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

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called φ -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 φ -FEM approach. We will first recall the idea of φ -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 φ -FEM paradigm. • A new ML-based approach called φ -FEM-FNO to solve PDEs taking the geometry as input. • Comparison of φ -FEM-FNO with other ML-based methods. • A new φ -FEM scheme to solve hyperelastic problems on complex geometries.