Applications of Machine Learning to Modelling and Analysing Dynamical Systems. (arXiv:2308.03763v1 [cs.LG])

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.