A Machine Learning accelerated geophysical fluid solver

TL;DR

The work addresses accelerating geophysical fluid simulations by integrating data-driven discretization with finite-volume solvers for hyperbolic PDEs. It implements classic shallow water and Euler solvers and compares them to PyClaw, then introduces four CNN-based ML approaches for embedding data-driven components into the SWE solver, with the fourth approach (CNN-generated reconstruction slopes) showing the strongest performance and stability. Across planar and spherical domains, the classic solvers generally outperform the PyClaw baseline, while the ML-augmented methods demonstrate potential for improved efficiency and accuracy under suitable training and constraints, particularly for boundary reconstruction. The study demonstrates a path toward efficient, high-fidelity CFD on structured grids and curved manifolds, with practical implications for geophysical modeling on spherical grids and in resource-constrained environments.

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.

Figures (42)

• Figure 1: A typical finite volume grid displaying cell averages and fluxes.
• Figure 2: Cell averages define Riemann problems at every interface.
• Figure 3: On the left is the computational grid and the sphere grid shown in the right picture.
• Figure 4: First NN structure: CNN output the boundaries' numerical flux directly.
• Figure 5: Second NN structure: Calculate boundary states with CNN generated linear coefficients.