Large-scale Neural Solvers for Partial Differential Equations

TL;DR

This paper explores mesh-free, physics-informed neural networks for solving PDEs, emphasizing domain decomposition and distributed training to improve efficiency and accuracy compared to traditional methods.

Contribution

It introduces a scalable, neural PDE solver that reduces data requirements and leverages domain decomposition and distributed computing for enhanced performance.

Findings

Domain decomposition improves neural solver runtime.

Distributed training effectively utilizes large GPU clusters.

GatedPINN achieves competitive accuracy with analytical and spectral solutions.

Abstract

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution. However, recent generative neural networks require lots of training data, i.e. full simulation runs making them costly. In contrast, we examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) solely requiring initial/boundary values and validation points for training but no simulation data. The induced curse of dimensionality is approached by learning a domain decomposition that steers the number of neurons per unit volume and significantly improves runtime. Distributed training on large-scale cluster systems also promises great utilization of large quantities of GPUs which we assess by a comprehensive evaluation study. Finally, we discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.