Neural Rough Differential Equations for Long Time Series

TL;DR

This paper introduces Neural RDEs, extending Neural CDEs using rough path theory, to effectively model long and irregular time series with faster training and lower memory usage.

Contribution

The paper presents Neural RDEs, a novel extension of Neural CDEs utilizing log-signatures from rough path theory for improved long time series modeling.

Findings

Effective on long time series up to 17,000 observations

Significant training speed-ups observed

Reduced memory requirements compared to existing methods

Abstract

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.