Smooth Flow Matching for Synthesizing Functional Data

Jianbin Tan; Anru R. Zhang

stat.MLAug 19, 2025

Smooth Flow Matching for Synthesizing Functional Data

Authors: Jianbin Tan, Anru R. Zhang

Organizations: Department of Biostatistics & Bioinformatics, Duke University, Durham, NC, USA · Department of Computer Science, Duke University, Durham, NC, USA

Abstract

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.

Figures & tables

Figure 1: A pictorial illustration of functional data generation via a three-dimensional flow. The left panel shows the base function input into the flow, while the right panel displays the output functions generated by applying the base function through the flow transformation.

Method	Deep Generative Models	Gaussian-based Methods	Smooth Flow Matching (this work)
Data Generator	Operator-form vector field (or score) estimated via deep learning ( Lim et al., 2023 ; Kerrigan et al., 2024 )	Gaussian process modelling via classical functional data approaches ( Yao et al., 2005 ; Yao, 2007 ; Hsing and Eubank, 2015 ) .	Three-dimensional vector field
Data Resolution	Dense	Dense or sparse	Dense or sparse
Computational Cost	High	Relatively low	Relatively low
Guaranteed Smoothness	No	Yes	Yes
Joint Distribution	Nonparametric	Parametric	Semiparametric

Table 1: Comparison of SFM with existing methods for functional data

Figure 2: The bias, variance, and mean squared error (MSE) of the estimated vector field

\hat{V}

, computed from

100

simulation replications. The estimator is obtained from ( 21 ) by replacing

\mathcal{L}_{n}(U)

with

\widehat{\mathcal{L}}_{n}(U)

, with

\lambda_{u}=\lambda_{t}=\lambda_{x}

taking different values. The simulation setting and the definitions of bias, variance, and MSE are given in Part C.1.1 of the Supplementary Materials.

\hat{V}:=\mathop{\rm arg\min}_{U\in\mathcal{B}_{L_{V},4}([0,1],\mathcal{S},\mathcal{X})}\big\{\hat{\mathcal{L}}_{n}(U)+\mathcal{J}(U)\big\}.

Algorithm 1 Smooth Flow Matching for Gaussian Copula Processes

Figure 3: Left panel: Box plots of Wasserstein distances from 100 simulation replications under varying sample sizes

n

(subtitles) and numbers of observed time points

J_{i}

(main titles). Right panel: average computation times.

Figure 4: Illustration of generated functional data under different numbers of observed time points (main titles) and different generation methods (subtitles). The sample size is set to

n=100

for data generation.

Figure 5: (A) Irregularly observed functional data for each clinical feature; (B) Synthetic functional data generated by SFM with Gaussian bases; (C) Synthetic functional data generated by SFM with Student-

t

bases; (D) Synthetic functional data generated by DSM; (E) Synthetic functional data generated by FM.

	SFM (Gaussian base)	SFM (Student- $t$ base)	DSM	FM
Arterial blood pressure	30.28	40.92	410.90	188.28
Respiratory rate	37.05	52.94	597.67	210.36
Heart rate	37.71	54.57	574.73	200.01

Table 2: Computational training time (in seconds) for different data generation methods. All experiments are conducted on a high performance machine with 40 CPU cores, 208 GB RAM, and a single NVIDIA RTX A5000 GPU.

Figure 6: Projected scores for the observed data, synthetic data, and Gaussian-based samples. The Wasserstein distances in the last four columns are computed between the corresponding samples and those from the observed data.

Figure 7: (A) Boxplots of privacy-loss values for different mixture proportions

a

(see legend) for datasets of three clinical features. For each setting, the privacy-loss values are computed over 200 replicated simulations. (B) Prediction errors of different methods across various prediction horizons

g

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

\displaystyle\hat{V}:=\operatorname{argmin}_{U\in{\mathcal{B}_{L_{V},4}([0,1],\mathcal{S},\mathcal{X})}}\big\{\hat{\mathcal{L}}_{n}(U)

\displaystyle+{\mathcal{J}(U)}\big\}.

