  We develop an unsupervised machine learning algorithm for the automated
discovery and identification of traveling waves in spatio-temporal systems
governed by partial differential equations (PDEs). Our method uses sparse
regression and subspace clustering to robustly identify translational
invariances that can be leveraged to build improved reduced order models
(ROMs). Invariances, whether translational or rotational, are well known to
compromise the ability of ROMs to produce accurate and/or low-rank
representations of the spatio-temporal dynamics. However, by discovering
translations in a principled way, data can be shifted into a coordinate systems
where quality, low-dimensional ROMs can be constructed. This approach can be
used on either numerical or experimental data with or without knowledge of the
governing equations. We demonstrate our method on a variety of PDEs of
increasing difficulty, taken from the field of fluid dynamics, showing the
efficacy and robustness of the proposed approach.
