Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables
Machine Learning
Summary
The authors developed a machine learning method to improve simplified simulations of complex systems described by partial differential equations, specifically using the Burgers' equation as an example. They designed a special neural network that separates different parts of the small-scale effects into two networks, ensuring the physical laws are preserved. Their approach can accurately recreate important behaviors of the full system, like how energy spreads and how patterns change over time. They also found that their model works well even when tested on conditions it wasn't trained on.
Authors
Aijaz Nazir, Ilya Timofeyev
Abstract
We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particular, we employ a decoupled neural network architecture explicitly separating the subgrid corrections into two distinct components: a conservative Flux Potential network and an Eddy Viscosity network. We demonstrate that this reduced-order framework maintains high physical fidelity, accurately reproducing the energy spectrum, spatial and temporal correlation functions, and dynamical characteristics of the full-scale system. Furthermore, we show that our approach is robust and applicable to parameters outside the training regime.