  For a large class of dynamical systems, the optimally time-dependent (OTD)
modes, a set of deformable orthonormal tangent vectors that track directions of
instabilities along any trajectory, 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. Implications for data-driven prediction and control of dynamical
instabilities are discussed.
