# Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics

arXiv: https://arxiv.org/abs/2307.13533 · HTML: https://mlsanity.com/paper/2307.13533/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2023.

## Built on by (most cited first, 9 of 9)

- [PICT -- A Differentiable, GPU-Accelerated Multi-Block PISO Solver for Simulation-Coupled Learning Tasks in Fluid Dynamics](https://mlsanity.com/paper/2505.16992/) (2025)
- [Harnessing Equivariance: Modeling Turbulence with Graph Neural Networks](https://mlsanity.com/paper/2504.07741/) (2025)
- [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)
- [A note on the error analysis of data-driven closure models for large eddy simulations of turbulence](https://mlsanity.com/paper/2405.17612/) (2024)
- [Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation](https://mlsanity.com/paper/2510.21852/) (2025)
- [Addressing A Posteriori Performance Degradation in Neural Network Subgrid Stress Models](https://mlsanity.com/paper/2511.17475/) (2025)
- [Modeling subgrid scale production rates on complex meshes using graph neural networks](https://mlsanity.com/paper/2603.19841/) (2026)
- [Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses](https://mlsanity.com/paper/2607.19387/) (2026)

## Abstract (arXiv)

Differentiable physical simulators are proving to be valuable tools for developing data-driven models for computational fluid dynamics (CFD). In particular, these simulators enable end-to-end training of machine learning (ML) models embedded within CFD solvers. This paradigm enables novel algorithms which combine the generalization power and low cost of physics-based simulations with the flexibility and automation of deep learning methods. In this study, we introduce a framework for embedding deep learning models within a finite element solver for incompressible Navier-Stokes equations, specifically applying this approach to learn a subgrid-scale (SGS) closure with a graph neural network (GNN). We first demonstrate the feasibility of the approach on flow over a two-dimensional backward-facing step, using it as a proof of concept to show that solver-consistent training produces stable and physically meaningful closures. Then, we extend this to a turbulent flow over a three-dimensional backward-facing step. In this setting, the GNN-based closure not only attains low prediction errors, but also recovers key turbulence statistics and preserves multiscale turbulent structures. We further demonstrate that the closure can be identified in data-limited learning scenarios as well. Overall, the proposed end-to-end learning paradigm offers a viable pathway toward physically consistent and generalizable data-driven SGS modeling on complex and unstructured domains.