Neural Network Approach to Construction of Classical Integrable Systems

TL;DR

This paper introduces a neural network-based method for systematically constructing classical integrable systems, enabling discovery without prior assumptions about transformations or Lax pairs.

Contribution

It presents a novel machine learning framework that learns Hamiltonians and canonical transformations simultaneously, facilitating the exploration of new integrable systems.

Findings

Successfully applied to Toda lattice

Enables unsupervised learning of integrable systems

Does not require prior Lax pair ansatz

Abstract

Integrable systems have provided various insights into physical phenomena and mathematics. The way of constructing many-body integrable systems is limited to few ansatzes for the Lax pair, except for highly inventive findings of conserved quantities. Machine learning techniques have recently been applied to broad physics fields and proven powerful for building non-trivial transformations and potential functions. We here propose a machine learning approach to a systematic construction of classical integrable systems. Given the Hamiltonian or samples in latent space, our neural network simultaneously learns the corresponding natural Hamiltonian in real space and the canonical transformation between the latent space and the real space variables. We also propose a loss function for building integrable systems and demonstrate successful unsupervised learning for the Toda lattice. Our approach enables exploring new integrable systems without any prior knowledge about the canonical transformation or any ansatz for the Lax pair.