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A PINN-enriched finite element method for linear elliptic problems

Xiao Chen, Yixin Luo, Jingrun Chen

TL;DR

The paper introduces a hybrid solver that couples physics-informed neural networks (PINNs) with the finite element method (FEM) to tackle linear elliptic PDEs. A PINN is trained to produce $u_{\theta}$, which enriches the FE space through additive $V_{h}^{+}=V_{h}+u_{\theta}$ or multiplicative $V_{h}^{*}=V_{h}\,u_{\theta}$, after which FEM solves the problem in this enriched space. The authors provide an a priori error analysis showing that the enriched methods retain the same convergence order as classical FEM, with potential accuracy gains tied to the PINN's derivative accuracy, and they address boundary conditions via a boundary operator and a two-phase training strategy. Numerical experiments in one to three dimensions validate the approach, illustrating both improved accuracy in some settings and limitations in others, and pointing to future work on adaptive meshes and extensions to nonlinear problems.

Abstract

In this paper, we propose a hybrid method that combines finite element method (FEM) and physics-informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an approximate solution $u_θ$; (2) enrich the finite element space with $u_θ$; (3) obtain the final solution by FEM in the enriched space. In the second step, the enriched space is constructed by addition $v + u_θ$ or multiplication $v \cdot u_θ$, where $v$ belongs to the standard finite element space. We conduct the convergence analysis for the proposed method. Compared to the standard FEM, the same convergence order is obtained and higher accuracy can be achieved when solution derivatives are well approximated in PINN. Numerical examples from one dimension to three dimensions verify these theoretical results. For some examples, the accuracy of the proposed method can be reduced by a couple of orders of magnitude compared to the standard FEM.

A PINN-enriched finite element method for linear elliptic problems

TL;DR

The paper introduces a hybrid solver that couples physics-informed neural networks (PINNs) with the finite element method (FEM) to tackle linear elliptic PDEs. A PINN is trained to produce , which enriches the FE space through additive or multiplicative , after which FEM solves the problem in this enriched space. The authors provide an a priori error analysis showing that the enriched methods retain the same convergence order as classical FEM, with potential accuracy gains tied to the PINN's derivative accuracy, and they address boundary conditions via a boundary operator and a two-phase training strategy. Numerical experiments in one to three dimensions validate the approach, illustrating both improved accuracy in some settings and limitations in others, and pointing to future work on adaptive meshes and extensions to nonlinear problems.

Abstract

In this paper, we propose a hybrid method that combines finite element method (FEM) and physics-informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an approximate solution ; (2) enrich the finite element space with ; (3) obtain the final solution by FEM in the enriched space. In the second step, the enriched space is constructed by addition or multiplication , where belongs to the standard finite element space. We conduct the convergence analysis for the proposed method. Compared to the standard FEM, the same convergence order is obtained and higher accuracy can be achieved when solution derivatives are well approximated in PINN. Numerical examples from one dimension to three dimensions verify these theoretical results. For some examples, the accuracy of the proposed method can be reduced by a couple of orders of magnitude compared to the standard FEM.

Paper Structure

This paper contains 21 sections, 9 theorems, 47 equations, 8 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Physics-informed neural networks
  4. classical finite element scheme
  5. Enriching Approximation Space of FEM with PINN
  6. Additive Space
  7. Multiplicative Space
  8. Dealing with Boundary Conditions
  9. Error Analysis
  10. Error Analysis of PINN
  11. Error Analysis of Modified FEM
  12. Numerical Results
  13. Poisson Equation in One Dimension
  14. biharmonic equation in One Dimension
  15. Linear Elliptic Equation in Two Dimension
  16. ...and 6 more sections

Key Result

Lemma 3.1

If $x$ is a zero point of $u_{\theta}$, any function $v \in V_{h}^{*}$ has the same zero point $x$.

Figures (8)

  • Figure 1: left: $L^{2} error$ using different spaces, right: $H^{1} error$ using different spaces.
  • Figure 2: left: mesh when $h=2^{-2}$, with 32 triangles and 25 nodes, right: mesh when $h=2^{-3}$, with 128 triangles and 81 nodes.
  • Figure 3: Comparison between solutions obtained using classical FEM and multiplicative space for cases $c(x)=-2\pi^{2}$+0.01 with $P^{2}$ Lagrange element and mesh size $h=2^{-5}$.
  • Figure 4: training with boundary operator. From left to right: training loss (A.2) with 10000 epochs using Adam; solution; error of $u-u_{\theta}$, error of second order derivative of $u-u_{\theta}$
  • Figure 5: training with penalty term. From left to right: training loss (A.3) with 10000 epochs using Adam; solution; $u-u_{\theta}$, second order derivative of $u-u_{\theta}$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Theorem 4.1
  • ...and 9 more