Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have 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 with 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 an affine linear transformation of the input to an affine 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 than 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 except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite depth neural networks the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs.

翻译：除了经典的前馈神经网络（如多层感知机），神经常微分方程近年来也受到特别关注。神经常微分方程可解释为前馈或残差神经网络的无限深度极限。我们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，输入是与有限个节点相关联的状态，经过多重非线性变换映射至单个输出节点的状态。类似地，神经常微分方程将输入的仿射线性变换映射至其时间$T$映射的仿射线性变换。我们证明，根据网络的具体结构，输入-输出映射在临界点存在性与正则性方面具有不同性质。这些性质可通过莫尔斯函数（即所有临界点均为非退化的标量函数）进行表征。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。研究表明：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，若所用仿射线性变换具有满秩，则除权重空间的勒贝格测度零集外，每个临界点均为非退化点。对于每种架构类型，所证明的性质在有限深度与无限深度情形中具有可比性。建立的定理使我们能够表述关于通用嵌入与通用逼近的结果，即关于神经网络与神经常微分方程对映射的精确表示与近似表示。