Appendix

Algorithm S1 Smooth Flow Matching for Student-

t

Copula Processes

\displaystyle\hat{V}:=\operatorname{argmin}_{U\in{\mathcal{B}_{L_{V},4}([0,1],\mathcal{S},\mathcal{X})}}\big\{\hat{\mathcal{L}}_{n}(U)+{\mathcal{J}(U)}\big\}.

Appendix

Algorithm S2 Smooth Flow Matching for General Copula Processes

Figure 8: Median values of the bias and variance of the estimated flow map

\hat{\phi}_{1,t}(x)

over dense grids of

t\in[0,1]

and

x\in[-3,3]

(A), and median values of the averaged empirical roughness norms and Sobolev norms across 100 simulation replications (B). In each replication, one of

\lambda_{u}

\lambda_{t}

, or

\lambda_{x}

is varied over a range of values, while the other two are fixed at

10^{-7}

when obtaining

\hat{V}

. The averaged norm is computed as the mean of the norms of all generated functions within that simulation. The simulations follow the Gamma–process setting, with the sample size fixed at 100 and the numbers of observed time points set to

\{2,\ldots,6\}

Figure 9: Normalized mean squared error (NMSE) between correlation functions. We compare the raw estimated correlation functions without NPD (denoted by Est), the estimated correlation functions after applying NPD (denoted by Est (NPD)), and the true correlation functions (denoted by True). For calculating NMSE, the denominator for normalization is the

\mathcal{L}^{2}

norm of the latter correlation function; that is, “

A

vs.

B

” represents

\|A-B\|_{\mathcal{L}^{2}}^{2}/\|B\|_{\mathcal{L}^{2}}^{2}

, where

A

and

B

are kernels defined on

\mathcal{S}\times\mathcal{S}

with norm

\|\cdot\|_{\mathcal{L}^{2}}

. The simulation follows the Gamma–process setting, with the numbers of observed time points set to

\{2,\ldots,6\}

Figure 10: Empirical Sobolev norms of the estimated correlation functions across 100 simulation replications. We compare the raw estimator without NPD (denoted by Est) and the estimator after applying NPD (denoted by Est (NPD)). The simulation follows the Gamma–process setting, with the numbers of observed time points set to

\{2,\ldots,6\}

Figure 11: Averaged empirical roughness norms from 100 simulation replications. For each replication, the averaged norm is computed as the mean of the norms of all generated functions within that simulation. The simulation follows the Gamma–process setting, with the sample size fixed at 100 and the numbers of observed time points set to

\{2,\ldots,6\}

Figure 12: Boxplots of the conditional variance from 100 simulation replications under each setting. (A) Varying the number of discretization points in

u

(

F

) with

H

fixed at

1000

; (B) varying the number of generated base samples (

H

) with

F

fixed at

30

. The simulations follow the Gamma–process setting, with sample size fixed at

100

and the numbers of observed time points set to

\{2,\ldots,6\}

Figure 13: Mean squared error

\frac{1}{6}\int_{0}^{1}\!\!\int_{-3}^{3}\bigl|\hat{\phi}_{1,t}(x)-\phi_{1,t}(x)\bigr|^{2}\,\mathrm{d}x\,\mathrm{d}t

from 100 simulation replications, using a Gaussian base with covariance function

K_{v}(t,s)=\exp(-|t-s|/v)

. Here,

v

is varied over a range of values. The simulation follows the Gamma–process setting, with the numbers of observed time points set to

\{2,\ldots,6\}

MSE			$\{2,\ldots,6\}$				$\{6,\ldots,10\}$
MSE			MF	EF1	EF2	MDF	MF	EF1	EF2	MDF
(A)	$n=100$	DSM	0.45	1.07	1.10		0.20	1.12	1.18
		FM	0.73	1.02	0.99		0.43	1.06	1.00
		GP	0.12	0.25	0.39		0.11	0.19	0.31
		KL	0.13	0.27	0.41		0.10	0.19	0.30
		SFM	0.12	0.29	0.39		0.12	0.20	0.30

Appendix

Table 3: Mean squared errors (MSE) between the true mean function (MF), eigenfunctions (EF), median function (MDF), and their estimates from raw/generated data. "Oracle" refers to MF/EF estimated directly from observed data using the smoothing method of Hsing and Eubank (2015) .

