In progress — ICMAI-MAC 2026
Tensor Basis Neural Networks vs. Random Forests for RANS Turbulence Closure
Two machine-learned RANS turbulence closures, trained on periodic hills and tested — out of distribution — on a NASA benchmark, explained with SHAP.
Inside the work
Where the flow stops behaving.
Schematic attached boundary layer, separation, recirculation, and downstream reattachment over a hump; not a computed velocity field.
Approach
The incoming boundary layer encounters a changing surface and pressure gradient.
Separate and reattach
A separated shear layer and downstream recirculation region create a demanding test for a learned turbulence closure.
Generalize
Compare models on the held-out hump, then inspect which flow invariants shape their predictions.
Overview
The Reynolds-averaged Navier–Stokes equations are unclosed: they require a model for the Reynolds-stress anisotropy tensor. Pope's (1975) tensor-basis formulation writes that tensor as a linear combination of ten invariant basis functions, with the combination weights left as scalar functions of five flow-state invariants — exactly the shape a Tensor Basis Neural Network (Ling et al., 2016) or Tensor Basis Random Forest (Kaandorp & Dwight, 2020) can learn.
This thesis trains both architectures with a matched 50-trial Bayesian-optimization budget and a 5-seed protocol, exclusively on a family of flow over periodic hills (PHLL29), then evaluates both — without any retraining — on the held-out, out-of-distribution NASA wall-mounted hump (NASAHUMP) benchmark. The central methodological contribution is a spatially- and state-resolved SHAP (SHapley Additive exPlanations) attribution of each model's closure decision, projected onto both the classical Lumley barycentric triangle and the Wnek et al. (2025) polar invariant map.
Dataset
PHLL29
Training set — periodic-hills flow family
NASAHUMP
Held-out OOD test — NASA wall-mounted hump
26 figures, 30 panels
Publication-quality graphical catalog
Chart
RMSE: in-distribution validation vs. out-of-distribution NASAHUMP test
Results
Validation (in-distribution) vs. generalization (NASAHUMP, out-of-distribution)
| Model | Validation RMSE | NASAHUMP RMSE | Realizability violations |
|---|---|---|---|
| TBNN | 0.0532 (better) | 0.2383 | ~1.19% of points |
| TBRF | 0.0616 | 0.1807 (better) | 0% of points |
Key Findings
What the work shows
The two architectures invert their ordering between validation and generalization: TBNN wins in-distribution, but TBRF is clearly superior out-of-distribution (signed-rank test across 5 seeds, p = 0.0). The random forest's depth-limited, bootstrap-averaged structure acts as an inductive bias against the aggressive extrapolation neural networks perform off-manifold.
TBRF never violates the Lumley-triangle realizability bounds — its piecewise-constant, convex-combination output can never leave the convex hull of physically admissible training states. TBNN violates realizability on ~1.19% of test points, concentrated in the separation bubble where extrapolation is most aggressive.
TBNN is also the only model with meaningful cross-seed variance (concentrated in the recirculation region); TBRF is statistically near-deterministic (cross-seed RMSE std 6.7×10⁻⁵).
SHAP attribution shows the two models are locally attending to different physical mechanisms: TBRF concentrates on invariants I1/I2 (strain and rotation magnitude) at the separation bubble, while TBNN spreads weight onto the higher-order I3–I5 invariants with a fatter attribution tail — and the TBNN's error hot-spots co-locate almost exactly with where its SHAP attribution diverges from TBRF's.
26-figure publication-quality graphical catalog (30 rendered panels), full metric tables, and per-figure interpretation are documented in the thesis.