Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks

TL;DR

This paper introduces Poisson neural networks (PNNs) that leverage the Darboux-Lie theorem and structured neural networks with physical priors to accurately learn Poisson systems and autonomous system trajectories from data.

Contribution

The work extends the Darboux-Lie theorem to unknotted trajectories and employs structured neural networks with physical priors for learning complex dynamical systems.

Findings

Accurately models particle motion in electromagnetic potential

Successfully predicts nonlinear Schrödinger equation dynamics

Handles pixel observations of the two-body problem effectively

Abstract

We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data. Based on the Darboux-Lie theorem, the phase flow of a Poisson system can be written as the composition of (1) a coordinate transformation, (2) an extended symplectic map and (3) the inverse of the transformation. In this work, we extend this result to the unknotted trajectories of autonomous systems. We employ structured neural networks with physical priors to approximate the three aforementioned maps. We demonstrate through several simulations that PNNs are capable of handling very accurately several challenging tasks, including the motion of a particle in the electromagnetic potential, the nonlinear Schr{\"o}dinger equation, and pixel observations of the two-body problem.