Figure 14: Boxplots of Wasserstein distances from 100 simulation replications under varying sample sizes

n

and numbers of observed time points

J_{i}\in\{27,\ldots,31\}

Figure 15: Averaged empirical roughness norms

\|f^{(2)}\|_{\mathcal{L}^{2}}

for a function

f

from 100 simulation replications. For each replication, the averaged norm is computed as the mean of the norms of all generated functions within that simulation.

Figure 16: Wasserstein distances from 100 simulation replications under varying sample sizes

n

. The simulation follows the Gamma–process setting, with the numbers of observed time points set to

\{6,\ldots,10\}

Figure 17: Estimated mean function (MF) and the first two eigenfunctions (EF1 and EF2) for the observed and synthesized functional data.

Functional Data Analysis

See all 10 papers

Censored Flow Matching for Synthetic Time-to-Event Generation
Oct 16, 2026Jianbin Tan, Xingran Chen, Kevin He +3Synthetic Data Generation Survival Analysis
Department of Biostatistics & Bioinformatics, Duke University · Department of Biostatistics, University of Michigan · Department of Biostatistics, Yale University +1
ChronoFlow: Hierarchical Flow Matching for Irregular Time Series Generation
Sep 27, 2026Changhun Kim, Sunguk Jang, Jeongjun Lee +5Irregular Time-Series Modeling Flow Matching
University of Wisconsin–Madison · AITRICS · KAIST
Interpretable Synthetic Medical Tabular Data Generation for Clinical Decision Support Using Fuzzy Cognitive Maps
Sep 30, 2026Michael Vasilakakis, Dimitris K. IakovidisSynthetic Data Generation Synthetic Tabular Data Generation
Department of Computer Science and Biomedical Informatics, University of Thessaly, Lamia, Greece

Oct 16, 2026J/K move · Enter open · S save

Censored Flow Matching for Synthetic Time-to-Event Generation

Jianbin Tan, Xingran Chen, Kevin He, Zhenke Wu+2

Department of Biostatistics & Bioinformatics, Duke University · Department of Biostatistics, University of Michigan · Department of Biostatistics, Yale University+1

Synthetic Data Generation Survival Analysis Flow-Based Generative Modeling Synthetic Clinical Data Generation

Open paper AskPDF arXiv

Generating time-to-event data is increasingly important in survival analysis, but it remains statistically challenging because event times are often censored and thus prevent flexible generative modeling. To fill this gap, we propose a novel method called censored flow matching (CFM), aiming at providing a flexible flow-based generative mechanism for synthesizing time-to-event data as well as learning their corresponding probabilistic densities. Unlike existing methods, CFM does not rely on survival likelihoods to handle censoring, but instead exploits the monotonicity property of differential flows to address censoring via pullback sampling. Under this framework, we implement an alternating procedure between pullback sampling and CFM steps, leading to a flow-based expectation--maximization (EM) algorithm tailored for time-to-event inference. Under mild distributional conditions, we establish the convergence of CFM to the true event-time distribution as the EM iterations proceed. We then apply CFM to generate time-to-event data and estimate their probabilistic densities, where the observed censoring time can be incorporated to provide truncated sampling or truncated density estimation for survival inference. Through extensive simulation studies, we demonstrate the benefits of CFM for complex and large-scale data generation. Finally, we apply CFM to analyze data from the Organ Procurement and Transplantation Network (OPTN) and the Medical Information Mart for Intensive Care IV (MIMIC-IV), showcasing its benefits for synthetic and predictive survival analyses in terms of data fidelity, predictive performance, and empirical privacy characteristics.

Abstract

Functional Data Analysis

Explore similar work

Censored Flow Matching for Synthetic Time-to-Event Generation

ChronoFlow: Hierarchical Flow Matching for Irregular Time Series Generation

Interpretable Synthetic Medical Tabular Data Generation for Clinical Decision Support Using Fuzzy Cognitive Maps