Validated Synthetic Patient Generation for Small Longitudinal Cohorts: Coagulation Dynamics Across Pregnancy

Jeffrey D. Varner; Maria Cristina Bravo; Carole McBride; Thomas Orfeo; Ira Bernstein

cs.LGApr 8, 2026

Validated Synthetic Patient Generation for Small Longitudinal Cohorts: Coagulation Dynamics Across Pregnancy

Authors: Jeffrey D. Varner, Maria Cristina Bravo, Carole McBride, Thomas Orfeo, Ira Bernstein

Organizations: Robert F. Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14850 · Department of Biochemistry, The Robert Larner, M.D. College of Medicine, University of Vermont Medical Center, Burlington, VT 05401 · Department of Obstetrics, Gynecology, and Reproductive Sciences, The Robert Larner, M.D. College of Medicine, University of Vermont Medical Center, Burlington, VT 05401

Abstract

Small longitudinal cohorts, common in maternal health, rare diseases, and early-phase trials, limit computational modeling because enrollment is slow and the data are too sparse to train reliable models. We present multiplicity-weighted Stochastic Attention (SA), a generative framework based on modern Hopfield networks. Stochastic attention stores real patient profiles as memory patterns in a continuous energy landscape. The resulting distribution is a finite mixture with one component centered on each stored profile. We generated the reported cohort with Langevin dynamics; direct sampling from the same distribution reproduced its fidelity and membership-inference results. Multiplicity weights amplify selected subgroups at inference time without retraining. We applied the method to longitudinal coagulation data from 23 patients, with 72 features measured before pregnancy and during the first and third trimesters. The cohort included three patients with polycystic ovary syndrome and five who developed preeclampsia. Across statistical, structural, and mechanistic tests, including a separately specified coagulation model that was blind to the generator but calibrated on the same cohort, the synthetic profiles closely matched the real profiles under the measures applied at this sample size. A model calibrated on synthetic profiles predicted held-out real-patient TGA measurements as well as one calibrated on real profiles. These results provide proof of concept for using SA in the tested modeling tasks, including mechanistic calibration and hypothesis generation, with very small longitudinal cohorts.

Figures & tables

Characteristic	Mean $\pm$ SD	Median (Range)	$n$ (%)
Age at enrollment (yrs)	30.2 $\pm$ 5.1	29 (20–41)
Race
White			22 (96)
Not disclosed			1 (4)
Prepregnancy BMI	26.6 $\pm$ 5.3	25 (19.0–37.8)
Parity		0 (0–2)

Table 1: Demographics and clinical characteristics of the 23 patients with complete longitudinal data across all three visits. Values are reported as mean

\pm

SD, median (range), or

n

(%) as appropriate.

Feature	Visit 1	Visit 2	Visit 3
Factor II (%)	0.018	0.017	0.007
Factor VIII (%)	0.018	0.008	0.005
AT (%)	0.025	0.001	0.008
Fibrinogen (mg/dL)	0.005	0.009	$<$ 0.001
TF Peak (nM)	0.025	0.002	0.017
TF ETP (nM $\cdot$ min)	0.013	0.008	0.023

Table 2: Per-visit mean relative error (MRE) between real and SA-generated synthetic patients for representative coagulation features, computed as

|\bar{x}_{\text{synth}}-\bar{x}_{\text{real}}|/|\bar{x}_{\text{real}}|

Figure 1: Physiological correlations. Scatter plots of coagulation factor levels versus thrombin generation parameters across all three visits. Filled markers are real patients; open markers are SA-generated synthetic patients. Colors encode visit: blue (V1/baseline), green (V2/first trimester), orange (V3/third trimester). Real and synthetic patients occupy the same joint regions across all four pairwise relationships. AT inversely correlates with TF-initiated peak thrombin (top left), FVIII positively correlates with peak (top right), and both FII (bottom left) and AT (bottom right) show the expected relationships with ETP. The visit-dependent shift in these relationships, reflecting the pregnancy-driven increase in procoagulant factors, is also preserved in the synthetic cohort.

