Continuous Temporal Learning of Probability Distributions via Neural ODEs with Applications in Continuous Glucose Monitoring Data

Abstract: Modeling the dynamics of probability distributions from time-dependent data samples is a fundamental problem in many fields, including digital health. The goal is to analyze how the distribution of a biomarker, such as glucose, changes over time and how these changes may reflect the progression of chronic diseases like diabetes. We introduce a probabilistic model based on a Gaussian mixture that captures the evolution of a continuous-time stochastic process. Our approach combines a non-parametric estimate of the distribution, obtained with Maximum Mean Discrepancy (MMD), and a Neural Ordinary Differential Equation (Neural ODE) that governs the temporal evolution of the mixture weights. The model is highly interpretable, detects subtle distribution shifts, and remains computationally efficient. Simulation studies show that our method matches or surpasses the estimation accuracy of state-of-the-art, less interpretable techniques such as normalizing flows and non-parametric kernel density estimators. We further demonstrate its utility using data from a digital clinical trial, revealing how interventions affect the time-dependent distribution of glucose levels. The proposed method enables rigorous comparisons between control and treatment groups from both mathematical and clinical perspectives, offering novel longitudinal characterizations that existing approaches cannot achieve.