MissPath-FM: Flow Matching with Structured Priors for Partially Observed Time Series

Genpei Zhang

cs.LGOct 13, 2026

MissPath-FM: Flow Matching with Structured Priors for Partially Observed Time Series

Authors: Genpei Zhang

Organizations: Department of Computer Sciences, University of Wisconsin–Madison

Abstract

Flow matching has emerged as a powerful alternative to diffusion models for generative modeling, but standard conditional flow matching is fundamentally misaligned with partially observed time series: starting from an uninformed Gaussian source forces the model to transport even already-observed entries from noise to the target, wasting capacity on trivially known regions. We propose MissPath-FM, a partial-observation-aware flow matching framework that redesigns the source prior and probability path for incomplete time series. MissPath-FM employs structured source priors (interpolation priors and Gaussian process posterior priors) to initialize the flow close to the observed signal, paired with a missingness-aware smooth path that allocates transport asymmetrically between observed and missing regions. We evaluate systematically across four datasets, four missingness mechanisms (MCAR, block, MNAR, irregular), two missing ratios, and three sequence lengths (T in {24, 48, 96}), totaling 48 settings against eight baselines. MissPath-FM wins 29 of 32 settings at T=48 and 45 of 48 overall (93.8%). Our ablations reveal a regime-dependent design hierarchy in which no single component suffices: under random missingness the source prior yields a large initial gain, but under structured block missingness it can fail entirely and the learned flow becomes indispensable; the missingness-aware path contributes a further 12%-26% across all regimes. Mechanistically, structured priors shorten observed-region transport by up to 15x, concentrating the velocity field on genuinely uncertain entries. Project page: https://zgp110.github.io/misspath-fm/

Figures & tables

Figure 1: Overview of MissPath-FM. (1) A structured source prior initializes $z_{0}$ close to the observed signal. (2) A missingness-aware path ( $s_{\mathrm{obs}}>s_{\mathrm{miss}}$ ) locks in observed entries early. (3) A context-conditioned Transformer predicts the velocity; at inference, $z_{0}$ is integrated with 20 Euler steps and observed entries are restored.

Data and context		Source prior (choose one)
$x\in\mathbb{R}^{T\times D}$	complete series	$\mu^{\mathrm{int}},\Sigma(m)$	Option A: interpolation (default)
$m\in\{0,1\}^{T\times D}$	mask, $1$ = observed	$\mu_{\star},\Sigma_{\star}$	Option B: GP posterior (optional)
$\delta$	time since last observation	$\sigma_{\mathrm{obs}},\sigma_{\mathrm{miss}}$	prior noise scales
$c=(x^{\mathrm{obs}},m,\delta)$	conditioning context
Flow matching		Path schedule
$z_{0}\sim p_{0}(\cdot\mid c)$	source endpoint ( $t{=}0$ )	$s_{\mathrm{obs}},s_{\mathrm{miss}}$	transport speeds

Table 1: Notation, grouped by role. The two source priors are alternative choices for the same object

p_{0}(\cdot\mid c)

, not sequential steps.

			Transport-based				Deterministic / Generative
Dataset	Mechanism	Rate	FlowTS	CLWF	TSFlow	PSW-I	SAITS	BRITS	CSDI	GP-VAE	Ours
ETTm1	MCAR	30%	0.393	0.190	0.121	0.144	0.096	0.103	0.159	0.454	0.075
	MCAR	50%	0.431	0.224	0.132	0.162	0.109	0.122	0.203	0.583	0.085
	Block	30%	0.396	0.197	0.245	0.265	0.152	0.160	0.248	0.511	0.106
	Block	50%	0.450	0.249	0.267	0.320	0.199	0.238	0.358	0.572	0.145
	MNAR	30%	0.439	0.366	0.186	0.243	0.143	0.186	0.219	0.746	0.117

Table 2: Imputation results (MAE

\downarrow

T{=}48

). Best in bold , second best underlined . MissPath-FM wins 29/32 settings. All methods share identical splits and masks. Baselines use official code where available; FlowTS, TSFlow, and CLWF include re-implemented components (Appendix C ).

Figure 3: Qualitative imputation on ETTm1. (a) MNAR 30%: MissPath-FM recovers masked extreme values where baselines deviate. (b) Block 30%: within the contiguous gap (shaded), TSFlow diverges and BRITS flattens, while MissPath-FM tracks the ground truth with calibrated uncertainty.

