This paper investigates the universal approximation capabilities of Hamiltonian Deep Neural Networks (HDNNs) that arise from the discretization of Hamiltonian Neural Ordinary Differential Equations. Recently, it has been shown that HDNNs enjoy, by design, non-vanishing gradients, which provide numerical stability during training. However, although HDNNs have demonstrated state-of-the-art performance in several applications, a comprehensive study to quantify their expressivity is missing. In this regard, we provide a universal approximation theorem for HDNNs and prove that a portion of the flow of HDNNs can approximate arbitrary well any continuous function over a compact domain. This result provides a solid theoretical foundation for the practical use of HDNNs.

翻译：本文研究了由哈密顿神经常微分方程离散化产生的哈密顿深度神经网络（HDNNs）的通用逼近能力。近期研究表明，HDNNs 在结构上天然具备非零梯度，这为训练过程提供了数值稳定性。然而，尽管 HDNNs 在多项应用中展现了最先进的性能，目前尚缺乏对其表达能力的系统性量化研究。为此，我们提出了 HDNNs 的通用逼近定理，证明了 HDNNs 流的一部分能够在紧致域上以任意精度逼近任意连续函数。该结果为 HDNNs 的实际应用奠定了坚实的理论基础。