The modelling of dynamical systems from discrete observations is a challenge faced by modern scientific and engineering data systems. Hamiltonian systems are one such fundamental and ubiquitous class of dynamical systems. Hamiltonian neural networks are state-of-the-art models that unsupervised-ly regress the Hamiltonian of a dynamical system from discrete observations of its vector field under the learning bias of Hamilton's equations. Yet Hamiltonian dynamics are often complicated, especially in higher dimensions where the state space of the Hamiltonian system is large relative to the number of samples. A recently discovered remedy to alleviate the complexity between state variables in the state space is to leverage the additive separability of the Hamiltonian system and embed that additive separability into the Hamiltonian neural network. Following the nomenclature of physics-informed machine learning, we propose three separable Hamiltonian neural networks. These models embed additive separability within Hamiltonian neural networks. The first model uses additive separability to quadratically scale the amount of data for training Hamiltonian neural networks. The second model embeds additive separability within the loss function of the Hamiltonian neural network. The third model embeds additive separability through the architecture of the Hamiltonian neural network using conjoined multilayer perceptions. We empirically compare the three models against state-of-the-art Hamiltonian neural networks, and demonstrate that the separable Hamiltonian neural networks, which alleviate complexity between the state variables, are more effective at regressing the Hamiltonian and its vector field.

翻译：从离散观测数据中对动力系统进行建模是现代科学与工程数据系统面临的挑战。哈密顿系统作为动力系统中一类基础且普遍存在的系统，其建模尤为重要。哈密顿神经网络是当前最先进的模型，能够借助哈密顿方程的归纳偏置，从向量场的离散观测中无监督地回归动力系统的哈密顿量。然而，哈密顿动力学通常复杂，尤其在状态空间维度较高且样本数相对有限的情况下更为显著。近期研究发现，利用哈密顿系统的可加可分离性并将其嵌入哈密顿神经网络，可有效缓解状态变量间的复杂度。遵循物理信息机器学习的术语体系，本文提出三种可分离哈密顿神经网络模型。这些模型将可加可分离性嵌入哈密顿神经网络：第一种模型利用可加可分离性使训练数据量呈二次方级扩展；第二种模型将可加可分离性嵌入损失函数；第三种模型通过联体多层感知器在架构层面实现可加可分离性。我们通过实验将三种模型与当前最优的哈密顿神经网络进行对比，结果表明，能够缓解状态变量间复杂度的可分离哈密顿神经网络在回归哈密顿量及其向量场方面更为高效。