			Prior-Only		Incremental Flow
Dataset	Setting	(A) Gaussian +Flow	(B) Interpolation Prior	(C) GP Prior	(D) +Flow Matching	(E) +Context Encoding	(F) +Missingness- Aware Path	$\Delta_{\mathrm{flow}}$
ETTm1	MCAR 30%	0.7387	0.1091	0.1214	0.0918	0.0856	0.0751	$-31\%$
ETTm1	Block 30%	0.7248	0.2162	0.4150	0.1335	0.1253	0.1057	$-51\%$
Synthetic	MCAR 30%	0.8165	0.1614	0.1089	0.1208	0.1182	0.0981	$-10\%$
Synthetic	Block 30%	0.8056	0.8574	0.6326	0.5578	0.4938	0.3737	$-41\%$
Electricity	MCAR 30%	0.4028	0.1897	0.2230	0.1238	0.1137	0.0910	$-52\%$

Table 3: Incremental ablation (MAE

\downarrow

, 4 d.p.). Configurations A–F add one component at a time. Best per row in bold .

\Delta_{\mathrm{flow}}

: gain of (F) over best Prior-Only.

Dataset	History	SAITS	BRITS	CSDI	GP-VAE	FlowTS	CLWF	TSFlow	Ours
ETTm1	Clean	0.489	0.692	0.763	0.694	0.702	0.483	0.355	0.266
ETTm1	30% MCAR	0.528	0.692	0.593	0.704	0.679	0.480	0.365	0.264
Synthetic	Clean	0.791	0.781	1.084	0.787	0.850	0.825	0.566	0.511
Synthetic	30% MCAR	0.806	0.778	1.071	0.794	0.850	0.837	0.573	0.572
Electricity	Clean	1.040	0.873	0.861	0.937	0.935	0.947	0.226	0.149
Electricity	30% MCAR	0.871	0.821	1.056	0.907	0.918	0.953	0.237	0.153

Table 4: Forecasting results at

h{=}24

(MAE

\downarrow

). Best in bold , second underlined .

	Synthetic			ETTm1			Electricity
Method	Disc.	Pred.	Spec.	Disc.	Pred.	Spec.	Disc.	Pred.	Spec.
CSDI	0.500	0.925	1.540	0.474	0.745	1.313	0.500	0.631	1.294
GP-VAE	0.500	0.809	1.657	0.500	0.887	1.062	0.477	0.581	1.341
FlowTS	0.275	0.266	0.126	0.184	0.214	0.045	0.250	0.279	0.112
TSFlow	0.500	0.929	1.538	0.500	0.906	1.699	0.500	0.734	1.655
Gauss.+Std	0.300	0.240	0.155	0.198	0.286	0.256	0.182	0.201	0.113

Table 5: Generation quality (all metrics

\downarrow

). Best in bold , second underlined .

Figure 5: Velocity field magnitude $\|v_{\theta}\|$ at $t{=}0.5$ on ETTm1 MCAR 30%. (a) Gaussian prior: velocity is spread across all entries. (b) Interpolation prior: observed entries are silenced and velocity concentrates on missing entries. (c) GP posterior prior: missing-entry velocity is further reduced, since the posterior mean starts closer to the target.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Samples	$T$	$D$	Description
Synthetic	2,000	48	8	Multivariate sinusoidal (freq. 0.5–2.0, amp. 0.5–2.0, noise $\sigma{=}0.1$ )
ETTm1	1,451	48	7	Electricity transformer temperatures
Electricity	3,000	48	20	UCI Electricity; 20 highest-variance clients
PhysioNet	4,000	48	36	ICU clinical variables (set-a); 48h hourly bins

Appendix

Table 7: Dataset summary.

			Transport-based				Deterministic / Generative
$T$	Mechanism	Rate	FlowTS	CLWF	TSFlow	PSW-I	SAITS	BRITS	CSDI	GP-VAE	Ours
$T$ = 24	MCAR	30%	0.338	0.155	0.127	0.175	0.096	0.117	0.162	0.427	0.079
	MCAR	50%	0.361	0.181	0.140	0.195	0.106	0.141	0.186	0.582	0.090
	Block	30%	0.396	0.203	0.237	0.339	0.181	0.196	0.325	0.560	0.122
	Block	50%	0.400	0.207	0.254	0.364	0.235	0.264	0.365	0.609	0.145
	MNAR	30%	0.319	0.253	0.176	0.262	0.127	0.200	0.215	0.632	0.118

Appendix

Table 8: Sequence length generalization on ETTm1 (MAE

\downarrow

). MissPath-FM wins all 16 settings.

