Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering

TL;DR

This paper introduces Neural Jump ODEs, a continuous-time model that learns the conditional expectation of stochastic processes with theoretical guarantees, outperforming existing methods on complex tasks and real-world data.

Contribution

The paper proposes Neural Jump ODEs, providing the first theoretical guarantees for continuous-time neural models predicting stochastic processes, with convergence to the optimal prediction.

Findings

Model converges to $L^2$-optimal prediction.

Outperforms baselines on complex tasks.

Effective on real-world datasets.

Abstract

Combinations of neural ODEs with recurrent neural networks (RNN), like GRU-ODE-Bayes or ODE-RNN are well suited to model irregularly observed time series. While those models outperform existing discrete-time approaches, no theoretical guarantees for their predictive capabilities are available. Assuming that the irregularly-sampled time series data originates from a continuous stochastic process, the $L^2$-optimal online prediction is the conditional expectation given the currently available information. We introduce the Neural Jump ODE (NJ-ODE) that provides a data-driven approach to learn, continuously in time, the conditional expectation of a stochastic process. Our approach models the conditional expectation between two observations with a neural ODE and jumps whenever a new observation is made. We define a novel training framework, which allows us to prove theoretical guarantees for the first time. In particular, we show that the output of our model converges to the $L^2$-optimal prediction. This can be interpreted as solution to a special filtering problem. We provide experiments showing that the theoretical results also hold empirically. Moreover, we experimentally show that our model outperforms the baselines in more complex learning tasks and give comparisons on real-world datasets.