# Learning Chaotic Systems and Long-Term Predictions with Neural Jump ODEs

arXiv: https://arxiv.org/abs/2407.18808 · HTML: https://mlsanity.com/paper/2407.18808/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 26 Jul 2024.

## Builds on (most cited first, 9 of 9)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems](https://mlsanity.com/paper/2207.02542/) (2022)
- [Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering](https://mlsanity.com/paper/2006.04727/) (2020)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [Optimal Estimation of Generic Dynamics by Path-Dependent Neural Jump ODEs](https://mlsanity.com/paper/2206.14284/) (2022)
- [Extending Path-Dependent NJ-ODEs to Noisy Observations and a Dependent Observation Framework](https://mlsanity.com/paper/2307.13147/) (2023)

## Built on by (most cited first, 3 of 3)

- [CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk](https://mlsanity.com/paper/2507.08150/) (2025)
- [Nonparametric Filtering, Estimation and Classification using Neural Jump ODEs](https://mlsanity.com/paper/2412.03271/) (2024)
- [Operator Neural Jump ODEs: L2-optimal prediction in function spaces](https://mlsanity.com/paper/2607.23110/) (2026)

## Abstract (arXiv)

The Path-dependent Neural Jump ODE (PD-NJ-ODE) is a model for online prediction of generic (possibly non-Markovian) stochastic processes with irregular (in time) and potentially incomplete (with respect to coordinates) observations. It is a model for which convergence to the L2-optimal predictor, which is given by the conditional expectation, is established theoretically. Thereby, the training of the model is solely based on a dataset of realizations of the underlying stochastic process, without the need of knowledge of the law of the process. In the case where the underlying process is deterministic, the conditional expectation coincides with the process itself. Therefore, this framework can equivalently be used to learn the dynamics of ODE or PDE systems solely from realizations of the dynamical system with different initial conditions. We showcase the potential of our method by applying it to the chaotic system of a double pendulum. When training the standard PD-NJ-ODE method, we see that the prediction starts to diverge from the true path after about half of the evaluation time. In this work we enhance the model with two novel ideas, which independently of each other improve the performance of our modelling setup. The resulting dynamics match the true dynamics of the chaotic system very closely. The same enhancements can be used to provably enable the PD-NJ-ODE to learn long-term predictions for general stochastic datasets, where the standard model fails. This is verified in several experiments.