	CRPS $\downarrow$				ACE $\downarrow$
Setting	Ours	CSDI	TSFlow	GP-VAE	Ours	CSDI	TSFlow	GP-VAE
ETTm1 MCAR30	0.053	0.075	0.081	0.524	0.018	0.073	0.123	0.500
ETTm1 Block30	0.076	0.121	0.173	0.559	0.050	0.167	0.025	0.500
Synth MCAR30	0.069	0.079	0.089	0.798	0.046	0.064	0.249	0.500
Synth Block30	0.245	0.324	0.475	0.799	0.104	0.111	0.203	0.500
Electr MCAR30	0.065	0.106	0.110	0.349	0.052	0.091	0.288	0.500

Appendix

Table 9: Uncertainty calibration (all metrics

\downarrow

). MissPath-FM wins 29/32 metric–setting combinations, including a clean sweep on the multivariate Energy Score.

Dataset	Setting	A $\to$ B (Prior)	B $\to$ F (Flow)
ETTm1	MCAR 30%	$-85\%$	$-31\%$
ETTm1	Block 30%	$-70\%$	$-51\%$
Synthetic	MCAR 30%	$-87\%$	$-10\%$
Synthetic	Block 30%	$-21\%$	$-41\%$
Electricity	MCAR 30%	$-53\%$	$-52\%$
Electricity	Block 30%	$+79\%$	$-71\%$

Appendix

Table 10: Prior and flow contributions (%). A

\to

B: best prior-only vs Gaussian+Flow; B

\to

F: Full MissPath-FM vs best prior-only.

Axis	Value	MAE	Dataset / Setting
Solver	Euler-10	0.550	Synthetic Block 50%
	Euler-20	0.470
	Euler-50	0.476
	Euler-100	0.482
	Adaptive RK23	0.481
GP $\ell$	2.0	0.517	Synthetic Block 50%

Appendix

Table 11: Hyperparameter sensitivity. Each block varies one axis; defaults in bold .

Prior+Path	Medium gaps	Long gaps
Gaussian+Std	0.890	0.877
Interp+Std	0.456	0.684
GP+Std	0.297	0.618
GP+MA	0.296	0.607

Appendix

Table 12: Gap-bucketed MAE (Synth. Block 50%). GP priors benefit medium gaps most; long gaps remain challenging for all configurations.

Setting	Seed 42	Seed 123	Seed 456	Mean $\pm$ Std
ETTm1 MCAR 30%	0.075	0.078	0.077	$0.077\pm 0.001$
Synth. Block 50%	0.470	0.453	0.473	$0.465\pm 0.009$
Electr. MCAR 30%	0.091	0.092	0.089	$0.091\pm 0.001$

Appendix

Table 13: Multi-seed evaluation (MAE

\downarrow

, 3 seeds). Low variance confirms reproducibility.

Setting	Full	$-$ Prior	$-$ MA path	$-$ Context	$-$ RandomRate
ETTm1 MCAR30	0.0754	0.7437 (+886%)	0.0862 (+14%)	0.0895 (+19%)	0.0860 (+14%)
ETTm1 Block30	0.1072	0.6953 (+548%)	0.1292 (+21%)	0.1332 (+24%)	0.1293 (+21%)
Synth. MCAR30	0.0999	0.8056 (+706%)	0.1191 (+19%)	0.1088 (+9%)	0.1262 (+26%)
Synth. Block30	0.3586	0.8090 (+126%)	0.4869 (+36%)	0.4911 (+37%)	0.5659 (+58%)
Electr. MCAR30	0.0920	0.4177 (+354%)	0.1112 (+21%)	0.1101 (+20%)	0.1184 (+29%)
Electr. Block30	0.2241	0.4027 (+80%)	0.2454 (+10%)	0.2751 (+23%)	0.2181 ( $-$ 3%)

Appendix

Table 14: Leave-one-out ablation (MAE

\downarrow

; in parentheses, change relative to Full). † PhysioNet uses a fresh draw of the artificial holdout mask in this harness, so its absolute MAE differs by 4–6% from Tables 2 and 3 .

	Observed-entry MAE $\downarrow$		Mean $\\|v_{\theta}\\|$ on observed entries
Setting	Gaussian source	Ours	Gaussian source	Ours
ETTm1 MCAR30	0.7418	0.0229	0.9536	0.0394
ETTm1 Block30	0.7663	0.0236	0.9661	0.0398
Electr. MCAR30	0.3993	0.0247	1.0313	0.0371
Physio Art10	0.6153	0.0253	0.8068	0.0362

Appendix

Table 15: Observed entries before deterministic restoration. Gaussian source: Gaussian prior with standard path. Ours: interpolation prior with MA path.

