# Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression

arXiv: https://arxiv.org/abs/2106.02875 · HTML: https://mlsanity.com/paper/2106.02875/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 104 references that are in the map.
- First posted on arXiv 5 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [RETAIN: An Interpretable Predictive Model for Healthcare using Reverse Time Attention Mechanism](https://mlsanity.com/paper/1608.05745/) (2016)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Learning Independent Causal Mechanisms](https://mlsanity.com/paper/1712.00961/) (2017)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods](https://mlsanity.com/paper/1909.10461/) (2019)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Generative ODE Modeling with Known Unknowns](https://mlsanity.com/paper/2003.10775/) (2020)
- [Learning from Clinical Judgments: Semi-Markov-Modulated Marked Hawkes Processes for Risk Prognosis](https://mlsanity.com/paper/1705.05267/) (2017)
- [Learning Insulin-Glucose Dynamics in the Wild](https://mlsanity.com/paper/2008.02852/) (2020)
- [Neural Pharmacodynamic State Space Modeling](https://mlsanity.com/paper/2102.11218/) (2021)
- [Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networks](https://mlsanity.com/paper/1912.00775/) (2019)

## Built on by (most cited first, 30 of 48)

- [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)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Robust Hybrid Learning With Expert Augmentation](https://mlsanity.com/paper/2202.03881/) (2022)
- [CF-GODE: Continuous-Time Causal Inference for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2306.11216/) (2023)
- [Automatically Learning Hybrid Digital Twins of Dynamical Systems](https://mlsanity.com/paper/2410.23691/) (2024)
- [Hybrid2 Neural ODE Causal Modeling and an Application to Glycemic Response](https://mlsanity.com/paper/2402.17233/) (2024)
- [Learning Differential Operators for Interpretable Time Series Modeling](https://mlsanity.com/paper/2209.01491/) (2022)
- [Deep Grey-Box Modeling With Adaptive Data-Driven Models Toward Trustworthy Estimation of Theory-Driven Models](https://mlsanity.com/paper/2210.13103/) (2022)
- [Learning Interacting Dynamical Systems with Latent Gaussian Process ODEs](https://mlsanity.com/paper/2205.11894/) (2022)
- [LaTiM: Longitudinal representation learning in continuous-time models to predict disease progression](https://mlsanity.com/paper/2404.07091/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Counterfactual Probabilistic Diffusion with Expert Models](https://mlsanity.com/paper/2508.13355/) (2025)
- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [Position: Biology is the Challenge Physics-Informed ML Needs to Evolve](https://mlsanity.com/paper/2510.25368/) (2025)
- [On the Generalization and Approximation Capacities of Neural Controlled Differential Equations](https://mlsanity.com/paper/2305.16791/) (2023)
- [Longitudinal Self-supervised Learning Using Neural Ordinary Differential Equation](https://mlsanity.com/paper/2310.10431/) (2023)
- [Uncertainty-Aware Optimal Treatment Selection for Clinical Time Series](https://mlsanity.com/paper/2410.08816/) (2024)
- [Benchmarking Continuous Time Models for Predicting Multiple Sclerosis Progression](https://mlsanity.com/paper/2302.07854/) (2023)
- [Combining Slow and Fast: Complementary Filtering for Dynamics Learning](https://mlsanity.com/paper/2302.13754/) (2023)
- [Hybrid additive modeling with partial dependence for supervised regression and dynamical systems forecasting](https://mlsanity.com/paper/2307.02229/) (2023)
- [Global Deep Forecasting with Patient-Specific Pharmacokinetics](https://mlsanity.com/paper/2309.13135/) (2023)
- [Towards Automated Knowledge Integration From Human-Interpretable Representations](https://mlsanity.com/paper/2402.16105/) (2024)
- [Variational Grey-Box Dynamics Matching](https://mlsanity.com/paper/2602.17477/) (2026)
- [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)
- [Probabilistic Temporal Prediction of Continuous Disease Trajectories and Treatment Effects Using Neural SDEs](https://mlsanity.com/paper/2406.12807/) (2024)
- [Learning and Current Prediction of PMSM Drive via Differential Neural Networks](https://mlsanity.com/paper/2412.09028/) (2024)
- [Exploring Neural Ordinary Differential Equations as Interpretable Healthcare classifiers](https://mlsanity.com/paper/2503.03129/) (2025)

## Abstract (arXiv)

Modeling a system's temporal behaviour in reaction to external stimuli is a fundamental problem in many areas. Pure Machine Learning (ML) approaches often fail in the small sample regime and cannot provide actionable insights beyond predictions. A promising modification has been to incorporate expert domain knowledge into ML models. The application we consider is predicting the progression of disease under medications, where a plethora of domain knowledge is available from pharmacology. Pharmacological models describe the dynamics of carefully-chosen medically meaningful variables in terms of systems of Ordinary Differential Equations (ODEs). However, these models only describe a limited collection of variables, and these variables are often not observable in clinical environments. To close this gap, we propose the latent hybridisation model (LHM) that integrates a system of expert-designed ODEs with machine-learned Neural ODEs to fully describe the dynamics of the system and to link the expert and latent variables to observable quantities. We evaluated LHM on synthetic data as well as real-world intensive care data of COVID-19 patients. LHM consistently outperforms previous works, especially when few training samples are available such as at the beginning of the pandemic.