Spatial-Temporal, Parallel, Physics-Informed Neural Networks for Solving Forward and Inverse PDE Problems via Overlapping Domain Decomposition

Physics-informed neural networks (PINNs) have emerged as an effective tool for solving both forward and inverse partial differential equation (PDE) problems. However, their application in large-scale problems is limited due to their expensive computational cost. In this study, we employed an overlapping domain decomposition technique to enable the spatial-temporal parallelism in PINNs to accelerate training. Moreover, we proposed a rescaling approach for PINN inputs in each subdomain, which is capable of migrating the spectral bias in vanilla PINNs. We demonstrated the accuracy of the PINNs with overlapping domain decomposition (overlapping PINNs) for spatial parallelism using several differential equations: a forward ODE with a high-frequency solution, a two-dimensional (2D) forward Helmholtz equation, and a 2D inverse heat conduction problem. In addition, we tested the accuracy of overlapping PINNs for spatial-temporal parallelism using two nonstationary PDE problems, i.e., a forward Burgers’ equation and an inverse heat transfer problem. The results demonstrate (1) the effectiveness of overlapping PINNs for spatial-temporal parallelism when solving forward and inverse PDE problems, and (2) the rescaling technique proposed in this work is able to migrate the spectral bias in vanilla PINNs. Finally, we demonstrated that the overlapping PINNs achieve approximately <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mn>90</mml:mn><mml:mi>%</mml:mi></mml:mrow></mml:math> efficiency with up to 8 GPUs using the example of the inverse time-dependent heat transfer problem.