Setting	SAITS	SAITS+RR	TSFlow	TSFlow+RR	Ours
ETTm1 MCAR30	0.0942	0.0950	0.1190	0.1173	0.0751
ETTm1 Block30	0.1432	0.1468	0.2401	0.3092	0.1067
Electr. MCAR30	0.1583	0.1599	0.1584	0.1465	0.0920
Electr. Block30	0.1836	0.1905	0.3412	0.5414	0.2241
Physio Art10	0.2336	0.2346	–	–	0.2271
Physio MNAR50	0.3138	0.3162	–	–	0.3283

Appendix

Table 16: Baselines retrained with RandomRate (+RR). MAE

\downarrow

; best per row in bold .

		MCAR			Block
Dataset	Rate	Ours	Gauss.	SAITS	Ours	Gauss.	SAITS
ETTm1	10%	0.069	0.752	0.085	0.084	0.671	0.120
	30%	0.076	0.751	0.098	0.106	0.709	0.149
	50%	0.086	0.742	0.110	0.144	0.712	0.210
	70%	0.103	0.738	0.132	0.219	0.714	0.252
	90%	0.189	0.740	0.212	0.389	0.728	0.302

Appendix

Table 17: Missing-rate sweep (MAE

\downarrow

). Gauss.: Gaussian prior with flow matching. Best per row and mechanism in bold .

Dataset	$\sigma_{\mathrm{noise}}$	$s_{\mathrm{obs}}{=}2.0$	$s_{\mathrm{obs}}{=}1.5$	$s_{\mathrm{obs}}{=}1.0$	SAITS
ETTm1	0.0	0.0754	0.0812	0.0861	0.1055
	0.1	0.0902	0.0961	0.0990	0.1109
	0.2	0.1256	0.1325	0.1363	0.1323
	0.3	0.1731	0.1794	0.1862	0.1712
	0.5	0.2888	0.2944	0.2963	0.2557
Electricity	0.0	0.0921	0.1033	0.1114	0.1557

Appendix

Table 18: Observation-noise sensitivity (MCAR 30%, MAE

\downarrow

). Best per row in bold .

Method	MCAR	Block	MNAR	Train (s)	Peak memory
SAITS	0.1222	0.1846	0.2032	95	565 MB
BRITS	0.1180	0.2294	0.2020	1541	242 MB
TSFlow	0.1149	0.2727	0.1967	696	306 MB
Ours	0.0775	0.1181	0.1235	466	1037 MB

Appendix

Table 19: ETTm1 at

T{=}192

, rate 30% (MAE

\downarrow

), with training time and peak GPU memory.

Length $T$	GP posterior prior	Interpolation prior	Ratio
48	601 ms	15.5 ms	$39\times$
192	1316 ms	56.5 ms	$23\times$

Appendix

Table 20: Source-prior construction time per batch (

B{=}64

D{=}7

Design choice	Outcome
OT-like path	No improvement over MA path
Gap-aware loss	$+1.9\%$ ; disrupts velocity field
Adaptive solver (RK23)	$5\times$ higher NFE, no improvement
Matérn kernels	Worse than RBF at matched $\ell$
EMA (exp. moving avg.)	Neutral or negative
Hybrid prior (Interp+GP)	Worse than either pure prior

Appendix

Table 21: Negative or neutral design choices.

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Flow Matching with Missing Data

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Open paper AskPDF arXiv

Flow matching assumes fully observed training data, which many real-world applications rarely provide. We study Missing-Data Flow Matching, which treats the missing coordinates of training samples as latent variables and averages the flow matching loss over the values they could take. We first prove that the oracle correction is exact: completing from the true pattern-conditioned posterior recovers the complete-data flow-matching objective under any missingness mechanism. Under missing completely at random (MCAR) and missing at random (MAR), this posterior equals the ordinary complete-data conditional, while under missing not at random (MNAR) it generally depends on the missingness mechanism and may require additional structure to identify. Our finite-sample analysis then answers design questions that the algorithm leaves open. Missingness transfers estimator variance rather than adding it, and for the fixed-parameter loss estimator one completion per example matches complete-data variance and is optimal under a fixed evaluation budget. A learned completion model contributes population completion bias, which we bound by its expected conditional Wasserstein distance to the true completion law. Experiments numerically validate the theoretical predictions, show that deterministic rather than frozen imputation reduces generated conditional variability, and place our method alongside strong classical and deep imputation baselines on real tabular data.

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