On the space-time expressivity of ResNets
This work provides a theoretical foundation for using ResNets as universal approximators of ODEs, which is incremental as it extends existing connections between ResNets and neural ODEs to more general ODEs.
The paper tackles the problem of approximating arbitrary ordinary differential equations (ODEs) using deep ReLU ResNets, showing that by increasing the number and expressivity of residual blocks, these networks can approximate ODE solutions in space and time simultaneously, with derived complexity estimates for prescribed accuracy under regularity assumptions.
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 $x_{k+1} = x_k + R_k(x_k)$ where $R_k$ 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.