CFD-driven symbolic regression of RANS transition models

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.

Deep learning (DL) has shown promise for flow simulations, but progress is constrained by the scarcity of high-fidelity training data, which is costly to generate and limited to few flow conditions. Closures trained offline on such data tend to overfit and fail to generalise to new regimes. We introduce an online optimisation framework for DL-based Reynolds-averaged Navier-Stokes (RANS) closures. Training data is generated by a simulation at direct numerical simulation (DNS) resolution embedded in a subdomain of the RANS domain. The RANS solution supplies boundary conditions to the embedded simulation, which provides the mean velocity and turbulence statistics used to update the closure, which thereby adapts to the target flow without precomputed datasets. The approach is developed on the stochastically forced Burgers equation and on turbulent channel flow at $Re_τ=180$, 270, 395 and 590 with embedded domain lengths $1\leq L_0/L\leq 8$, and is then applied to spanwise-rotating channel flow at $Ro_b=0.15$ and $0.5$, a flow in which the Coriolis force vanishes from the equations of a linear eddy-viscosity closure. Online-optimised RANS models outperform offline-trained and literature-calibrated closures with modest embedded subdomains, and in the rotating channel recover the rotation-induced asymmetry from non-rotating inflow turbulence and a rotation-blind closure. The learned coefficients are interpretable, recovering near-wall damping and, under rotation, the collapse of the eddy viscosity on the stabilised side. Performance degrades when boundary contamination dominates or domains are too short to capture low-wavenumber modes. This framework provides a route to data-adaptive machine learning closures without large precomputed training datasets.

Aerodynamic surrogate models trained on high-fidelity CFD data reproduce numerical predictions of both scalar outputs and entire fields accurately, yet their predictive fidelity is limited by systematic discrepancies between CFD and experimental observations. We present an experimentally grounded correction framework that adapts a CFD-trained deep learning surrogate using wind-tunnel PSP measurements. A Geotransolver surrogate trained on 2,300 high-fidelity CFD simulations of the NASA CRM wing-body configuration, spanning geometric variation, Mach 0.70-0.85, and angles of attack 0 to 4 degrees, reproduces the CFD integrated aerodynamic forces and pitching moment to R2 > 0.99 but does not match the experimental data. To incorporate experimental information without retraining the surrogate, a correction network is trained on spatially registered PSP measurements at two freestream Mach numbers (0.70 and 0.85) across the same angle-of-attack range, learning the discrepancy between the surrogate-predicted and experimentally measured surface-pressure distributions. At Mach 0.85 the correction substantially improves agreement with PSP, particularly at the wing suction peak, shock location, and subsequent pressure recovery, reducing both the magnitude of the prediction error and the fraction of wetted surface on which it exceeds 0.05 in Cp, and it does so from a limited experimental dataset without modifying the pretrained surrogate parameters. On held-out angles of attack the grounded surrogate agrees with measurement to within 2.3-2.7% of the measured Cp range, and outperforms direct interpolation between the measured conditions at every state tested. Experimental measurements can therefore ground a large-scale simulation-trained surrogate by learning systematic CFD-to-experiment discrepancies while preserving its generalization capability and computational efficiency.