# Augmenting Neural Differential Equations to Model Unknown Dynamical Systems with Incomplete State Information

arXiv: https://arxiv.org/abs/2008.08226 · HTML: https://mlsanity.com/paper/2008.08226/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2020.

## Built on by (most cited first, 4 of 4)

- [Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems](https://mlsanity.com/paper/2207.02542/) (2022)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Enhancing Human-Machine Teaming for Medical Prognosis Through Neural Ordinary Differential Equations (NODEs)](https://mlsanity.com/paper/2102.04121/) (2021)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)

## Abstract (arXiv)

Neural Ordinary Differential Equations replace the right-hand side of a conventional ODE with a neural net, which by virtue of the universal approximation theorem, can be trained to the representation of any function. When we do not know the function itself, but have state trajectories (time evolution) of the ODE system we can still train the neural net to learn the representation of the underlying but unknown ODE. However if the state of the system is incompletely known then the right-hand side of the ODE cannot be calculated. The derivatives to propagate the system are unavailable. We show that a specially augmented Neural ODE can learn the system when given incomplete state information. As a worked example we apply neural ODEs to the Lotka-Voltera problem of 3 species, rabbits, wolves, and bears. We show that even when the data for the bear time series is removed the remaining time series of the rabbits and wolves is sufficient to learn the dynamical system despite the missing the incomplete state information. This is surprising since a conventional ODE system cannot output the correct derivatives without the full state as the input. We implement augmented neural ODEs and differential equation solvers in the julia programming language.