Interpretable comparisons of multistable dynamical systems

The comparison of time series data is an increasingly important problem in neuroscience and machine learning, with a growing collection of high dimensional temporal datasets being collected from neural recordings and large sequence models. Comparison is necessary for validating computational models of physical systems, tracking changes over time or context, and many other fundamental steps in the scientific pipeline. Dynamical Similarity Analysis (DSA) is a popular class of methods that computes distance metrics on dynamical systems. DSA involves a two-step approach, first finding a linear dynamical embedding of each system (an approximation of the Koopman operator), and subsequently quantifying distance between the linearized systems. These globally linear representations can be challenging to interpret when applied to multistable dynamics, defined as having multiple basins of attraction. For these settings, we developed an interpretable and robust variant of DSA called multistable DSA (mDSA). mDSA divides and conquers basins of attraction by first decomposing the data via a global Koopman operator, then estimating local operators for each basin as well as the global basin topology. mDSA compares these multiscale representations jointly or individually via an optimal transport metric called the fused Gromov--Wasserstein distance. On stochastic multistable dynamics and RNNs, we demonstrate that mDSA provides an informative and interpretable decomposition of dynamical similarity.