Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networks

TL;DR

This paper develops a rotationally equivariant super-resolution method for 2D fluid velocity fields using CNNs, emphasizing the importance of dataset consistency and weight sharing to achieve covariance with fluid system rotations.

Contribution

It introduces the concept of rotational consistency in datasets and demonstrates how imposing rotational equivariance through weight sharing improves super-resolution of velocity fields.

Findings

Rotational consistency in datasets is crucial for learning rotational equivariance.

Weight sharing in CNNs enforces rotational equivariance without large datasets.

Equivariant CNNs outperform standard CNNs in super-resolution tasks for fluid velocity fields.

Abstract

This paper investigates the super-resolution (SR) of velocity fields in two-dimensional fluids from the viewpoint of rotational equivariance. SR refers to techniques that estimate high-resolution images from those in low resolution and has lately been applied in fluid mechanics. The rotational equivariance of SR models is defined as the property in which the super-resolved velocity field is rotated according to a rotation of the input, which leads to the inference covariant to the orientation of fluid systems. Generally, the covariance in physics is related to symmetries. To clarify a relationship to symmetries, the rotational consistency of datasets for SR is newly introduced as the invariance of pairs of low- and high-resolution velocity fields with respect to rotation. This consistency is sufficient and necessary for SR models to acquire rotational equivariance from large datasets with supervised learning. Such a large dataset is not required when rotational equivariance is imposed on SR models through weight sharing of convolution kernels as prior knowledge. Even if a fluid system has rotational symmetry, this symmetry may not carry over to a velocity dataset, which is not rotationally consistent. This inconsistency can occur when the rotation does not commute with the generation of low-resolution velocity fields. These theoretical suggestions are supported by the results from numerical experiments, where two existing convolutional neural networks (CNNs) are converted into rotationally equivariant CNNs and the inferences of the four CNNs are compared after the supervised training.