Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to represent critical points of the input-output map. We prove that this has global consequences for target function approximations and show that the manifestation of this defect is typically a shift of the critical point to infinity, which we call the ``tunnel effect'' in the context of classification tasks. While ResNets offer greater expressivity than standard multilayer perceptrons (MLPs), their capability strongly depends on the signal ratio between the skip and residual channels. We establish quantitative approximation bounds for both the residual-dominant (close to MLP) and skip-dominant (close to neural ODE) regimes. These estimates depend explicitly on the channel ratio and uniform network weight bounds. Low-dimensional examples further provide a detailed analysis of the different ResNet regimes and how architecture-target incompatibility influences the approximation error.

The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property

Neural Ordinary Differential Equations (Neural ODEs), which are the continuous-time analog of Residual Neural Networks (ResNets), have gained significant attention in recent years. Similarly, Neural Delay Differential Equations (Neural DDEs) can be interpreted as an infinite depth limit of Densely Connected Residual Neural Networks (DenseResNets). In contrast to traditional ResNet architectures, DenseResNets are feed-forward networks that allow for shortcut connections across all layers. These additional connections introduce memory in the network architecture, as typical in many modern architectures. In this work, we explore how the memory capacity in neural DDEs influences the universal approximation property. The key parameter for studying the memory capacity is the product $K \tau$ of the Lipschitz constant and the delay of the DDE. In the case of non-augmented architectures, where the network width is not larger than the input and output dimensions, neural ODEs and classical feed-forward neural networks cannot have the universal approximation property. We show that if the memory capacity $K\tau$ is sufficiently small, the dynamics of the neural DDE can be approximated by a neural ODE. Consequently, non-augmented neural DDEs with a small memory capacity also lack the universal approximation property. In contrast, if the memory capacity $K\tau$ is sufficiently large, we can establish the universal approximation property of neural DDEs for continuous functions. If the neural DDE architecture is augmented, we can expand the parameter regions in which universal approximation is possible. Overall, our results show that by increasing the memory capacity $K\tau$, the infinite-dimensional phase space of DDEs with positive delay $\tau>0$ is not sufficient to guarantee a direct jump transition to universal approximation, but only after a certain memory threshold, universal approximation holds.

Analysis of the Geometric Structure of Neural Networks and Neural ODEs via Morse Functions

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.

Embedding Capabilities of Neural ODEs

A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension. The embedding capability of a neural ODE architecture can be increased by adding, for example, a linear layer, or augmenting the phase space. Yet, there is currently no systematic theory available and our work contributes towards this goal by developing various embedding results as well as identifying situations, where no embedding is possible. The mathematical techniques used include as main components iterative functional equations, Morse functions and suspension flows, as well as several further ideas from analysis. Although practically, mainly universal approximation theorems are used, our geometric dynamical systems viewpoint on universal embedding provides a fundamental understanding, why certain neural ODE architectures perform better than others.