用不连续伽辽金法构建神经网络，高效求解复杂方程
DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods
- 用分片神经网络做试函数，分片多项式作检验函数
- 在稳态与时变问题上精度高、训练快，抗扰动强
- 适合求解带间断解或复杂几何的偏微分方程
我们提出一种基于内罚不连续伽辽金法（IPDGM）的通用框架——不连续伽辽金诱导神经网络（DGNN）。该方法中，试函数空间由定义在计算域上的分片神经网络构成，检验函数空间则由分片多项式组成。我们在多个数值例子中验证了DGNN的优势，涵盖稳态与时变问题。DGNN能有效处理高扰动、间断解以及复杂几何区域，表现出优异的精度与训练效率。
原文摘要 · Abstract (English)
We propose a general framework for the Discontinuous Galerkin-induced Neural Network (DGNN), inspired by the Interior Penalty Discontinuous Galerkin Method (IPDGM). In this approach, the trial space consists of piecewise neural network space defined over the computational domain, while the test function space is composed of piecewise polynomials. We demonstrate the advantages of DGNN in terms of accuracy and training efficiency across several numerical examples, including stationary and time-dependent problems. Specifically, DGNN easily handles high perturbations, discontinuous solutions, and complex geometric domains.
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