Learning Poisson Systems and Trajectories of Autonomous Systems via Poisson Neural Networks

Pengzhan Jin, Zhen Zhang, Ioannis G. Kevrekidis, George Em Karniadakis

2022IEEE Transactions on Neural Networks and Learning Systems55 citationsDOI

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 Schrödinger equation, and pixel observations of the two-body problem.

Topics & Concepts

Poisson distributionArtificial neural networkNonlinear systemCoordinate systemTransformation (genetics)Symplectic geometryDiscrete Poisson equationComputer scienceDynamical systems theoryApplied mathematicsMathematicsArtificial intelligenceUniqueness theorem for Poisson's equationMathematical analysisPhysicsUniquenessQuantum mechanicsGeneStatisticsChemistryBiochemistryModel Reduction and Neural NetworksComputational Physics and Python ApplicationsNeural Networks and Applications