Many natural systems, including neural circuits involved in decision making, are modeled as high-dimensional dynamical systems with multiple stable states. While existing analytical tools primarily describe behavior near stable equilibria, characterizing separatrices--the manifolds that delineate boundaries between different basins of attraction--remains challenging, particularly in high-dimensional settings. Here, we introduce a numerical framework leveraging Koopman Theory combined with Deep Neural Networks to effectively characterize separatrices. Specifically, we approximate Koopman Eigenfunctions (KEFs) associated with real positive eigenvalues, which vanish precisely at the separatrices. Utilizing these scalar KEFs, optimization methods efficiently locate separatrices even in complex systems. We demonstrate our approach on synthetic benchmarks, ecological network models, and high-dimensional recurrent neural networks trained on either neuroscience-inspired tasks or fit to real neural data. Moreover, we illustrate the practical utility of our method by designing optimal perturbations that can shift systems across separatrices, enabling predictions relevant to optogenetic stimulation experiments in neuroscience.

Predicting the occurence of rare and extreme events in complex systems is a well-known problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in the last ten years, which better estimate tail distributions. They often use large deviation concepts without the need to perform heavy direct ensemble simulations. In particular, a well-known approach is to derive a minimum action principle and to find its minimizers. The analysis of rare reactive events in non-equilibrium systems without detailed balance is notoriously difficult either theoretically and computationally. They are described in the limit of small noise by the Freidlin-Wentzell action. We propose here a new method which minimizes the geometrical action instead using neural networks: it is called deep gMAM. It relies on a natural and simple machine-learning formulation of the classical gMAM approach. We give a detailed description of the method as well as many examples. These include bimodal switches in complex stochastic (partial) differential equations, quasi-potential estimates, and extreme events in Burgers turbulence.