Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

TL;DR

This paper explores the use of machine learning to develop preconditioners that enhance the efficiency of linear solvers in geophysical fluid flow models, demonstrating competitive performance and robustness.

Contribution

It introduces a flexible machine learning-based preconditioning approach embedded within linear solvers, eliminating the need for traditional preconditioners and maintaining robustness.

Findings

Machine learning preconditioners are competitive with conventional ones.

The approach remains effective outside the training data range.

Preconditioning improves solver performance in geophysical models.

Abstract

It is tested whether machine learning methods can be used for preconditioning to increase the performance of the linear solver -- the backbone of the semi-implicit, grid-point model approach for weather and climate models. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices. Several machine learning methods are used to learn the optimal preconditioner for a shallow-water model with semi-implicit timestepping that is conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset.