用神经网络+不连续有限元法求解含间断系数的偏微分方程。
Discontinuous Galerkin finite element operator network for solving non-smooth PDEs
- 用SIPG弱形式残差最小化，通过神经网络预测单元解系数。
- 无需配对数据，在一维和二维问题上准确恢复间断点，收敛率稳定。
- 适合处理含奇点、参数多变的复杂偏微分方程，特别适合工程仿真场景。
我们提出Discontinuous Galerkin Finite Element Operator Network（DG-FEONet），一种无需数据的算子学习框架，结合不连续伽辽金（DG）方法与神经网络，用于求解具有间断系数和非光滑解的参数化偏微分方程（PDE）。与DeepONet、Fourier Neural Operator等需大规模配对数据且在尖锐特征附近表现不佳的传统模型不同，本方法基于对称内罚伽辽金（SIPG）方案，最小化DG弱形式残差。通过神经网络预测每个单元的解系数，实现无需预计算输入输出对的数据自由训练。我们提供了收敛性理论分析，并在一系列一维和二维PDE问题上验证了模型性能，结果表明其能准确恢复间断，对参数空间具有强泛化能力，且收敛率可靠。研究展示了将局部离散化方法与机器学习结合，在挑战性PDE场景中实现鲁棒、奇点感知算子近似的潜力。
原文摘要 · Abstract (English)
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.
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