Neural PDE Solvers for Irregular Domains

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2022
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Khara, Biswajit
Herron, Ethan
Jiang, Zhanhong
Balu, Aditya
Yang, Chih-Hsuan
Saurabh, Kumar
Jignasu, Anushrut
Sarkar, Soumik
Hegde, Chinmay
Krishnamurthy, Adarsh
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arXiv
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Mechanical Engineering
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Abstract
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.
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This is a pre-print of the article Khara, Biswajit, Ethan Herron, Zhanhong Jiang, Aditya Balu, Chih-Hsuan Yang, Kumar Saurabh, Anushrut Jignasu et al. "Neural PDE Solvers for Irregular Domains." arXiv preprint arXiv:2211.03241 (2022). DOI: 10.48550/arXiv.2211.03241. Copyright 2022 The Authors. Posted with permission.
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