Enriching continuous Lagrange finite element approximation spaces using neural networks

TL;DR

This paper introduces a novel method that combines neural networks with finite element spaces to improve PDE solutions, achieving higher accuracy with coarser meshes and providing error estimates to certify performance.

Contribution

It proposes enriching FEM approximation spaces with neural network predictions, leading to faster, more accurate solutions with proven error bounds.

Findings

Enriched FEM reduces computational time for parametric PDE problems.

The approach outperforms classical FEM based on error estimates.

Numerical validation on 1D, 2D, and 3D problems confirms effectiveness.

Abstract

In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.