# Learning the Tangent Space of Dynamical Instabilities from Data

## TL;DR

This paper introduces a neural network approach to learn the pointwise mapping of dynamical instabilities in phase space, enabling better data-driven prediction and control of complex systems.

## Contribution

It presents a novel method to learn the tangent space of dynamical instabilities directly from data using neural networks, capturing the local instability directions.

## Key findings

- Successfully learned the pointwise tangent space mapping from data.
- Provided a new data-driven way to identify and analyze instabilities.
- Implications for improved prediction and control of dynamical systems.

## Abstract

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.