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

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.

24 pages, 9 figures

References in corpus (10)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DGM: A deep learning algorithm for solving partial differential equations
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• Residual-based attention in physics-informed neural networks
• Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology
• Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies
• Nonlinear Field-split Preconditioners for Solving Monolithic Phase-field Models of Brittle Fracture
• Two-level Group Convolution
• Parareal Neural Networks Emulating a Parallel-in-time Algorithm
• Activations and Gradients Compression for Model-Parallel Training