Enriching continuous Lagrange finite element approximation spaces using neural networks
Hélène Barucq, Michel Duprez, Florian Faucher, Emmanuel Franck, Frédérique Lecourtier, Vanessa Lleras, Victor Michel-Dansac, Nicolas Victorion
TL;DR
The paper develops a hybrid PDE solver that enriches coarse continuous FEM spaces with priors from neural networks, specifically PINNs, to achieve fast, accurate, and provably convergent solutions for parametric linear elliptic problems. It proposes two enrichment strategies—additive and multiplicative priors—each with rigorous error analyses that bound the FEM error by the prior quality, enabling reliable gains on coarse meshes. Priors are constructed from parametric PINNs, with improvements such as exact boundary condition imposition, Sobolev training, and Fourier features to handle high-frequency solutions. Extensive numerical experiments in 1D, 2D, and 3D demonstrate that the enriched bases can match or surpass standard FEM accuracy on coarser meshes while reducing computational cost, especially for parametric studies. The approach provides a principled framework for integrating physics-informed networks with classical FEM, preserving convergence guarantees while accelerating solutions in complex, parametric settings.
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.