Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

By Benjamin Moseley, Andrew Markham, T. Nissen‐Meyer. Advances in Computational Mathematics, 2021

400 citations recorded by the source corpus.

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs) . FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems.

Subfields: Electronic, Optical and Magnetic Materials, Mechanical Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/abb565c8fe70dbddaa491dfc274f9e4827ab30a1

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In Electronic, Optical and Magnetic Materials

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Citations in the corpus

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “the entire domain, Moseley et al [116] suggests using multiple neural networks, one for each subdomain.”