# Operator Neural Jump ODEs: L2-optimal prediction in function spaces

arXiv: https://arxiv.org/abs/2607.23110 · HTML: https://mlsanity.com/paper/2607.23110/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 16 of 16)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering](https://mlsanity.com/paper/2006.04727/) (2020)
- [Black box variational inference for state space models](https://mlsanity.com/paper/1511.07367/) (2015)
- [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)
- [Learning Chaotic Systems and Long-Term Predictions with Neural Jump ODEs](https://mlsanity.com/paper/2407.18808/) (2024)
- [Neural Jump ODEs as Generative Models](https://mlsanity.com/paper/2510.02757/) (2025)
- [Nonparametric Filtering, Estimation and Classification using Neural Jump ODEs](https://mlsanity.com/paper/2412.03271/) (2024)

## Abstract (arXiv)

In this paper, we study the extension of Neural Jump ODEs to infinite-dimensional function spaces. In particular, the underlying process X now takes values in L2(Xi, ℝdX) instead of ℝdX and the Operator NJ-ODE approximates the optimal predictor of this process by producing a representative of the conditional expectation. The NJ-ODE model is a framework for online learning the optimal prediction of continuous-time stochastic processes, given discrete, possibly irregular and incomplete past observations. In a series of works, this model has been extended to deal with generic path-dependent processes, with observation noise and dependent observations, with long-term predictions, and with input-output systems. However, throughout all of these works, the underlying processes were restricted to be finite-dimensional. In particular, function-valued problems, like yield curve or volatility surface predictions, could only be handled through discretization, which inherently leads to a loss of information. In this work, we build on ideas from Neural Operator methods that allow us to extend the NJ-ODE framework to an infinite-dimensional output process. To prove convergence of the NJ-ODE to the optimal prediction process, we develop a new approximation strategy that also generalizes previous works in the finite-dimensional setting by considerably weakening the assumptions.