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.
@article{arxiv.2402.18512,
title = {Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference},
author = {Benjamin Walker and Andrew D. McLeod and Tiexin Qin and Yichuan Cheng and Haoliang Li and Terry Lyons},
journal= {arXiv preprint arXiv:2402.18512},
year = {2025}
}