将哈密顿网络拓展至非完整约束系统，实现物理规律与神经网络结合建模。
Hamiltonian-based neural networks for systems under nonholonomic constraints
- 设计三网络并行结构，同步学习哈密顿量、约束及其乘子
- 在带噪声数据下仍能准确建模滚动圆盘和旋转台上的球体运动
- 适用于需物理一致性约束的机器人、力学系统建模任务
近年来，将物理先验融入神经网络架构的方法日益受到关注。其中，哈密顿神经网络（HNN）及其变体通过结构和损失函数显式编码哈密顿力学，展现出良好建模能力。尽管带有非完整约束的系统通常不是哈密顿系统，但可将其表述为伪哈密顿形式，配备近泊松李括号。这为在非完整约束系统中应用部分HNN原理提供了可能。本文旨在开发一种改进的哈密顿神经网络架构，用于建模满足完整与非完整约束的哈密顿系统。提出一个三网络并行架构，同时学习系统的哈密顿量、约束条件及其关联乘子。以滚动圆盘和旋转台上的球体作为典型例子，评估所提架构性能。实验还采用含噪声训练集，在更贴近现实的条件下测试建模表现。
原文摘要 · Abstract (English)
There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. A family of these methodologies that has gained traction are Hamiltonian neural networks (HNN) and their variations. These architectures explicitly encode Hamiltonian mechanics both in their structure and loss function. Although Hamiltonian systems under nonholonomic constraints are in general not Hamiltonian, it is possible to formulate them in pseudo-Hamiltonian form, equipped with a Lie bracket which is almost Poisson. This opens the possibility of using some principles of HNNs in systems under nonholonomic constraints. The goal of the present work is to develop a modified Hamiltonian neural network architecture capable of modeling Hamiltonian systems under holonomic and nonholonomic constraints. A three-network parallel architecture is proposed to simultaneously learn the Hamiltonian of the system, the constraints, and their associated multipliers. A rolling disk and a ball on a spinning table are considered as canonical examples to assess the performance of the proposed Hamiltonian architecture. The experiments are then repeated with a noisy training set to study modeling performance under more realistic conditions.
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