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

TL;DR

The paper tackles fast PDE solving on variable geometries by marrying phi-FEM with Fourier Neural Operators to create a geometry-aware neural operator. By treating the domain boundary via a level-set $\varphi$ as an input and training on synthetic phi-FEM data, the method delivers near-FEM accuracy with massive speedups on multiple test cases, including Poisson problems and nonlinear elasticity with holes. Key contributions include a detailed phi-FEM-FNO architecture, loss formulations that enforce boundary conditions and reduce Gibbs effects, and demonstrated scalability across elliptic and hyperelastic settings. The approach offers real-time capability for complex geometries, with precise boundary handling and potential for extension to broader PDE classes and higher-order discretizations.

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.

Figures (17)

• Figure 1: Left: example of $\varphi$-FEM meshes. In red, the exact boundary $\Gamma$ of an ellipse $\Omega$, in white $\mathcal{T}_h^{\mathcal{O}}$, in gray $\mathcal{T}_h^{\Gamma}$ and in blue, $\mathcal{T}_h \setminus \mathcal{T}_h^{\Gamma}$. Right: Convergence curves of $\varphi$-FEM and a standard finite element method, to solve \ref{['eq:governing_poisson']} for 5 combinations of domain, force, and boundary conditions.
• Figure 2: Construction of a prediction of $\varphi$-FEM-FNO to solve \ref{['eq:governing_poisson']}.
• Figure 3: The $\varphi$-FEM-FNO pipeline to solve \ref{['eq:governing_poisson']}. Illustration based on the representation of paper_FNO. The upper part represents the entire pipeline, and the lower part is a zoom on a Fourier layer. The red circles correspond to the inputs provided by the user and the output returned by our $\varphi$-FEM-FNO. We represent the inputs and outputs seen by the FNO in purple, where $X = (f_h,\varphi_h, g_h)$. In orange, $P_\theta$ and $Q_\theta$ are two transformations parameterized by neural networks. Moreover, $\mathcal{F}$ and $\mathcal{F}^{-1}$ are respectively the Fourier and inverse Fourier transforms. In blue, $\sigma$ is the activation function. Finally, black arrows correspond to steps inside our FNO, and purple arrows to steps outside the FNO.
• Figure 4: In red, the real boundary of an example domain. In blue and gray, the set $\mathcal{S}_0$. In gray, $\mathcal{S}_1$.
• Figure 5: Test case 1. On the left-hand side (resp. right-hand side), we represent the evolution of the cost function $\mathcal{L}$ (resp. the relative $L^2$ error) on a subset of the training set and on the validation set.

Theorems & Definitions (10)

• Remark
• Remark : Number of parameters
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• Remark : Data generation
• Remark : Implementation aspect.
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• Remark : Calibration of the learning rate.