Supervised dynamic mode decomposition via multitask learning

Understanding dynamical systems by extracting spatiotemporal patterns from data is fundamental in a variety of fields of engineering and science. Dynamic mode decomposition (DMD) has recently attracted attention in these fields as a way of obtaining a global modal description of a nonlinear dynamical system from data, without requiring explicit prior knowledge. However, DMD is in principle an unsupervised dimensionality reduction algorithm; it is not endowed with the mechanism to utilize label information even if a set of data with different labels is given. In this paper, we propose the algorithm that incorporates label information into DMD via multitask learning by solving sparse-group Lasso. To this end, we estimate sparse weights over dynamic modes in a label-wise manner by regarding data with different labels as different tasks. Modal descriptions estimated by this approach share a part of the global modes, resulting in the extraction of label-specific and common (or mixed) dynamical structures, which could be useful in understanding mechanisms in the spatiotemporal behavior behind data. We investigate the empirical performance using synthetic and real-world datasets, and validate that our algorithm can extract and visualize common and label-specific spatiotemporal structures.