Neural PDE Solvers for Irregular Domains

Abstract

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, most neural PDE solvers only apply to rectilinear domains and do not systematically address the imposition of boundary conditions over irregular domain boundaries. In this paper, we present a neural framework to solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Given the shape of the domain as an input (represented as a binary mask), our network is able to predict the solution field, and can generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a physics-informed loss function that directly incorporates the interior-exterior information of the geometry. 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 various applications in 2D and 3D, along with favorable comparisons with ground truth solutions.