# Smooth Flow Matching for Synthesizing Functional Data

arXiv: https://arxiv.org/abs/2508.13831 · HTML: https://mlsanity.com/paper/2508.13831/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 10 of its 67 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 10 of 10)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Convergence of Continuous Normalizing Flows for Learning Probability Distributions](https://mlsanity.com/paper/2404.00551/) (2024)
- [Score Matching With Missing Data](https://mlsanity.com/paper/2506.00557/) (2025)
- [Boosting Data Analytics With Synthetic Volume Expansion](https://mlsanity.com/paper/2310.17848/) (2023)
- [Denoising Diffused Embeddings: a Generative Approach for Hypergraphs](https://mlsanity.com/paper/2501.01541/) (2025)
- [Functional Principal Component Analysis for Distribution-Valued Processes](https://mlsanity.com/paper/2310.20088/) (2023)
- [Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems](https://mlsanity.com/paper/2507.20072/) (2025)

## Abstract (arXiv)

Functional data, i.e., random functions observed over a continuous domain, are increasingly available in areas such as biomedical research, health informatics, and epidemiology. However, effective statistical analysis for functional data is often hindered by challenges such as privacy constraints, sparse and irregular sampling, infinite-dimensionality, and non-Gaussian structures. To address these challenges, we introduce a novel framework named Smooth Flow Matching (SFM), tailored for generative modeling of functional data that enables statistical analysis without exposing sensitive real data. Under a copula framework, SFM constructs a semiparametric smooth flow to generate infinite-dimensional functional data, free of Gaussianity and low-rank assumptions. It is computationally efficient, handles irregular observations, and guarantees the smoothness of the generated functions, offering a practical and flexible solution in scenarios where existing deep generative methods are not applicable. Through extensive simulation studies, we demonstrate the advantages of SFM in terms of both synthetic data quality and computational efficiency. We then apply SFM to generate clinical trajectory data from the MIMIC-IV patient electronic health records (EHR) longitudinal database. Our analysis showcases the ability of SFM to produce high-quality surrogate data for downstream tasks, highlighting its potential to boost the utility of EHR data for clinical applications.