# Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolution

arXiv: https://arxiv.org/abs/2509.20683 · HTML: https://mlsanity.com/paper/2509.20683/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Single-snapshot machine learning for super-resolution of turbulence](https://mlsanity.com/paper/2409.04923/) (2024)
- [Turbulence Scaling from Deep Learning Diffusion Generative Models](https://mlsanity.com/paper/2311.06112/) (2023)
- [A Regularization-Guided Equivariant Approach for Image Restoration](https://mlsanity.com/paper/2505.19799/) (2025)
- [Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networks](https://mlsanity.com/paper/2202.11099/) (2022)

## Abstract (arXiv)

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.