Turbulence teaches equivariance to neural networks

TL;DR

This work investigates how the rotational symmetry of turbulence influences learned mappings between Navier–Stokes quantities and their generalization to unseen coordinate frames and flows. By defining an equivariance error $E(\mathbf{x}_n;g)=|\mathbf{f}(g\cdot \mathbf{x}_n)-g\cdot \mathbf{f}(\mathbf{x}_n)|$ and averaging it over a discrete rotation group $G=O$ with $|G|=24$, the authors assess how well a compact 3D CNN super-resolution model respects NS symmetries on a six-box turbulent channel dataset (Re$_{\tau}=1000$). They find a strong link between lower equivariance error and better generalization across time extrapolation, anisotropy shifts, and Reynolds-number transfer, and reveal that turbulence provides implicit data augmentation that strengthens with dataset size and isotropy in line with Kolmogorov’s local isotropy hypothesis, observable in scale-dependent equivariance spectra. The results imply that respecting NS symmetries improves transferability of tensorial flow mappings and motivate incorporating isotropy-aware training and equivariance biases in turbulence ML applications.

Abstract

We investigate how the rotational nature of turbulence affects learned mappings between quantities governed by the Navier-Stokes equations. By varying the degree of anisotropy in a turbulence dataset, we explore how statistical symmetry affects these mappings. To do this, we train super-resolution models at different wall-normal locations in a channel flow, where anisotropy varies naturally, and test their generalization. By evaluating the learned mappings on new coordinate frames and new flow conditions, we find that coordinate-frame generalization is a key part of the generalization problem. Turbulent flows naturally present a wide range of local orientations, so respecting the symmetries of the Navier-Stokes equations improves generalization to new flows. Importantly, turbulence's rotational structure can embed these symmetries into learned mappings -- an effect that strengthens with isotropy and dataset size. This is because a more isotropic dataset samples a wider range of orientations, more fully covering the rotational symmetries of the Navier-Stokes equations. The dependence on isotropy means equivariance error is also scale-dependent, consistent with Kolmogorov's hypothesis. Therefore, turbulence provides its own data augmentation (we term this implicit data augmentation). We expect this effect to apply broadly to learned mappings between tensorial flow quantities, making it relevant to most machine learning applications in turbulence.

Figures (6)

• Figure 1: (a) Approximate location of the two sub-boxes (near-wall and middle) in the turbulent channel flow, and (b) test loss vs equivariance error for three generalization tasks: extrapolating in time, extrapolating to a new level of anisotropy (by changing the wall-normal location), and generalizing to a $5\times$ increase in Reynolds number (single box, explicit augmentation). Colour in (b) is by training dataset. Across all models and training configurations, there is a consistently high correlation between a model's ability to generalize to new coordinate frames (equivariance error), and generalization error.
• Figure 2: Anisotropy tensor magnitude for the three sub-boxes in the near-wall (blue) and middle (red) locations in the channel. The $\sqrt{2/3}$ bound represents the one-component limit Banerjee2007. The near-wall region is more anisotropic due to the proximity of the wall.
• Figure 3: Example outputs from the CNN (1500 samples, single box, rotational augmentation). Colour here is by velocity magnitude, but the models super-resolve each velocity component separately.
• Figure 4: Test equivariance error vs number of training samples for various training configurations. While rotational augmentation reduces the equivariance error, adding more training samples also provides a degree of rotational data augmentation in turbulence.
• Figure 5: Example test set equivariance error field for the (a) near-wall and (b) middle datasets. We show the case of 1500 training examples from a single box, without any explicit data augmentation (i.e., implicit augmentation only). Hatching indicates the direction of the top wall for reference (not the wall boundary). $[\cdot]_z$ denotes the $z$-component of a field. The $z$ velocity component on an $xy$ plane is shown, with $g \equiv C_4z$ (a 90$^\circ$ rotation about the z axis). The $z$ velocity component is not transformed by this rotation, so the transformed image appears as a straightforward rotation. Comparing the difference contours, we see that the model trained on the more isotropic mid-plane data (b) has a significantly lower equivariance error.