Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.

翻译：偏微分方程（PDE）的数值求解器对科学与工程具有根本性意义。迄今为止，对传统技术的历史性依赖限制了大数据知识的可能整合，并且对某些PDE公式表现出次优效率，而数据驱动的神经方法通常缺乏收敛性与正确性的数学保证。本文阐述了一种数学上严谨的线性PDE神经求解器。所提出的UGrid求解器建立在U-Net与MultiGrid的原则性集成之上，展现了收敛性与正确性的数学严谨证明，并展示了高数值精度，以及对各种输入几何/值和多种PDE公式的强大泛化能力。此外，我们设计了一种新的残差损失度量，该度量支持无监督训练，并在传统损失函数基础上提供了更高的稳定性和更大的解空间。