Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systems
This provides a new perspective for understanding the approximation properties of feedforward neural networks, though it is incremental as it extends an existing concept from residual networks to a classical structure.
The paper tackles the problem of linking deep neural networks to dynamical systems by proving that vanilla feedforward networks, not just residual networks, can be viewed as a numerical discretization of dynamical systems, with the width matching input and output dimensions.
Deep learning has made significant applications in the field of data science and natural science. Some studies have linked deep neural networks to dynamic systems, but the network structure is restricted to the residual network. It is known that residual networks can be regarded as a numerical discretization of dynamic systems. In this paper, we back to the classical network structure and prove that the vanilla feedforward networks could also be a numerical discretization of dynamic systems, where the width of the network is equal to the dimension of the input and output. Our proof is based on the properties of the leaky-ReLU function and the numerical technique of splitting method to solve differential equations. Our results could provide a new perspective for understanding the approximation properties of feedforward neural networks.