arXiv:2605.26358v1 Announce Type: cross 
Abstract: Turbulence is ubiquitous in engineering and science, yet direct simulation is prohibitively expensive. The Reynolds-averaged Navier-Stokes (RANS) equations provide savings exceeding ten orders of magnitude but introduce unclosed terms (the closure problem). Offline-trained machine-learning (ML) closures suffer distribution shift in predictive simulations, while ML methods that bypass the governing equations struggle to generalise from scarce high-fidelity data. We develop a physics-derived deep learning closure model for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on small datasets and accurately generalise across Reynolds numbers, to unseen geometries, and to different flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak-equilibrium assumption, imposing physics-based structure on the ML closure. End-to-end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit-explicit structure for efficient optimisation. On canonical square-duct and periodic-hill benchmarks, DARSM reduces average test velocity error over baseline RANS by $2$-$4\times$ across Reynolds number, geometries, and flow regimes, with peak case-level reductions of $12\times$. The model trained on attached, anisotropy-dominated flows (square duct) accurately generalises without retraining to separated flows (periodic hills), a regime change in the underlying physics. DARSM also outperforms five established ML methods: offline training, tensor-basis neural networks, field-inversion machine learning, DeepONets, and physics-informed neural networks.

تم تطوير نموذج جديد يعتمد على التعلم العميق، وهو نموذج إجهاد رينولدز الجبري العميق (DARSM)، لتحسين محاكاة التدفقات المضطربة باستخدام معادلات نافير-ستوكس المتوسطة رينولدز (RANS). يتناول هذا النموذج مشكلة الإغلاق في نمذجة الاضطراب من خلال التعميم بدقة عبر أرقام رينولدز المختلفة وأنظمة التدفق، حتى مع وجود بيانات محدودة.

Se ha desarrollado un nuevo modelo basado en aprendizaje profundo, el Deep Algebraic Reynolds Stress Model (DARSM), para mejorar las simulaciones de flujos turbulentos utilizando las ecuaciones de Navier-Stokes promediadas por Reynolds (RANS). Este modelo aborda el problema de cierre en la modelación de turbulencias al generalizar con precisión a través de varios números de Reynolds y regímenes de flujo, incluso con datos limitados.

Un nouveau modèle basé sur l'apprentissage profond, le Deep Algebraic Reynolds Stress Model (DARSM), a été développé pour améliorer les simulations de flux turbulents selon les équations de Navier-Stokes moyennées de Reynolds (RANS). Ce modèle aborde le problème de fermeture dans la modélisation de la turbulence en généralisant avec précision à travers divers nombres de Reynolds et régimes d'écoulement, même avec des données limitées.

A new deep learning-based model, the Deep Algebraic Reynolds Stress Model (DARSM), has been developed to enhance Reynolds-averaged Navier-Stokes (RANS) simulations of turbulent flows. This model addresses the closure problem in turbulence modeling by accurately generalizing across various Reynolds numbers and flow regimes, even with limited data.