# Neural ODE enhanced linear mixed effect models for estimating complex association patterns of time-varying covariates with the marker trajectory

arXiv: https://arxiv.org/abs/2608.29714 · HTML: https://mlsanity.com/paper/2608.29714/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 32 references that are in the map.
- First posted on arXiv 30 Aug 2026.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Using simulation studies to evaluate statistical methods](https://mlsanity.com/paper/1712.03198/) (2017)
- [Integrating Random Effects in Deep Neural Networks](https://mlsanity.com/paper/2206.03314/) (2022)
- [A New PHO-rmula for Improved Performance of Semi-Structured Networks](https://mlsanity.com/paper/2306.00522/) (2023)
- [Estimation of extended mixed models using latent classes and latent processes: the R package lcmm](https://mlsanity.com/paper/1503.00890/) (2015)
- [Multivariate Functional Additive Mixed Models](https://mlsanity.com/paper/2103.06606/) (2021)
- [Time-varying exposure history and subsequent health outcomes: a two-stage approach to identify critical windows](https://mlsanity.com/paper/2008.11866/) (2020)

## Abstract (arXiv)

Longitudinal cohort studies produce repeated data that enable the assessment of time-varying association patterns between exposures and health outcomes. Classical linear mixed-effects models (LMMs) can accommodate a large variety of association patterns while accounting for the irregularly spaced, partially observed measurement. But they require the analyst to pre-specify the functional form linking the exposure history to the outcome. We propose the Neural ODE-LMM, which embeds a Neural Ordinary Differential Equation (Neural ODE) within the linear mixed-effects framework: a learned vector field encodes covariate trajectories into a continuous-time latent state that drives both the fixed- and random-effect design, while preserving the standard LMM observation model. This retains classical likelihood-based inference while learning complex, potentially cumulative, covariate effects flexibly. All parameters are estimated by maximising a penalised marginal likelihood. To quantify covariate effects, we introduce contrasts of counterfactual predictions that compare the expected outcome under alternative covariate trajectories with variance estimated via the delta method. In simulations, the model recovers both instantaneous and cumulative-burden effects without prior specification of the functional form. Applied to the Trois-Cit'es (3C) cohort, a population-based study of 7,324 participants, the method reveals trajectory-dependent associations of BMI and fasting glucose with cognitive decline.