On the space-time expressivity of ResNets

Residual networks (ResNets) are a deep learning architecture that substantially improved the state of the art performance in certain supervised learning tasks. Since then, they have received continuously growing attention. ResNets have a recursive structure xk+1 = xk + Rk(xk) where Rk is a neural network called a residual block. This structure can be seen as the Euler discretisation of an associated ordinary differential equation (ODE) which is called a neural ODE. Recently, ResNets were proposed as the space-time approximation of ODEs which are not of this neural type. To elaborate this connection we show that by increasing the number of residual blocks as well as their expressivity the solution of an arbitrary ODE can be approximated in space and time simultaneously by deep ReLU ResNets. Further, we derive estimates on the complexity of the residual blocks required to obtain a prescribed accuracy under certain regularity assumptions.