Integrability ex machina

TL;DR

This paper introduces an automated machine learning approach to identify integrable structures like Lax pairs in dynamical systems, reducing manual effort and discovering new integrable deformations.

Contribution

It formulates the search for Lax pairs as an optimization problem and demonstrates the method's effectiveness on classical systems, uncovering novel structures.

Findings

Successfully identified Lax pairs in standard integrable systems

Discovered new integrable deformations of classical systems

Automated approach reduces manual analysis effort

Abstract

Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of relevant structures, the Lax pair and Lax connection respectively. By formulating this search as an optimization problem, we are able to identify appropriate structures via machine learning techniques. We test our method on standard systems of classical integrability and find that we can single out some integrable deformations of a system. Due to the ambiguity in defining a Lax pair our algorithm identifies novel Lax pairs which can be easily verified analytically.