Learning Subgrid-scale Models with Neural Ordinary Differential Equations

TL;DR

This paper introduces a novel neural ordinary differential equation-based method for learning subgrid-scale models in PDE simulations, enhancing accuracy and efficiency in computational fluid dynamics and chaotic systems.

Contribution

It presents a continuous-level learning strategy using NODEs to improve subgrid-scale parameterization and coupling in PDE solvers, addressing tuning challenges.

Findings

Effective in Lorenz 96 ODE, convection-diffusion PDE, and Burgers' PDE

Improves accuracy of coarse-grid simulations

Enhances efficiency of low-order solvers

Abstract

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.