Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes

Abstract

Dynamical modelling in pharmacometrics, systems biology, and epidemiology often involves subjects with related but distinct dynamics, observed through temporally sparse and irregular measurements. Classical nonlinear mixed-effects ODE models represent this heterogeneity but rely on parametric vector fields, making them vulnerable to structural misspecification. Existing nonparametric ODE methods typically assume shared dynamics rather than an explicit mixed-effect hierarchy. We propose MEGPODE, a Bayesian nonparametric mixed-effect ODE model that decomposes each subject's vector field into a shared population component and a subject-specific deviation, both endowed with Gaussian process priors. To avoid repeated ODE solves during training, we represent latent trajectories with state-space GP priors and penalize discrepancies between their derivatives and the vector fields on a finite time grid. A single variational objective enables approximate joint inference over trajectories and vector fields, accounting for state uncertainty and dependence between population and subject-specific dynamics. Across heterogeneous oscillatory and biomedical ODE benchmarks, MEGPODE improves population-field recovery and subject-level trajectory prediction relative to strong baselines.