Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries

TL;DR

This paper introduces Phi-FEM-FNO, a novel method combining Fourier Neural Operators with the Phi-FEM finite element approach to efficiently solve PDEs on variable geometries, demonstrating promising numerical results.

Contribution

It presents a new hybrid approach integrating neural operators with Phi-FEM to handle PDEs on changing domains, especially for hyperelastic materials.

Findings

Effective solution for PDEs on varying geometries

Numerical results show high accuracy and efficiency

New scheme for hyperelastic materials

Abstract

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping operator. The purpose of this paper is to provide numerical evidence to show the effectiveness of this technique. We will focus here on the resolution of two equations: the Poisson-Dirichlet equation and the non-linear elasticity equations. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. The considered domains are defined by level-set functions due to the use of the Phi-FEM approach. We will first recall the idea of $\varphi$-FEM and of the Fourier Neural Operator. Then, we will explain how to combine these two methods. We will finally illustrate the efficiency of this combination with some numerical results on three test cases. In addition, in the last test case, we propose a new numerical scheme for hyperelastic materials following the Phi-FEM paradigm.