An Extrinsic Approach Based on Physics-Informed Neural Networks for PDEs on Surfaces

Zhuochao Tang, Zhuojia Fu () and Sergiy Reutskiy
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Zhuochao Tang: Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China
Zhuojia Fu: Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing 210098, China
Sergiy Reutskiy: Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

Mathematics, 2022, vol. 10, issue 16, 1-14

Abstract: In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces.

Keywords: machine learning; extrinsic; embedding; intrinsic; surfaces; Laplace–Beltrami operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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