# Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics

arXiv: https://arxiv.org/abs/2604.23874 · HTML: https://mlsanity.com/paper/2604.23874/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 26 Apr 2026.

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

- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows](https://mlsanity.com/paper/2203.13760/) (2022)
- [Explaining the physics of transfer learning a data-driven subgrid-scale closure to a different turbulent flow](https://mlsanity.com/paper/2206.03198/) (2022)
- [Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies](https://mlsanity.com/paper/2208.03498/) (2022)
- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics](https://mlsanity.com/paper/2307.13533/) (2023)
- [Differentiable physics-enabled closure modeling for Burgers' turbulence](https://mlsanity.com/paper/2209.11614/) (2022)
- [JAX-Fluids 2.0: Towards HPC for Differentiable CFD of Compressible Two-phase Flows](https://mlsanity.com/paper/2402.05193/) (2024)
- [What You See is Not What You Get: Neural Partial Differential Equations and The Illusion of Learning](https://mlsanity.com/paper/2411.15101/) (2024)
- [A note on the error analysis of data-driven closure models for large eddy simulations of turbulence](https://mlsanity.com/paper/2405.17612/) (2024)

## Built on by (most cited first, 3 of 3)

- [Improving Reduced-Order Rotating Detonation Engine Models with Data Assimilation and Machine Learning](https://mlsanity.com/paper/2609.16237/) (2026)
- [Staying on the Attractor: Supervising Neural Surrogates of 3D Turbulence Where They Leave It](https://mlsanity.com/paper/2609.32864/) (2026)
- [The Symbol of the Surrogate: Measuring Numerical Provenance in Neural PDE Solvers](https://mlsanity.com/paper/2610.09255/) (2026)

## Abstract (arXiv)

The differentiable physics paradigm may be leveraged as an a-posteriori approach for discovering turbulence closure models by embedding a neural network parameterization directly inside the solver and optimizing it given potentially sparse target data. This addresses a key limitation of a-priori learning where direct numerical simulation (DNS) data is used to approximate the subgrid stress with the assumption of a low-pass filter. Closures trained in this a-priori manner frequently lead to unstable deployments due to the mismatch between the assumed filter and the effect of numerical discretizations and coarse-graining. In comparison, while typically stable during deployment, a-posteriori learning incurs high computational costs due to the need to backpropagate through a large eddy simulation (LES) solver. Furthermore, a-posteriori methods are challenging to apply broadly since they require significant modification of existing solvers. Finally, both approaches are limited when generalization is desired across different numerical schemes with their implicit filtering characteristics. In this work, we present a deep-learning approach for turbulence closure modeling built on the continuous data assimilation framework. Our approach enables the a-priori training of closures using sparsely observed DNS data without modifying or differentiating through the LES solver, while preserving stability during deployment for the recovery of invariant statistics. We focus on the model's ability to adapt to different discretizations by explicitly conditioning it on the numerical scheme. We use two- and three-dimensional canonical cases to test our framework and show that the learned correction systematically tracks the discretization error of the coarse solver.