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

TL;DR

This paper introduces DG-FEONet, a data-free neural operator framework that couples Discontinuous Galerkin Finite Element Methods with neural networks to learn parametric PDE solution operators in the presence of discontinuities. The network predicts local DG coefficients and is trained by minimizing the SIPG weak-form residual, bypassing the need for labeled input-output data. A rigorous convergence analysis is provided, showing coefficient-space convergence as network capacity grows and generalization improves with sample size, leading to overall convergence of the predicted PDE solution in $L^2$ over parameter and physical domains. Numerical experiments in 1D and 2D demonstrate accurate resolution of discontinuities and robust generalization across complex interface geometries, outperforming CG-based methods and PINNs on non-smooth problems.

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.

Figures (11)

• Figure 1: Schematic overview of the DG-FEONet framework. The PDE inputs, which we emphasize that these can be discontinuous, are passed to a neural network that predicts the coefficients for DG-FEM solutions. These coefficients are used to reconstruct the DG solution using basis functions defined over a triangulated domain. The network is trained in an data-free manner using the residual of the DG weak formulation as the loss function.
• Figure 1: Performance of DG-FEONet for a discontinuous diffusion coefficient (Experiment 1). (a) Solution plots for $m=5,10,$ and $100$. (b) Convergence of the average relative $\ell^2$-error $E_{\mathrm{rel}}$ for different values of $m$.
• Figure 2: Comparison of the DG reference solution, DG-FEONet prediction, and PINN approximation for Experiment 1 with a discontinuous diffusion coefficient $\varepsilon(x)$ and jump factor $m=10$.
• Figure 3: Three representative examples showing the input reaction coefficient $c(x)$ with discontinuities (left) and the DG-FEONet solution $u^\text{nn}$ vs. FEM solution $u^{\text{FEM}}$ (right) (Experiment 2). Annotated arrows indicate the magnitude of the jumps $|\Delta c|$ across elements. DG-FEONet successfully resolves both the discontinuous input and the resulting non-smooth output.
• Figure 4: Comparison of CG-FEONet and DG-FEONet for a sample with two discontinuities (Experiment 2). The CG-FEONet solution oversmooths the jumps, while DG-FEONet resolves them and agrees with the DG reference solution.

Theorems & Definitions (13)

• Definition 4.1: DG energy norm
• Lemma 4.3
• Theorem 4.4
• Definition 4.5
• Theorem 4.6
• Proof 1
• Lemma 4.7
• Theorem 4.8
• Lemma 4.9
• Theorem 4.10