Figure 2: Pregnancy-related longitudinal trajectories. Mean

\pm

standard deviation bands for six key coagulation features across visits (BL = baseline, 1st = first trimester, 3rd = third trimester) for real (blue) and SA-generated synthetic (orange) patients. The characteristic pregnancy-driven increases in fibrinogen, Factor VIII, and vWF are captured, as are the stable-to-declining patterns in Factor II and AT.

Figure 3: Cross-visit correlation structure. Top row: full $216\times 216$ Pearson correlation matrices for Real ( $K{=}23$ , left), SA ( $N{=}100$ , center), and MVN ( $N{=}100$ , right) populations. Each matrix is organized as a $3\times 3$ grid of $72\times 72$ blocks (delineated by black lines), where the diagonal blocks capture within-visit feature correlations (V1–V1, V2–V2, V3–V3) and the off-diagonal blocks capture cross-visit dependencies (e.g., V1–V3 correlations between baseline and third-trimester Factor VIII). Stochastic attention preserves both the within-visit and cross-visit block structure. Bottom row: pairwise residual matrices (element-wise subtraction), all plotted on the same $\pm 1$ color scale as the top row. The SA $-$ Real residual shows small, unstructured deviations. The MVN $-$ Real residual is smoother, particularly in the off-diagonal blocks, reflecting the Ledoit–Wolf regularization that systematically shrinks cross-visit correlations toward zero. The SA $-$ MVN residual shows the largest differences in these off-diagonal blocks, where SA retains longitudinal dependencies that MVN’s rank-deficient covariance estimate cannot represent. A version with amplified residual color scale ( $\pm 0.5$ ) and Frobenius norms is provided in Fig. S1 .

Figure 4: Population recovery and clinical-cohort subsampling. (A) For each low-rank Gaussian setting, the stochastic-attention memory matrix was constructed and a sample-covariance Gaussian generator was estimated from the same simulated training cohort; both generators produced $100$ profiles. Color represented the ratio of their relative Frobenius errors against the known population covariance (Gaussian error divided by stochastic-attention error), averaged over latent ranks $3$ , $5$ , and $10$ . Values above one (blue) indicated lower error for stochastic attention. Because the target covariance was fixed before the training cohort was drawn, this measured population recovery rather than reproduction of the training data. (B) A stochastic-attention memory matrix was constructed from each of ten random subsets of the clinical cohort at $n{=}20,15,10,$ and $8$ , and each matrix generated $100$ profiles. Points and bars represented the mean and standard deviation of the relative cross-visit correlation error against the same full $23$ -patient reference; the star represented generation using all $23$ patients.

Figure 5: Condition-specific feature preservation. Grouped bar charts comparing real (dark bars) and SA-generated synthetic (light bars) patient means (

\pm

SD error bars) for eight coagulation features, shown separately for each clinical subgroup: Uncomplicated (

n{=}18

, left, blue), PCOS (

n{=}3

, center, orange), and Developed PE (

n{=}5

, right, red). Each bar is the mean of that assay pooled across all three visits (V1–V3); factor activities (FVIII, vWF, FIX, FV,

\alpha_{2}

-AP, FVII, plasminogen) are expressed as % of normal pooled reference plasma, and fibrinogen (Fbgn) in mg/dL. Gray percentages indicate the mean relative error (MRE). Green annotations show a descriptive bootstrap Mann–Whitney diagnostic: the fraction of 1,000 bootstrap replicates (synthetic subsampled to

n_{\text{real}}

) in which

p>0.05

, meaning that this test detected no difference at the available sample size. This is not an equivalence test. Features achieving

\geq

90% are shown in green;

<

90% in red. Across all 24 feature–condition pairs, 20 (83%) had no difference detected in

\geq

90% of replicates (median: 98.6%). The four pairs below this threshold were FIX and plasminogen in the PCOS and Developed PE subgroups, where small subgroup sizes and large inter-patient variability reduced test power.

