Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions

TL;DR

This paper introduces a deep learning-based numerical method to identify separatrices in high-dimensional dynamical systems using Koopman Eigenfunctions, aiding in understanding system boundaries and transitions.

Contribution

It develops a novel framework combining Koopman Theory with deep neural networks to efficiently locate separatrices in complex, high-dimensional systems.

Findings

Successfully applied to synthetic benchmarks and neural network models.

Able to predict system transitions relevant to neuroscience experiments.

Demonstrates practical utility in designing interventions to shift system states.

Abstract

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.