# Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Data

arXiv: https://arxiv.org/abs/2202.09463 · HTML: https://mlsanity.com/paper/2202.09463/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 18 Feb 2022.

## Builds on (most cited first, 11 of 11)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Efficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee](https://mlsanity.com/paper/2011.07439/) (2020)
- [An Adaptive Empirical Bayesian Method for Sparse Deep Learning](https://mlsanity.com/paper/1910.10791/) (2019)
- [Bayesian inference for diffusion driven mixed-effects models](https://mlsanity.com/paper/1507.06807/) (2015)
- [Inferring Multidimensional Rates of Aging from Cross-Sectional Data](https://mlsanity.com/paper/1807.04709/) (2018)
- [Conditional Recurrent Flow: Conditional Generation of Longitudinal Samples with Applications to Neuroimaging](https://mlsanity.com/paper/1811.09897/) (2018)

## Built on by (most cited first, 3 of 3)

- [Deep Geometric Learning with Monotonicity Constraints for Alzheimer's Disease Progression](https://mlsanity.com/paper/2310.03353/) (2023)
- [Amortized In-Context Mixed Effect Transformer Models: A Zero-Shot Approach for Pharmacokinetics](https://mlsanity.com/paper/2508.15659/) (2025)
- [Using Intermediate Forward Iterates for Intermediate Generator Optimization](https://mlsanity.com/paper/2302.02336/) (2023)

## Abstract (arXiv)

Panel data involving longitudinal measurements of the same set of participants taken over multiple time points is common in studies to understand childhood development and disease modeling. Deep hybrid models that marry the predictive power of neural networks with physical simulators such as differential equations, are starting to drive advances in such applications. The task of modeling not just the observations but the hidden dynamics that are captured by the measurements poses interesting statistical/computational questions. We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing such panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms using MC based sampling methods and numerical ODE solvers. We demonstrate ME-NODE's utility on tasks spanning the spectrum from simulations and toy data to real longitudinal 3D imaging data from an Alzheimer's disease (AD) study, and study its performance in terms of accuracy of reconstruction for interpolation, uncertainty estimates and personalized prediction.