  The vector field of a controlled differential equation (CDE) describes the
relationship between a control path and the evolution of a solution path.
Neural CDEs (NCDEs) treat time series data as observations from a control path,
parameterise a CDE's vector field using a neural network, and use the solution
path as a continuously evolving hidden state. As their formulation makes them
robust to irregular sampling rates, NCDEs are a powerful approach for modelling
real-world data. Building on neural rough differential equations (NRDEs), we
introduce Log-NCDEs, a novel, effective, and efficient method for training
NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the
study of rough paths for approximating a CDE's solution. Log-NCDEs are shown to
outperform NCDEs, NRDEs, the linear recurrent unit, S5, and MAMBA on a range of
multivariate time series datasets with up to $50{,}000$ observations.
