  Nonlinear principal component analysis (NLPCA) via autoencoders has attracted
attention in the dynamical systems community due to its larger compression rate
when compared to linear principal component analysis (PCA). These model
reduction methods experience an increase in the dimensionality of the latent
space when applied to datasets that exhibit invariant samples due to the
presence of symmetries. In this study, we introduce a novel machine learning
embedding for autoencoders, which uses Siamese networks and spatial transformer
networks to account for discrete and continuous symmetries, respectively. The
Siamese branches autonomously find a fundamental domain to which all samples
are transformed, without introducing human bias. The spatial transformer
network discovers the optimal slicing template for continuous translations so
that invariant samples are aligned in the homogeneous direction. Thus, the
proposed symmetry-aware autoencoder is invariant to predetermined input
transformations. This embedding can be employed with both linear and nonlinear
reduction methods, which we term symmetry-aware PCA (s-PCA) and symmetry-aware
NLPCA (s-NLPCA). We apply the proposed framework to the Kolmogorov flow to
showcase the capabilities for a system exhibiting both a continuous symmetry as
well as discrete symmetries.
