# Learning Governing Equations of Unobserved States in Dynamical Systems

arXiv: https://arxiv.org/abs/2404.18572 · HTML: https://mlsanity.com/paper/2404.18572/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 29 Apr 2024.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes](https://mlsanity.com/paper/2009.07444/) (2020)
- [Gaussian Process Assisted Active Learning of Physical Laws](https://mlsanity.com/paper/1910.03120/) (2019)
- [Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference](https://mlsanity.com/paper/2401.16943/) (2024)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)

## Built on by (most cited first, 7 of 7)

- [Position: Biology is the Challenge Physics-Informed ML Needs to Evolve](https://mlsanity.com/paper/2510.25368/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)
- [Learning partially observed systems with neural Hamiltonian ordinary differential equations](https://mlsanity.com/paper/2605.23510/) (2026)
- [RTS Smoother-Guided Learning of Physics-Based Neural Differential Models](https://mlsanity.com/paper/2607.15180/) (2026)
- [FluxDisco: Symbolic Regression for Stoichiometric Dynamical Systems via Monte Carlo Graph Search](https://mlsanity.com/paper/2609.05207/) (2026)
- [Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism](https://mlsanity.com/paper/2609.34566/) (2026)

## Abstract (arXiv)

Data-driven modelling and scientific machine learning have been responsible for significant advances in determining suitable models to describe data. Within dynamical systems, neural ordinary differential equations (ODEs), where the system equations are set to be governed by a neural network, have become a popular tool for this challenge in recent years. However, less emphasis has been placed on systems that are only partially-observed. In this work, we employ a hybrid neural ODE structure, where the system equations are governed by a combination of a neural network and domain-specific knowledge, together with symbolic regression (SR), to learn governing equations of partially-observed dynamical systems. We test this approach on two case studies: A 3-dimensional model of the Lotka-Volterra system and a 5-dimensional model of the Lorenz system. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems, with robustness to measurement noise.