Charts and atlases for nonlinear data-driven dynamics on a manifold

We present a method that learns minimal-dimensional dynamical models directly from time series data. The key enabling assumption is that the data, although nominally high-dimensional, lives on a lower-dimensional manifold. Our method is based on the description of a manifold as an atlas of charts—overlapping patches that can be invertibly mapped to low-dimensional Euclidean spaces. We learn the dynamics in each chart's low-dimensional Euclidean coordinates, and sew these local models together to produce a dynamical system for the global dynamics on the manifold. Our examples—ranging from simple low-dimensional periodic dynamics to complex non-periodic bursting dynamics of the Kuramoto-Sivashinsky equation—will bear out three major benefits of our atlas-of-charts-based framework: (1) the ability to obtain dynamical models of the lowest possible dimension, which previous methods are incapable of; (2) a divide-and-conquer approach that leads to computational benefits including scalability, the ability to adapt models locally, and an embarrassingly parallelizable algorithm; and (3) the ability to separate state space into region of distinct behaviors.