This letter deals with neural networks as dynamical systems governed by finite difference equations. It shows that the introduction of k-many skip connections into network architectures, such as residual networks and additive dense networks, defines kth order dynamical equations on the layer-wise transformations. Closed-form solutions for the state-space representations of general kth order additive dense networks, where the concatenation operation is replaced by addition, as well as kth order smooth networks, are found. The developed provision endows deep neural networks with an algebraic structure. Furthermore, it is shown that imposing kth order smoothness on network architectures with d-many nodes per layer increases the state-space dimension by a multiple of k, and so the effective embedding dimension of the data manifold by the neural network is k·d-many dimensions. It follows that network architectures of these types reduce the number of parameters needed to maintain the same embedding dimension by a factor of k2 when compared to an equivalent first-order, residual network. Numerical simulations and experiments on CIFAR10, SVHN, and MNIST have been conducted to help understand the developed theory and efficacy of the proposed concepts.

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