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

arXiv: https://arxiv.org/abs/2607.14855 · HTML: https://mlsanity.com/paper/2607.14855/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 52 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 7 of 7)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws](https://mlsanity.com/paper/2507.01795/) (2025)
- [Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting](https://mlsanity.com/paper/2602.00378/) (2026)
- [Application of Machine Learning and Convex Limiting to Subgrid Flux Modeling in the Shallow-Water Equations](https://mlsanity.com/paper/2407.17214/) (2024)

## Abstract (arXiv)

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.