Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables

2026-07-16

AI Summary

This work addresses the physical inaccuracies introduced by coarse-grid simulations due to their inability to resolve small-scale turbulence. The authors propose a decoupled dual-network architecture based on structure-preserving neural networks and entropy variables to learn a subgrid-scale parameterization for the Burgers equation. In this framework, the subgrid flux is decomposed into a conservative flux potential network and an eddy-viscosity network. The approach rigorously preserves conservation properties while significantly enhancing model robustness and generalization in extrapolation scenarios. Numerical experiments demonstrate that the model faithfully reproduces the energy spectrum, spatiotemporal correlation functions, and dynamical characteristics of the fully resolved system, maintaining high fidelity even when applied beyond the range of training parameters.

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.

Problem

• subgrid-scale parameterization
• Burgers' equation
• coarse simulation
• partial differential equations
• physical fidelity

Innovation

• structure-preserving neural networks
• entropy variables
• subgrid-scale parameterization
• decoupled neural architecture
• Burgers' equation