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

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.