A Machine Learning accelerated geophysical fluid solver

TL;DR

This paper explores the integration of machine learning with traditional PDE solvers, demonstrating improved accuracy and stability in fluid simulations through data-driven discretization and neural network approaches.

Contribution

It introduces ML-based methods for PDE solving that enhance traditional schemes, including implementing classic fluid solvers and proposing neural network models for better performance.

Findings

ML-based solvers outperform traditional methods in accuracy

Two neural network approaches yield satisfactory solutions

Enhanced stability and conservation in fluid simulations

Abstract

Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs. Among various approaches to applying ML techniques to solving PDEs, the data-driven discretization method presents a promising way of accelerating and improving existing PDE solver on structured grids where it predicts the coefficients of quasi-linear stencils for computing values or derivatives of a function at given positions. It can improve the accuracy and stability of low-resolution simulation compared with using traditional finite difference or finite volume schemes. Meanwhile, it can also benefit from traditional numerical schemes like achieving conservation law by adapting finite volume type formulations. In this thesis, we have implemented the shallow water equation and Euler equation classic solver under a different framework. Experiments show that our classic solver performs much better than the Pyclaw solver. Then we propose four different deep neural networks for the ML-based solver. The results indicate that two of these approaches could output satisfactory solutions.