Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

Integrablity is vital in the study of dynamical systems since it is a mathematically rich topic and often the starting point for analyzing more complex equations. However, it is difficult to even recognize if a given system is integrable. Therefore, we formulate the automated discovery of integrability in dynamical systems as a symbolic regression problem. Specifically, we seek to maximize the compatibility between the known Hamiltonian of the system and a pair of matrix/differential operators known as Lax pairs. Our approach is tested on a variety of systems ranging from nonlinear oscillators to canonical Hamiltonian PDEs. We test robustness of the framework against nonintegrable perturbations, and, in all examples, reliably confirm or deny integrability. Moreover, using a thresholded $l^0$ regularization to promote sparsity, we recover expected and discover new Lax pairs despite wide hypotheses on the operators. We will discuss future directions for adapting our framework to studying exactly solvable models in statistical and quantum field theories and building an all purpose data-driven nonintegrability filter of complex systems.