Roto-Translation Equivariant Super-Resolution of Two-Dimensional Flows Using Convolutional Neural Networks

Abstract

Convolutional neural networks (CNNs) often process vectors as quantities having no direction like colors in images. This study investigates the effect of treating vectors as geometrical objects in terms of super-resolution of velocity on two-dimensional fluids. Vector is distinguished from scalar by the transformation law associated with a change in basis, which can be incorporated as the prior knowledge using the equivariant deep learning. We convert existing CNNs into equivariant ones by making each layer equivariant with respect to rotation and translation. The training data in the low-and high-resolution are generated with the downsampling or the spectral nudging. When the data inherit the rotational symmetry, the equivariant CNNs show comparable accuracy with the non-equivariant ones. Since the number of parameters is smaller in the equivariant CNNs, these models are trainable with a smaller size of the data. In this case, the transformation law of vector should be incorporated as the prior knowledge, where vector is explicitly treated as a quantity having direction. Two examples demonstrate that the symmetry of the data can be broken. In the first case, a downsampling method makes the correspondence between low-and high-resolution patterns dependent on the orientation. In the second case, the input data are insufficient to recognize the rotation of coordinates in the experiment with the spectral nudging. In both cases, the accuracy of the CNNs deteriorates if the equivariance is forced to be imposed, and the usage of conventional CNNs may be justified even though vector is processed as a quantity having no direction.