Bayesian Kolmogorov-Arnold networks for trustworthy Reynolds-stress closures
Abstract
Data-driven turbulence closures can improve Reynolds-averaged Navier-Stokes (RANS) predictions, but their use is limited by two issues: learned constitutive relations are often difficult to inspect, and predictions typically lack reliable uncertainty estimates. We develop a Bayesian tensor-basis Kolmogorov-Arnold network (BKAN) for Reynolds-stress anisotropy that addresses both within a single closure. Learnable univariate spline functions on network edges provide an inspectable representation that can, where sufficiently simple, be approximated symbolically. Variational posteriors over spline coefficients yield a predictive distribution separating epistemic uncertainty, due to limited training data, from aleatoric uncertainty, the irreducible scatter of high-fidelity targets at given inputs. Galilean and rotational invariance, symmetry, and tracelessness are enforced through the tensor basis, while physical realisability is treated separately through soft regularisation. A controlled regression benchmark assesses predictive accuracy, calibration, and Bayesian parameterisation. Placing the posterior on spline coefficients gives the best overall balance of accuracy, calibration, parameter cost, and interpretability. Applied to Reynolds-stress anisotropy, the BKAN is accurate in distribution, with calibrated, informative uncertainty that tends to identify larger prediction errors. Under geometric extrapolation, however, the model becomes overconfident: epistemic uncertainty increases but does not fully account for growing prediction errors. Finally, the framework is extended to three-dimensional wind-farm wakes, where the learned correction transfers to unseen layouts and posterior samples are propagated through the RANS solver, with the aleatoric component injected as a spatially correlated field, to obtain uncertainty bounds on the predicted wake flow.
Related
- Granger causality detection with Kolmogorov-Arnold networks
- Finite basis Kolmogorov-Arnold networks: Domain decomposition for data-driven and physics-informed problems
- Physics-Informed Kolmogorov-Arnold Networks for Grad-Shafranov Tokamak Equilibria
- Physics-Informed Kolmogorov-Arnold networks for viscoelastic fluid equations
- Tractable Inference with Kolmogorov-Arnold Networks
BibTeX
@misc{eidi2026bayesian,
title = {{Bayesian Kolmogorov-Arnold networks for trustworthy Reynolds-stress closures}},
author = {Ali Eidi and Kherlen Jigjid and Tyler Buchanan and Richard P. Dwight},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07582},
eprint = {2611.07582},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07582},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bayesian Kolmogorov-Arnold networks for trustworthy Reynolds-stress closures
AU - Ali Eidi
AU - Kherlen Jigjid
AU - Tyler Buchanan
AU - Richard P. Dwight
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07582
UR - https://arxiv.org/abs/2611.07582
AB - Data-driven turbulence closures can improve Reynolds-averaged Navier-Stokes (RANS) predictions, but their use is limited by two issues: learned constitutive relations are often difficult to inspect, and predictions typically lack reliable uncertainty estimates. We develop a Bayesian tensor-basis Kolmogorov-Arnold network (BKAN) for Reynolds-stress anisotropy that addresses both within a single closure. Learnable univariate spline functions on network edges provide an inspectable representation that can, where sufficiently simple, be approximated symbolically. Variational posteriors over spline coefficients yield a predictive distribution separating epistemic uncertainty, due to limited training data, from aleatoric uncertainty, the irreducible scatter of high-fidelity targets at given inputs. Galilean and rotational invariance, symmetry, and tracelessness are enforced through the tensor basis, while physical realisability is treated separately through soft regularisation. A controlled regression benchmark assesses predictive accuracy, calibration, and Bayesian parameterisation. Placing the posterior on spline coefficients gives the best overall balance of accuracy, calibration, parameter cost, and interpretability. Applied to Reynolds-stress anisotropy, the BKAN is accurate in distribution, with calibrated, informative uncertainty that tends to identify larger prediction errors. Under geometric extrapolation, however, the model becomes overconfident: epistemic uncertainty increases but does not fully account for growing prediction errors. Finally, the framework is extended to three-dimensional wind-farm wakes, where the learned correction transfers to unseen layouts and posterior samples are propagated through the RANS solver, with the aleatoric component injected as a spatially correlated field, to obtain uncertainty bounds on the predicted wake flow.
ER -