Besides classical feed-forward neural networks, also neural ordinary differential equations (neural ODEs) gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input-output dynamics of finite and infinite depth neural networks with scalar output. In the finite depth case, the input is a state associated to a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps a linear transformation of the input to a linear transformation of its time-$T$ map. We show that depending on the specific structure of the network, the input-output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions, where every critical point is non-degenerate. We prove that critical points cannot exist, if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that each critical point is non-degenerate, if for finite depth neural networks the underlying graph has no bottleneck, and if for neural ODEs, the linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and in the infinite depth case. The established theorems allow us to formulate results on universal embedding, i.e.\ on the exact representation of maps by neural networks and neural ODEs. Our dynamical systems viewpoint on the geometric structure of the input-output map provides a fundamental understanding, why certain architectures perform better than others.

翻译：除经典前馈神经网络外，神经常微分方程（neural ODEs）近年来也引起了特别关注。神经常微分方程可被解释为前馈或残差神经网络的无限深度极限。本文研究具有标量输出的有限深度与无限深度神经网络的输入-输出动力学。在有限深度情形中，输入是与有限节点数相关联的状态，该状态经过多次非线性变换映射至一个输出节点的状态。类比而言，神经常微分方程将输入的线性变换映射为其时间$T$映射的线性变换。我们证明，根据网络的具体结构，输入-输出映射在临界点的存在性与正则性方面具有不同性质。这些性质可通过Morse函数（一种每个临界点均非退化的标量函数）进行刻画。我们证明：若隐藏层维度单调递减，或相空间维度小于等于输入维度，则临界点不可能存在。在临界点存在的情形中，我们根据网络的具体架构对其正则性进行分类。我们证明：对于有限深度神经网络，若底层图无瓶颈；对于神经常微分方程，若所用线性变换具有满秩，则每个临界点均为非退化。对于每种架构类型，有限深度与无限深度情形下所证明的性质具有可比性。所建立的定理使我们能够得出关于通用嵌入（即神经网络与神经常微分方程对映射的精确表示）的结论。我们对输入-输出映射几何结构的动力系统观点，从根本上解释了为何某些架构的性能优于其他架构。