# Generative ODE Modeling with Known Unknowns

arXiv: https://arxiv.org/abs/2003.10775 · HTML: https://mlsanity.com/paper/2003.10775/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Mar 2020.

## Built on by (most cited first, 24 of 24)

- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Predicting the impact of treatments over time with uncertainty aware neural differential equations](https://mlsanity.com/paper/2202.11987/) (2022)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [Physics-informed Variational Autoencoders for Improved Robustness to Environmental Factors of Variation](https://mlsanity.com/paper/2210.10418/) (2022)
- [Uncertainty-Aware Optimal Treatment Selection for Clinical Time Series](https://mlsanity.com/paper/2410.08816/) (2024)
- [Multi-Parameter Molecular MRI Quantification using Physics-Informed Self-Supervised Learning](https://mlsanity.com/paper/2411.06447/) (2024)
- [Physically Interpretable World Models via Weakly Supervised Representation Learning](https://mlsanity.com/paper/2412.12870/) (2024)
- [A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments](https://mlsanity.com/paper/2507.10792/) (2025)
- [CONFIDE: Contextual Finite Differences Modelling of PDEs](https://mlsanity.com/paper/2303.15827/) (2023)
- [Effective Latent Differential Equation Models via Attention and Multiple Shooting](https://mlsanity.com/paper/2307.05735/) (2023)
- [Adversarial Disentanglement by Backpropagation with Physics-Informed Variational Autoencoder](https://mlsanity.com/paper/2506.13658/) (2025)
- [ECG synthesis with Neural ODE and GAN models](https://mlsanity.com/paper/2111.00314/) (2021)
- [Complementing Brightness Constancy with Deep Networks for Optical Flow Prediction](https://mlsanity.com/paper/2207.03790/) (2022)
- [Learning Hybrid Dynamics Models With Simulator-Informed Latent States](https://mlsanity.com/paper/2309.02873/) (2023)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations](https://mlsanity.com/paper/2202.12932/) (2022)
- [Structured Hybrid Mechanistic Models for Robust Estimation of Time-Dependent Intervention Outcomes](https://mlsanity.com/paper/2602.11350/) (2026)
- [Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics](https://mlsanity.com/paper/2607.28939/) (2026)
- [Identifying ODEs from Unstructured Data with Causal Representation Learning](https://mlsanity.com/paper/2609.37083/) (2026)

## Abstract (arXiv)

In several crucial applications, domain knowledge is encoded by a system of ordinary differential equations (ODE), often stemming from underlying physical and biological processes. A motivating example is intensive care unit patients: the dynamics of vital physiological functions, such as the cardiovascular system with its associated variables (heart rate, cardiac contractility and output and vascular resistance) can be approximately described by a known system of ODEs. Typically, some of the ODE variables are directly observed (heart rate and blood pressure for example) while some are unobserved (cardiac contractility, output and vascular resistance), and in addition many other variables are observed but not modeled by the ODE, for example body temperature. Importantly, the unobserved ODE variables are known-unknowns: We know they exist and their functional dynamics, but cannot measure them directly, nor do we know the function tying them to all observed measurements. As is often the case in medicine, and specifically the cardiovascular system, estimating these known-unknowns is highly valuable and they serve as targets for therapeutic manipulations. Under this scenario we wish to learn the parameters of the ODE generating each observed time-series, and extrapolate the future of the ODE variables and the observations. We address this task with a variational autoencoder incorporating the known ODE function, called GOKU-net for Generative ODE modeling with Known Unknowns. We first validate our method on videos of single and double pendulums with unknown length or mass; we then apply it to a model of the cardiovascular system. We show that modeling the known-unknowns allows us to successfully discover clinically meaningful unobserved system parameters, leads to much better extrapolation, and enables learning using much smaller training sets.