Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows

TL;DR

This paper introduces LDSolver, a learnable and differentiable finite volume solver that accelerates fluid flow simulations on coarse grids, combining traditional methods with machine learning for improved efficiency and accuracy.

Contribution

The paper presents LDSolver, a novel hybrid approach integrating differentiable finite volume methods with learnable modules, enhancing simulation speed and accuracy with limited training data.

Findings

LDSolver outperforms baseline models in various flow simulations.

It maintains high accuracy even with limited training data.

The method generalizes well across different flow systems.

Abstract

Simulation of fluid flows is crucial for modeling physical phenomena like meteorology, aerodynamics, and biomedicine. Classical numerical solvers often require fine spatiotemporal grids to satisfy stability, consistency, and convergence conditions, leading to substantial computational costs. Although machine learning has demonstrated better efficiency, they typically suffer from issues of interpretability, generalizability, and data dependency. Hence, we propose a learnable and differentiable finite volume solver, called LDSolver, designed for efficient and accurate simulation of fluid flows on spatiotemporal coarse grids. LDSolver comprises two key components: (1) a differentiable finite volume solver, and (2) an learnable module providing equivalent approximation for fluxes (derivatives and interpolations), and temporal error correction on coarse grids. Even with limited training data (e.g., only a few trajectories), our model could accelerate the simulation while maintaining a high accuracy with superior generalizability. Experiments on different flow systems (e.g., Burgers, decaying, forced and shear flows) show that LDSolver achieves state-of-the-art performance, surpassing baseline models with notable margins.