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

• URL: http://arxiv.org/abs/2601.03668v1
• Date: Wed, 07 Jan 2026 07:43:30 GMT
• Title: Discontinuous Galerkin finite element operator network for solving non-smooth PDEs
• Authors: Kapil Chawla, Youngjoon Hong, Jae Yong Lee, Sanghyun Lee,
• Abstract summary: We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework.<n>It combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations.<n>Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation.
• Score: 15.286345729268149
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

Related papers

• Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations [65.80144621950981]
  We build on Wiener chaos expansions (WCE) to design neural operator (NO) architectures for SPDEs and SDEs.<n>We show that WCE-based neural operators provide a practical and scalable way to learn SDE/SPDE solution operators.
  arXiv  Detail & Related papers  (2026-01-03T00:59:25Z)
• Physics-informed low-rank neural operators with application to parametric elliptic PDEs [0.0]
  We present PILNO, a neural operator framework for approximating solution operators of partial differential equations (PDEs) on point cloud data.<n>PILNO combines low-rank kernel approximations with an encoder--decoder architecture, enabling fast, continuous one-shot predictions while remaining independent of specific discretizations.<n>We demonstrate its effectiveness on diverse problems, including function fitting, the Poisson equation, the screened Poisson equation with variable coefficients, and parameterized Darcy flow.
  arXiv  Detail & Related papers  (2025-09-09T12:54:06Z)
• Accelerating PDE Solvers with Equation-Recast Neural Operator Preconditioning [9.178290601589365]
  Minimal-Data Parametric Neural Operator Preconditioning (MD-PNOP) is a new paradigm for accelerating parametric PDE solvers.<n>It recasts the residual from parameter deviation as additional source term, where trained neural operators can be used to refine the solution in an offline fashion.<n>It consistently achieves 50% reduction in computational time while maintaining full order fidelity for fixed-source, single-group eigenvalue, and multigroup coupled eigenvalue problems.
  arXiv  Detail & Related papers  (2025-09-01T12:14:58Z)
• Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum [78.27945336558987]
  Decentralized server (DFL) eliminates reliance on client-client architecture.<n>Non-smooth regularization is often incorporated into machine learning tasks.<n>We propose a novel novel DNCFL algorithm to solve these problems.
  arXiv  Detail & Related papers  (2025-04-17T08:32:25Z)
• Spectral operator learning for parametric PDEs without data reliance [6.7083321695379885]
  We introduce a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques.
  arXiv  Detail & Related papers  (2023-10-03T12:37:15Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs [0.8594140167290099]
  We perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet) ULGNet is a deep-learning-based numerical method for solving partial differential equations (PDEs)
  arXiv  Detail & Related papers  (2022-11-16T13:31:03Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.