Manifold-Adapted Sparse RBF-SINDy: Unbiased Library Construction and Unsupervised Discovery of Dynamical States in Turbulent Wall Flows

TL;DR

This paper introduces a manifold-adapted sparse RBF-SINDy method that accurately uncovers the skeleton of turbulent wall flows from wall measurements, correcting biases in library construction to identify flow states and dynamics.

Contribution

It proposes a novel library construction approach that respects the intrinsic geometry of turbulent attractors, enabling unsupervised discovery of flow states without prior labels.

Findings

Successfully recovers flow skeleton from wall data

Reproduces invariant measure and Lyapunov horizon

Identifies stable streaks and burst instabilities

Abstract

The turbulent attractor of wall bounded flows is not a structureless strange set but contains a skeleton of dynamically distinct states connected by rare directed transitions whose geometry is reflected in the invariant measure of the phase space trajectory. We show that this skeleton can be recovered from wall measurements alone, namely wall pressure and wall shear stress, without physical labels or prior knowledge, provided that the data driven function library used to identify the dynamics respects the intrinsic geometry of the attractor rather than the variance hierarchy of the POD representation. Standard sparse identification approaches introduce two structural biases during library construction. First, the steep decay of POD spectra causes Euclidean distances in k means clustering to be dominated by leading modes, collapsing basis function centres into a low dimensional subspace and leaving transitional dynamics poorly represented. Second, turbulent trajectories slow near quasi invariant states, so uniform time sampling over represents these regions and under samples rapid transitions. Both biases are corrected by resampling the trajectory uniformly in arc length and replacing the Euclidean metric with a Mahalanobis metric derived from the local cluster covariance. A single sparse regression on this corrected library yields a reduced model. Applied to a minimal turbulent channel at low Reynolds number, unsupervised clustering reveals two phases of the near wall cycle: stable streak states and burst initiating instabilities corresponding to the coherent structure skeleton of the flow. The model reproduces the invariant measure, reaches the Lyapunov predictability horizon and provides a differentiable vector field on which invariant solutions can be located by Newton iteration.