Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering
This provides a theoretically guaranteed method for continuous-time prediction and filtering in irregularly sampled time series, addressing a bottleneck in neural ODE and RNN combinations.
The paper tackles the problem of predicting irregularly observed time series by introducing Neural Jump ODE (NJ-ODE), a model that learns the conditional expectation of a stochastic process continuously in time, and proves theoretical convergence to the L^2-optimal prediction while showing empirical outperformance over baselines in complex tasks and real-world datasets.
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.