Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes

2026-05-13

AI Summary

This work proposes MEGPODE, a novel nonparametric ordinary differential equation (ODE) model that integrates mixed-effects hierarchical structure to explicitly capture both population-level dynamics and individual heterogeneity. Unlike traditional nonlinear mixed-effects ODE models that rely on parametric vector fields and are prone to structural misspecification, MEGPODE decomposes each individual’s dynamics into a shared population component and an individual-specific deviation, both assigned Gaussian process priors. Efficient Bayesian inference is achieved by combining state-space trajectory priors with pseudo-observation collocation points. The method effectively balances commonality across the population with individual variation, significantly outperforming strong baselines on multiple heterogeneous ODE benchmarks and achieving state-of-the-art performance in both population vector field recovery and individual trajectory prediction.

Abstract

Dynamical modelling is central to many scientific domains, including pharmacometrics, systems biology, physiology, and epidemiology. In these settings, heterogeneity is often intrinsic: different subjects or units follow related but distinct continuous-time dynamics. Classical nonlinear mixed-effects Ordinary Differential Equation (ODE) models address this by combining population-level structure with subject-specific effects, but they rely on a parametric vector field and are therefore vulnerable to structural misspecification and unmodelled mechanisms. This motivates nonparametric approaches that can retain principled uncertainty quantification, yet existing nonparametric ODE methods typically assume a single shared dynamical system rather than an explicit mixed-effect hierarchy over subject-specific dynamics. We propose MEGPODE, a Bayesian nonparametric mixed-effect ODE model in which each subject's vector field is decomposed into a shared population component and a subject-specific deviation, both endowed with Gaussian process (GP) priors. To avoid repeated ODE solves per subject during training, we combine state-space GP trajectory priors with virtual collocation observations, yielding Kalman-smoothing trajectory updates and closed-form regressions for the vector fields. Across controlled heterogeneous ODE benchmarks spanning oscillatory, biomedical systems, MEGPODE improves population-field recovery and subject-level trajectory prediction relative to strong baselines.

Problem

• mixed-effect ODE
• heterogeneity
• structural misspecification
• nonparametric modeling
• subject-specific dynamics

Innovation

• Bayesian nonparametrics
• mixed-effect ODEs
• Gaussian processes
• virtual collocation
• heterogeneous dynamics