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

arXiv: https://arxiv.org/abs/2505.08698 · HTML: https://mlsanity.com/paper/2505.08698/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 56 references that are in the map.
- First posted on arXiv 13 May 2025.

## Builds on (most cited first, 25 of 25)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting](https://mlsanity.com/paper/2405.14616/) (2024)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Maximum Mean Discrepancy Test is Aware of Adversarial Attacks](https://mlsanity.com/paper/2010.11415/) (2020)
- [Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence](https://mlsanity.com/paper/1912.05737/) (2019)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Functional Models for Time-Varying Random Objects](https://mlsanity.com/paper/1907.10829/) (2019)
- [Distributional data analysis of accelerometer data from the NHANES database using nonparametric survey regression models](https://mlsanity.com/paper/2104.01165/) (2021)
- [Estimation of copulas via Maximum Mean Discrepancy](https://mlsanity.com/paper/2010.00408/) (2020)
- [Multilevel functional distributional models with application to continuous glucose monitoring in diabetes clinical trials](https://mlsanity.com/paper/2403.10514/) (2024)
- [Distributional data analysis via quantile functions and its application to modelling digital biomarkers of gait in Alzheimer's Disease](https://mlsanity.com/paper/2102.10783/) (2021)
- [Multivariate Scalar on Multidimensional Distribution Regression](https://mlsanity.com/paper/2310.10494/) (2023)
- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Constructive approximate transport maps with normalizing flows](https://mlsanity.com/paper/2412.19366/) (2024)
- [Distributional outcome regression via quantile functions and its application to modelling continuously monitored heart rate and physical activity](https://mlsanity.com/paper/2301.11399/) (2023)
- [Interpretable Causal Inference for Analyzing Wearable, Sensor, and Distributional Data](https://mlsanity.com/paper/2312.10569/) (2023)
- [Beyond fixed thresholds: optimizing summaries of wearable device data via piecewise linearization of quantile functions](https://mlsanity.com/paper/2501.11777/) (2025)

## Abstract (arXiv)

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 such as 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 nonparametric 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. We illustrate the broad utility of our approach in a 26-week clinical trial that treats all continuous glucose monitoring (CGM) time series as the primary outcome. This method enables rigorous longitudinal comparisons between the treatment and control arms and yields characterizations that conventional summary-based clinical trials analytical methods typically do not capture.