Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations (Communications in Computational Physics, 2020)
We propose a generalized space-time domain decomposition approach forthe physics-informed neural networks (PINNs) to solve nonlinear partial differentialequations (PDEs) on arbitrary complex-geometry domains. The proposed framework,named eXtended PINNs ($XPINNs$), further pushes the boundaries of both PINNs aswell as conservative PINNs (cPINNs), which is a recently proposed domain decomposition approach in the PINN framework tailored to conservation laws. Compared toPINN, the XPINN method has large representation and parallelization capacity due tothe inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, thedomain can be decomposed in any arbitrary way (in space and time), which is notpossible in cPINN. Thus, XPINN offers both space and time parallelization, therebyreducing the training cost more effectively. In each subdomain, a separate neural network is employed with optimally selected hyperparameters, e.g., depth/width of thenetwork, number and location of residual points, activation function, optimizationmethod, etc. A deep network can be employed in a subdomain with complex solution,whereas a shallow neural network can be used in a subdomain with relatively simpleand smooth solutions. We demonstrate the versatility of XPINN by solving both forward and inverse PDE problems, ranging from one-dimensional to three-dimensionalproblems, from time-dependent to time-independent problems, and from continuousto discontinuous problems, which clearly shows that the XPINN method is promising in many practical problems. The proposed XPINN method is the generalization ofPINN and cPINN methods, both in terms of applicability as well as domain decomposition approach, which efficiently lends itself to parallelized computation. The XPINNcode is available on $https://github.com/AmeyaJagtap/XPINNs$.