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。

机器学习在动态系统建模与分析中的应用

Applications of Machine Learning to Modelling and Analysing Dynamical  Systems

We introduce Sparse Symplectically Integrated Neural Networks (SSINNs), a
novel model for learning Hamiltonian dynamical systems from data. SSINNs
combine fourth-order symplectic integration with a learned parameterization of
the Hamiltonian obtained using sparse regression through a mathematically
elegant function space. This allows for interpretable models that incorporate
symplectic inductive biases and have low memory requirements. We evaluate
SSINNs on four classical Hamiltonian dynamical problems: the H\'enon-Heiles
system, nonlinearly coupled oscillators, a multi-particle mass-spring system,
and a pendulum system. Our results demonstrate promise in both system
prediction and conservation of energy, often outperforming the current
state-of-the-art black-box prediction techniques by an order of magnitude.
Further, SSINNs successfully converge to true governing equations from highly
limited and noisy data, demonstrating potential applicability in the discovery
of new physical governing equations.

利用稀疏回归方法，结合四阶辛普森积分法实现了对哈密顿动力学系统的建模，能够在数据有限噪声较大的情况下较好地预测和求出系统规律。

稀疏辛合成神经网络

Sparse Symplectically Integrated Neural Networks

Accelerated gradient methods have had significant impact in machine learning
-- in particular the theoretical side of machine learning -- due to their
ability to achieve oracle lower bounds. But their heuristic construction has
hindered their full integration into the practical machine-learning algorithmic
toolbox, and has limited their scope. In this paper we build on recent work
which casts acceleration as a phenomenon best explained in continuous time, and
we augment that picture by providing a systematic methodology for converting
continuous-time dynamics into discrete-time algorithms while retaining oracle
rates. Our framework is based on ideas from Hamiltonian dynamical systems and
symplectic integration. These ideas have had major impact in many areas in
applied mathematics, but have not yet been seen to have a relationship with
optimization.

本研究提出了一种基于 Hamiltonian dynamical systems 和 symplectic integration 的框架，用于将加速梯度方法中连续时间动态转换成离散时间算法，从而实现 oracle 下界，这一框架将加速梯度方法引入了不同的优化领域。