EGU General Assembly 2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

Jan Ackmann1, Peter Düben2, Tim Palmer1, and Piotr Smolarkiewicz3
Jan Ackmann et al.
  • 1University of Oxford, AOPP, Oxford, United Kingdom of Great Britain – England, Scotland, Wales (
  • 2European Centre For Medium Range Weather Forecasts, Reading, UK
  • 3National Center for Atmospheric Research, Boulder, USA

Semi-implicit grid-point models for the atmosphere and the ocean require linear solvers that are working efficiently on modern supercomputers. The huge advantage of the semi-implicit time-stepping approach is that it enables large model time-steps. This however comes at the cost of having to solve a computationally demanding linear problem each model time-step to obtain an update to the model’s pressure/fluid-thickness field. In this study, we investigate whether machine learning approaches can be used to increase the efficiency of the linear solver.

Our machine learning approach aims at replacing a key component of the linear solver—the preconditioner. In the preconditioner an approximate matrix inversion is performed whose quality largely defines the linear solver’s performance. 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 of different complexity from simple linear regression to deep feed-forward neural networks are used to learn the optimal preconditioner for a shallow-water model with semi-implicit time-stepping. The shallow-water model is specifically designed to be 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.

How to cite: Ackmann, J., Düben, P., Palmer, T., and Smolarkiewicz, P.: Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-5507,, 2021.

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