# Differentiable physics-enabled closure modeling for Burgers' turbulence

arXiv: https://arxiv.org/abs/2209.11614 · HTML: https://mlsanity.com/paper/2209.11614/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 68 references that are in the map.
- First posted on arXiv 23 Sep 2022.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Turbulent scalar flux in inclined jets in crossflow: counter gradient transport and deep learning modelling](https://mlsanity.com/paper/2001.04600/) (2020)

## Built on by (most cited first, 7 of 7)

- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Gradient-free online learning of subgrid-scale dynamics with neural emulators](https://mlsanity.com/paper/2310.19385/) (2023)
- [Learning Subgrid-scale Models with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2212.09967/) (2022)
- [Improved deep learning of chaotic dynamical systems with multistep penalty losses](https://mlsanity.com/paper/2410.05572/) (2024)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics](https://mlsanity.com/paper/2604.23874/) (2026)
- [Offline Reinforcement Learning for Fluid Controls: Data-based Multi-observational Policy Extraction](https://mlsanity.com/paper/2606.31025/) (2026)

## Abstract (arXiv)

Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers' turbulence. We consider the 1D Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models across a range of system parameters, including viscosity, time, and grid resolution. We find that constraining models with inductive biases in the form of partial differential equations that contain known physics or existing closure approaches produces highly data-efficient, accurate, and generalizable models, outperforming state-of-the-art baselines. Addition of structure in the form of physics information also brings a level of interpretability to the models, potentially offering a stepping stone to the future of closure modeling.