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.

翻译：湍流在工程和科学领域普遍存在，但直接模拟成本过高。雷诺平均纳维-斯托克斯（RANS）方程可将计算量降低超过十个数量级，却引入了未封闭项（即封闭问题）。离线训练的机器学习（ML）封闭模型在预测模拟中会出现分布偏移，而绕过控制方程的机器学习方法难以从稀缺的高保真数据中泛化。我们开发了一种基于物理的深度学习RANS封闭模型——深度代数雷诺应力模型（DARSM），该模型可在小数据集上训练，并能准确泛化至不同雷诺数、未见几何构型及不同流动状态。神经网络将流动不变量映射至隐式代数雷诺应力方程中的经验参数，该方程基于弱平衡假设从雷诺应力输运方程推导而来，从而为机器学习封闭模型赋予基于物理的结构。通过控制偏微分方程及耦合隐式封闭模型进行端到端优化可消除分布偏移，但展开式自动微分和隐式自动微分均无法处理刚性耦合求解器。我们推导了伴随方程，利用求解器的隐式-显式结构实现高效优化。在标准方管和周期性山丘基准测试中，DARSM在雷诺数、几何构型和流动状态变化下，将平均测试速度误差相比基线RANS降低了2-4倍，单案例最大降幅达12倍。该模型在附着型各向异性主导流（方管）上训练后，无需重新训练即可准确泛化至分离流（周期性山丘），实现了底层物理机制的跨状态迁移。DARSM还优于五种现有机器学习方法：离线训练、张量基神经网络、场反演机器学习、DeepONet及物理信息神经网络。