On learning discretization schemes of partial differential equations in geoscience

Abstract

Solving partial differential equations (PDEs) stably and accurately is essential in simulation analysis of a variety of geophysical phenomena. Designing appropriate discretization schemes for PDEs requires careful and rigorous mathematical treatment and has been a long-term research topic. The computational efficiency is additionally a long-standing challenge when what-if hazard scenario analysis is considered. The data-driven discretization is a hybrid approach to combine machine learning and physics-based simulations, which provides a methodology to derive better discretization schemes from reliable references obtained typically using known stable schemes with higher resolution grids. As the resultant schemes may inherit the physics described by the PDEs, surrogate models employing them are expected to be in good agreement with expensive simulations. It is also argued that the learnt schemes by neural network models can exhibit similar characteristics to known sophisticated algorithms and outperform them in terms of accuracy. However, the method has currently been assessed with only limited examples and the detailed mechanisms of the learnt schemes are not well understood. In this presentation, thorough assessment and investigation of learning discretization schemes are conducted by applying the methodology to several types of differential equations with different learning models for the schemes. Whether the methodology has the potential to derive new schemes is also discussed.