Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applications

TL;DR

This paper presents a new two-level overlapping additive Schwarz preconditioner that accelerates training in scientific machine learning by improving convergence speed and accuracy, especially for physics-informed neural networks and operator learning.

Contribution

The paper introduces a novel two-level overlapping additive Schwarz preconditioner tailored for scientific machine learning training, combining domain decomposition with a new synchronization strategy.

Findings

Significantly speeds up convergence of LBFGS optimizer.

Yields more accurate machine learning models.

Enhances compatibility with model-parallel computations.

Abstract

We introduce a novel two-level overlapping additive Schwarz preconditioner for accelerating the training of scientific machine learning applications. The design of the proposed preconditioner is motivated by the nonlinear two-level overlapping additive Schwarz preconditioner. The neural network parameters are decomposed into groups (subdomains) with overlapping regions. In addition, the network's feed-forward structure is indirectly imposed through a novel subdomain-wise synchronization strategy and a coarse-level training step. Through a series of numerical experiments, which consider physics-informed neural networks and operator learning approaches, we demonstrate that the proposed two-level preconditioner significantly speeds up the convergence of the standard (LBFGS) optimizer while also yielding more accurate machine learning models. Moreover, the devised preconditioner is designed to take advantage of model-parallel computations, which can further reduce the training time.