Applications of Machine Learning to Modelling and Analysing Dynamical Systems
This work addresses the problem of accurately modeling and predicting complex dynamical systems for researchers in physics and machine learning, though it appears incremental as it builds on existing Hamiltonian Neural Network structures.
The paper tackled modeling nonlinear Hamiltonian dynamical systems by proposing Adaptable Symplectic Recurrent Neural Networks that preserve Hamilton's equations and symplectic structure, significantly outperforming previous neural networks, especially in multi-parameter potentials, and demonstrated robustness in chaotic, quasiperiodic, and periodic conditions. It also addressed predicting dynamics from partial information using Long Short Term Memory networks with Takens' embedding theorem, layered with the adaptable nets to preserve structure, showing efficient and accurate predictions for single-parameter potentials over long periods.
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.