Discontinuous Galerkin finite element operator network for solving non-smooth PDEs
This work addresses the challenge of solving non-smooth PDEs in computational science and engineering, offering a robust, singularity-aware operator approximation method, though it is incremental as it builds on existing operator learning and DG techniques.
The authors tackled the problem of solving parametric PDEs with discontinuous coefficients and non-smooth solutions by introducing DG-FEONet, a data-free operator learning framework that combines discontinuous Galerkin methods with neural networks, achieving accurate recovery of discontinuities and strong generalization across parameter space.
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.