Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data

In science, we are often interested in obtaining a generative model of the underlying system dynamics from observed time series. While powerful methods for dynamical systems reconstruction (DSR) exist when data come from a single domain, how to best integrate data from multiple dynamical regimes and leverage it for generalization is still an open question. This becomes particularly important when individual time series are short, and group-level information may help to fill in for gaps in single-domain data. Here we introduce a hierarchical framework that enables to harvest group-level (multi-domain) information while retaining all single-domain characteristics, and showcase it on popular DSR benchmarks, as well as on neuroscience and medical data. In addition to faithful reconstruction of all individual dynamical regimes, our unsupervised methodology discovers common low-dimensional feature spaces in which datasets with similar dynamics cluster. The features spanning these spaces were further dynamically highly interpretable, surprisingly in often linear relation to control parameters that govern the dynamics of the underlying system. Finally, we illustrate transfer learning and generalization to new parameter regimes, paving the way toward DSR foundation models.