Figure 6: Mechanistic validation (TF-only). Top row: BZ2012 ODE-predicted TGA values (vertical axis) versus dataset TGA values (horizontal axis) for five thrombin generation parameters. The dataset values are the TGA measurements from each patient’s record, present for both real and synthetic patients. The ODE-predicted values are computed by running each patient’s coagulation factor levels through the 58-species BZ2012 model. The dashed line indicates where the ODE model would perfectly reproduce the dataset value (

y{=}x

); systematic deviations from this line (e.g., lagtime underprediction) reflect ODE model bias, not synthetic data failure. Filled markers are real patients, open markers are SA-generated synthetic patients, colored by visit: blue (V1/baseline, training set), green (V2/first trimester), orange (V3/third trimester). Real and synthetic patients occupy the same cloud with the same pattern of ODE model bias. Bottom row: ODE-predicted/dataset ratio distributions for real (blue) and synthetic (orange) patients. The distributions overlap substantially. Each panel is annotated with cloud overlap (OL), the two-sample Kolmogorov–Smirnov statistic (

D

), and

p

-value. All five features had

p>0.35

; these tests did not detect a difference between real and synthetic ratio distributions.

Figure 7: Downstream utility: synthetic and real calibrations on held-out real profiles. Each panel compares the BZ2012 ODE predictions for one TGA feature when the model is calibrated on real V1 patients (horizontal axis) versus synthetic V1 patients (vertical axis), evaluated on the same held-out real V2 and V3 patients. Points near the

y{=}x

line indicate similar predictions from the two calibrations. Green: V2 (first trimester); orange: V3 (third trimester). The synth-calibrated model had slightly lower median relative error on all five features (overall synthetic-to-real error ratio

0.94

). Both calibrations used 11 random restarts; all starts converged.

$\beta/\beta^{*}$	$\beta$	Noise scale	PC1 std ratio	Novelty	Med MRE (%)
0.1	0.29	0.261	0.538	0.542	0.7
0.3	0.88	0.151	0.550	0.536	0.7
0.5	1.47	0.117	0.562	0.528	0.7
1.0	2.94	0.082	0.589	0.502	0.6
1.5	4.41	0.067	0.613	0.474	0.7
2.0	5.88	0.058	0.638	0.446	0.7

Table S1 : Inverse-temperature sensitivity. Generation metrics across

\beta/\beta^{*}

, where

\beta^{*}\approx 2.94

was selected from the attention-entropy transition in the 18-dimensional PCA space. The bold row marks the reported value. Med MRE is the median relative error of feature means; novelty is

1-\max_{k}\cos(\hat{\xi},\hat{m}_{k})

Condition	$K_{\text{sub}}$	$K_{\text{bg}}$	$\rho$	$f_{\text{eff}}$	$K_{\text{eff}}$
Uncomplicated	18	5	1.11	0.80	23.0
PCOS	3	20	26.67	0.80	4.6
Developed PE	5	18	14.40	0.80	7.7

Table S2 : Multiplicity weighting parameters for conditional generation. For the unit-normalized memories used here, the multiplicity ratio

\rho

was computed to place a target fraction

f_{\text{target}}=0.80

of finite-mixture component probability on the designated patterns.

K_{\text{eff}}

is the participation ratio of those component probabilities and describes their concentration across repeated draws.

Parameter	Value	Description
$\alpha$	0.01	Langevin step size (see Table S8 )
$T$	2,000	Langevin iterations per sample
$\beta^{*}$	2.94	Entropy-transition point (unconditioned)
PCA threshold	95%	Variance retained
$d_{\text{PCA}}$	18	PCA dimensionality
$N$	100	Synthetic patients per cohort

Table S3 : SA generation hyperparameters. The same values were used for unconditioned and conditioned generation except that

\beta^{*}

was recomputed with the multiplicity bias for conditioned cohorts.

