Learning the Tangent Space of Dynamical Instabilities from Data

The optimally time-dependent (OTD) modes are a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory of a dynamical system. Traditionally, these modes are computed by a time-marching approach that involves solving multiple initial-boundary-value problems concurrently with the state equations. However, for a large class of dynamical systems, the OTD modes are known to depend ``pointwise'' on the state of the system on the attractor, and not on the history of the trajectory. We leverage the power of neural networks to learn this ``pointwise'' mapping from phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in phase space, as well as accurate estimates for the leading Lyapunov exponents.