • ...and 1 more figures

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2