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

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...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Walker, Benjamin, McLeod, Andrew D, Qin, Tiexin, Cheng, Yichuan, Li, Haoliang, Lyons, Terry
Format: Article
Language:English
Subjects:
Differential equations
Fields (mathematics)
Irregular sampling
Mathematical models
Neural networks
Time series
Online Access:Get full text
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Summary: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.
ISSN:2331-8422