Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference

Walker Benjamin, University of Oxford

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) parameterise a CDE's vector field using a neural network, treat time series data as observations from a control path, and use the solution path as their hidden state. Given a partition of time, the Log-ODE method approximates the solution of a CDE by replacing it with an autonomous ODE on each interval. Building on the approach of neural rough differential equations (NRDEs), we introduce Log-NCDEs, which are the first application of the Log-ODE method to NCDEs. On a range of multivariate time series classification benchmarks, Log-NCDEs are shown to achieve an average test set accuracy higher than both NCDEs and NRDEs, and inline with current state-of-the-art deep learning methods.

Area: CS42 - Rough paths and data science (Christian Bayer, Paul Hager and Sebastian Riedel)

Keywords: Time Series, Neural Differential Equations, Machine Learning, Rough Paths

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