Year: 2022
Author: Muhammad I. Zafar, Jiequn Han, Xu-Hui Zhou, Heng Xiao
Communications in Computational Physics, Vol. 32 (2022), Iss. 2 : pp. 336–363
Abstract
Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0256
Communications in Computational Physics, Vol. 32 (2022), Iss. 2 : pp. 336–363
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Neural operators graph neural networks constitutive modeling inverse modeling deep learning.