Jeremiah Bailey | Searcharxiv

arXiv2026

Turbulence teaches equivariance to neural networks

We show that the rotational nature of turbulence affects how neural networks learn mappings between quantities governed by the Navier-Stokes equations. We train super-resolution models at different wall-normal locations in a turbulent channel flow, where anisotropy varies naturally, and test their generalization to new coordinate frames, new anisotropy regimes, and a higher Reynolds number. Our findings inform both the design of equivariant machine learning models for turbulence and our understanding of how turbulence shapes what those models learn. First, mappings that better respect the rotational symmetries of the Navier-Stokes equations generalize better to new flows. Coordinate-frame generalization is therefore a key part of the broader generalization problem, since turbulent flows contain a wide range of local orientations. Second, turbulence itself partially teaches equivariance to learned mappings, an effect we call implicit data augmentation. The effect strengthens with dataset size and with isotropy, since a more isotropic dataset samples more orientations under which the Navier-Stokes equations are covariant. Implicit augmentation is also scale-dependent, with smaller scales exhibiting lower equivariance error. This scale-dependency is consistent with Kolmogorov's hypothesis of local isotropy. Third, enforcing equivariance as an architectural inductive bias is the limit of these effects: an exactly equivariant network outperforms unconstrained CNNs on all generalization tests, with roughly an order of magnitude fewer parameters. We expect these effects to apply broadly to learned mappings between tensorial flow quantities, making them relevant to most machine learning applications in turbulence.

physics.flu-dyn↗

arXiv2025

Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolution

The immense computational cost of simulating turbulence has motivated the use of machine learning approaches for super-resolving turbulent flows. A central challenge is ensuring that learned models respect physical symmetries, such as rotational equivariance. We show that standard convolutional neural networks (CNNs) can partially acquire this symmetry without explicit augmentation or specialized architectures, as turbulence itself provides implicit rotational augmentation in both time and space. Using 3D channel-flow subdomains with differing anisotropy, we find that models trained on more isotropic mid-plane data achieve lower equivariance error than those trained on boundary layer data, and that greater temporal or spatial sampling further reduces this error. We show a distinct scale-dependence of equivariance error that occurs regardless of dataset anisotropy that is consistent with Kolmogorov's local isotropy hypothesis. These results clarify when rotational symmetry must be explicitly incorporated into learning algorithms and when it can be obtained directly from turbulence, enabling more efficient and symmetry-aware super-resolution.

physics.flu-dyn↗