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：本文研究汉密尔顿深度神经网络（HDNNs）的通用逼近能力，这些网络由离散化的汉密尔顿神经常微分方程引发。最近研究表明，HDNNs本质上具有不消失的梯度，使其训练具有数值稳定性。然而，虽然HDNNs在许多应用中展现了最先进的性能，但缺乏对其表达能力的全面研究。为此，我们提供了HDNNs的通用逼近定理，并证明了其流中的一部分可逼近紧致域上的任意连续函数。该结果为实际使用HDNNs提供了坚实的理论基础。