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

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