论文标题
不规则域的神经PDE求解器
Neural PDE Solvers for Irregular Domains
论文作者
论文摘要
基于神经网络的解决部分微分方程（PDE）的方法最近受到了特别关注。但是，绝大多数神经PDE求解器仅适用于直线域，并且在不规则的域边界上没有系统地解决Dirichlet/Neumann边界条件的强加。在本文中，我们为神经求解的框架求解了偏微分方程，该方程在具有不规则形状（非线性）几何边界的域上。我们的网络将域的形状作为输入（使用非结构化点云表示代表，或任何其他参数表示，例如不均匀的有理B-Splines），并能够推广到新颖（未见）不规则的域；实现此模型的关键技术成分是一种新颖的方法，用于以可区分的方式识别计算网格的内部和外部。我们还进行了仔细的错误分析，该分析揭示了对模型构建过程中产生的多种错误来源的理论见解。最后，我们展示了各种各样的应用程序，以及与地面真相解决方案的有利比较。
Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically address the imposition of Dirichlet/Neumann boundary conditions over irregular domain boundaries. In this paper, we present a framework to neurally solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Our network takes in the shape of the domain as an input (represented using an unstructured point cloud, or any other parametric representation such as Non-Uniform Rational B-Splines) and is able to generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a novel approach for identifying the interior and exterior of the computational grid in a differentiable manner. We also perform a careful error analysis which reveals theoretical insights into several sources of error incurred in the model-building process. Finally, we showcase a wide variety of applications, along with favorable comparisons with ground truth solutions.
