Neural Canonical Transformation with Symplectic Flows

Abstract

Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. Intriguingly, it has a natural correspondence to normalizing flows with a symplectic constraint. Building on this key insight, we design a neural canonical transformation approach to automatically identify independent slow collective variables in general physical systems and natural datasets. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model based either on the Hamiltonian function or phase-space samples. The learned model maps physical variables onto an independent representation where collective modes with different frequencies are separated, which can be useful for various downstream tasks such as compression, prediction, control, and sampling. We demonstrate the ability of this method first by analyzing toy problems and then by applying it to real-world problems, such as identifying and interpolating slow collective modes of the alanine dipeptide molecule and MNIST database images.