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.

使用物理上知悉的神经网络方法来分析含有一种运动第一积分的非线性哈密顿系统，并提出了一种结构，将现有的哈密顿神经网络结构与 Adaptable Symplectic 循环神经网络相结合，可以在整个参数空间内预测动力学，保留哈密顿方程以及相空间的辛结构。同时，利用神经网络的高维非线性能力，结合 Long Short Term Memory 网络进行判断嵌入定理的实现，构造系统的延迟嵌入，并将拓扑不变吸引子映射到真实形式。该方法对于单参数势能有效，并且即使在较长时间内也能提供准确的预测结果。