Discontinuous Galerkin finite element operator network for solving non-smooth PDEs

TL;DR

DG--FEONet is a novel data-free neural network framework that integrates the discontinuous Galerkin method to effectively solve parametric PDEs with discontinuities and non-smooth solutions, outperforming traditional models.

Contribution

The paper introduces a data-free operator learning approach combining DG methods with neural networks, enabling accurate, generalizable solutions for non-smooth PDEs without precomputed datasets.

Findings

Accurately captures discontinuities in PDE solutions

Demonstrates strong generalization across parameter space

Achieves reliable convergence rates

Abstract

We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations (PDEs) with discontinuous coefficients and non-smooth solutions. Unlike traditional operator learning models such as DeepONet and Fourier Neural Operator, which require large paired datasets and often struggle near sharp features, our approach minimizes the residual of a DG-based weak formulation using the Symmetric Interior Penalty Galerkin (SIPG) scheme. DG-FEONet predicts element-wise solution coefficients via a neural network, enabling data-free training without the need for precomputed input-output pairs. We provide theoretical justification through convergence analysis and validate the model's performance on a series of one- and two-dimensional PDE problems, demonstrating accurate recovery of discontinuities, strong generalization across parameter space, and reliable convergence rates. Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation in challenging PDE settings.