Applications of Machine Learning to Modelling and Analysing Dynamical Systems

TL;DR

This paper introduces advanced neural network architectures that incorporate physical principles to accurately model and predict complex Hamiltonian dynamical systems, including cases with partial information and chaotic behavior.

Contribution

It proposes Adaptable Symplectic Recurrent Neural Networks and a method combining Takens' embedding with neural networks to preserve physical structure and improve long-term predictions.

Findings

Outperforms previous neural networks in predicting Hamiltonian dynamics.

Successfully models systems with multiple parameters and chaotic behavior.

Accurately predicts dynamics from partial information over long time periods.

Abstract

We explore the use of Physics Informed Neural Networks to analyse nonlinear Hamiltonian Dynamical Systems with a first integral of motion. In this work, we propose an architecture which combines existing Hamiltonian Neural Network structures into Adaptable Symplectic Recurrent Neural Networks which preserve Hamilton's equations as well as the symplectic structure of phase space while predicting dynamics for the entire parameter space. This architecture is found to significantly outperform previously proposed neural networks when predicting Hamiltonian dynamics especially in potentials which contain multiple parameters. We demonstrate its robustness using the nonlinear Henon-Heiles potential under chaotic, quasiperiodic and periodic conditions. The second problem we tackle is whether we can use the high dimensional nonlinear capabilities of neural networks to predict the dynamics of a Hamiltonian system given only partial information of the same. Hence we attempt to take advantage of Long Short Term Memory networks to implement Takens' embedding theorem and construct a delay embedding of the system followed by mapping the topologically invariant attractor to the true form. This architecture is then layered with Adaptable Symplectic nets to allow for predictions which preserve the structure of Hamilton's equations. We show that this method works efficiently for single parameter potentials and provides accurate predictions even over long periods of time.