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

arXiv: https://arxiv.org/abs/2006.04727 · HTML: https://mlsanity.com/paper/2006.04727/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 30 of 34)

- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
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- [Attentive Neural Controlled Differential Equations for Time-series Classification and Forecasting](https://mlsanity.com/paper/2109.01876/) (2021)
- [Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2301.11308/) (2023)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Symplectic Adjoint Method for Exact Gradient of Neural ODE with Minimal Memory](https://mlsanity.com/paper/2102.09750/) (2021)
- [Fractional SDE-Net: Generation of Time Series Data with Long-term Memory](https://mlsanity.com/paper/2201.05974/) (2022)
- [Optimal Estimation of Generic Dynamics by Path-Dependent Neural Jump ODEs](https://mlsanity.com/paper/2206.14284/) (2022)
- [Learning Differential Operators for Interpretable Time Series Modeling](https://mlsanity.com/paper/2209.01491/) (2022)
- [Extending Path-Dependent NJ-ODEs to Noisy Observations and a Dependent Observation Framework](https://mlsanity.com/paper/2307.13147/) (2023)
- [CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk](https://mlsanity.com/paper/2507.08150/) (2025)
- [SeqLink: A Robust Neural-ODE Architecture for Modelling Partially Observed Time Series](https://mlsanity.com/paper/2212.03560/) (2022)
- [Transformers Can Solve Non-Linear and Non-Markovian Filtering Problems in Continuous Time For Conditionally Gaussian Signals](https://mlsanity.com/paper/2310.19603/) (2023)
- [Learning the Dynamics of Sparsely Observed Interacting Systems](https://mlsanity.com/paper/2301.11647/) (2023)
- [Learning Chaotic Systems and Long-Term Predictions with Neural Jump ODEs](https://mlsanity.com/paper/2407.18808/) (2024)
- [A General Framework for Uncertainty Quantification via Neural SDE-RNN](https://mlsanity.com/paper/2306.01189/) (2023)
- [IMTS is Worth Time × Channel Patches: Visual Masked Autoencoders for Irregular Multivariate Time Series Prediction](https://mlsanity.com/paper/2505.22815/) (2025)
- [A Jump-Diffusion Framework for Irregular Time Series Generation](https://mlsanity.com/paper/2505.23215/) (2025)
- [Neural MJD: Neural Non-Stationary Merton Jump Diffusion for Time Series Prediction](https://mlsanity.com/paper/2506.04542/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Nonparametric Filtering, Estimation and Classification using Neural Jump ODEs](https://mlsanity.com/paper/2412.03271/) (2024)
- [Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups](https://mlsanity.com/paper/2509.20599/) (2025)
- [Continuous time recurrent neural networks: overview and application to forecasting blood glucose in the intensive care unit](https://mlsanity.com/paper/2304.07025/) (2023)
- [On the Parallel Complexity of Multilevel Monte Carlo in Stochastic Gradient Descent](https://mlsanity.com/paper/2310.02402/) (2023)
- [Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces](https://mlsanity.com/paper/2402.05576/) (2024)
- [ODEStream: A Buffer-Free Online Learning Framework with ODE-based Adaptor for Streaming Time Series Forecasting](https://mlsanity.com/paper/2411.07413/) (2024)
- [Improving the Noise Estimation of Latent Neural Stochastic Differential Equations](https://mlsanity.com/paper/2412.17499/) (2024)
- [Anticipatory Reinforcement Learning: From Generative Path-Laws to Distributional Value Functions](https://mlsanity.com/paper/2604.04662/) (2026)

## Abstract (arXiv)

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 L2-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 L2-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.