  Neural controlled differential equations (CDEs) are the continuous-time
analogue of recurrent neural networks, as Neural ODEs are to residual networks,
and offer a memory-efficient continuous-time way to model functions of
potentially irregular time series. Existing methods for computing the forward
pass of a Neural CDE involve embedding the incoming time series into path
space, often via interpolation, and using evaluations of this path to drive the
hidden state. Here, we use rough path theory to extend this formulation.
Instead of directly embedding into path space, we instead represent the input
signal over small time intervals through its \textit{log-signature}, which are
statistics describing how the signal drives a CDE. This is the approach for
solving \textit{rough differential equations} (RDEs), and correspondingly we
describe our main contribution as the introduction of Neural RDEs. This
extension has a purpose: by generalising the Neural CDE approach to a broader
class of driving signals, we demonstrate particular advantages for tackling
long time series. In this regime, we demonstrate efficacy on problems of length
up to 17k observations and observe significant training speed-ups, improvements
in model performance, and reduced memory requirements compared to existing
approaches.
