Parallel Physics-Informed Neural Networks via Domain Decomposition

TL;DR

This paper introduces a parallel framework for physics-informed neural networks using domain decomposition, enabling efficient multi-scale problem solving and hyperparameter tuning across subdomains with MPI+X programming.

Contribution

It develops a distributed parallel algorithm for cPINNs and XPINNs, enhancing scalability, flexibility, and efficiency in solving complex multi-physics problems with domain decomposition.

Findings

cPINNs are more communication-efficient for space decomposition.

XPINNs handle time decomposition and complex subdomains effectively.

Parallel XPINNs successfully solve an inverse diffusion problem on the US map.

Abstract

We develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI $+$ X, where X $\in \{\text{CPUs},~\text{GPUs}\}$. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.