Rate constant	Role	Default	Calibrated	Scale
prothrombinase $k_{\text{cat}}$	Thrombin burst	63.5 s -1	21.94 s -1	0.35 $\times$
intrinsic Xase $k_{\text{cat}}$	Amplification Xa	8.2 s -1	0.169 s -1	0.021 $\times$
extrinsic Xase $k_{\text{cat}}$	Initiation Xa	6.0 s -1	93.2 s -1	15.5 $\times$
PC activation $k_{\text{cat}}$	Protein C activation	0.41 s -1	0.890 s -1	2.17 $\times$
mIIa conversion $k$	mIIa $\to$ IIa	$2.3{\times}10^{8}$	$1.15{\times}10^{9}$	5.01 $\times$
Units for mIIa conversion: M -1 s -1

Table S4 : BZ2012 ODE model calibration. Five rate constants were calibrated on Visit 1 real patients (

n{=}23

) using bounded Nelder–Mead optimization in log-scale-factor space. The cost function was the mean normalized relative squared error across five TGA features under TF-only conditions. Bounds:

\pm\log(100)

. Convergence:

f_{\text{tol}}=10^{-6}

, 500 iterations, 11 random restarts.

Pooled median MRE 95% bootstrap CI: (0.98%, 1.60%) over 10,000 resamples (seed 42).
Visit	Median	Mean	25th pct.	75th pct.	Maximum	MRE below 5%
V1 (baseline)	0.022	0.029	0.008	0.039	0.158	59/72 (81.9%)
V2 (1st tri)	0.009	0.021	0.004	0.022	0.224	65/72 (90.3%)
V3 (3rd tri)	0.011	0.016	0.005	0.024	0.062	69/72 (95.8%)
Pooled (all 3 visits)	0.012	0.022	0.005	0.028	0.224	193/216 (89.4%)

Table S5 : Summary of mean relative errors across all 72 features. Per-visit distribution of MREs between real and SA-generated synthetic patient means. The full feature-by-feature table (72 features

\times

3 visits = 216 entries) is provided in the code repository as paper_summary_statistics.csv .

Method	Epochs	Med MRE	Med KS	Corr MAE
SA	–	0.019	0.169	0.200
MVN	–	0.024	0.161	0.090
CTGAN	300	0.189	0.364	0.261
CTGAN	1,000	0.195	0.349	0.265
CTGAN	3,000	0.187	0.341	0.180
TVAE	300	0.037	0.174	0.152

Table S6 : Comparison with deep generative baselines. CTGAN and TVAE were trained on the same 90 per-visit records (23 patients

\times

3 visits, 72 features) at three epoch counts. Results for SA and MVN are shown for reference. The CTGAN model had median MRE near 19%, consistent with mode collapse at this sample size. The TVAE model matched marginal fidelity at high epoch counts but could not capture cross-visit covariance or perform conditional generation.

Threshold	$d_{\text{PCA}}$	$K/d$	$\beta^{*}$	Novelty	Med MRE	Corr MAE
85%	13	1.77	2.94	0.420	0.0098	0.124
90%	15	1.53	2.94	0.477	0.0160	0.126
95%	18	1.28	2.94	0.500	0.0124	0.143
97.5%	20	1.15	2.94	0.543	0.0120	0.135
99%	21	1.10	2.94	0.541	0.0172	0.138

Table S7 : PCA variance threshold sensitivity analysis. SA generation quality is stable across thresholds from 85% to 99%. Median MRE remains between 1.0% and 1.7% across all thresholds. The 95% threshold used in the main text (bold) is not a critical design choice.

Burn-in: 1,000 iterations discarded. $\tau_{\text{int}}$ : integrated autocorrelation time estimated with a windowed estimator (threshold 0.05). ESS = $(T-T_{\text{burn}})/\tau_{\text{int}}$ .
Metric	ULA	MALA
Acceptance rate (%)	– (no rejection)	$99.9\pm 0.03$
Mean energy (post-burn-in)	$1.932\pm 0.141$	$1.886\pm 0.231$
$\tau_{\text{int}}$	$109.5\pm 49.5$	$106.7\pm 30.4$
Effective sample size	$41.8\pm 14.2$	$40.0\pm 10.2$

Table S8 : ULA and MALA sampling diagnostics. Ten chains of 5,000 iterations each were run with both the Unadjusted Langevin Algorithm (ULA) and the Metropolis-Adjusted Langevin Algorithm (MALA) at step size

\alpha=0.01

and

\beta^{*}=2.94

on the 18-dimensional PCA memory (

K{=}23

patterns). The MALA acceptance rate was

99.9\%

, and the energy and effective sample-size estimates were similar between methods, indicating little discretization bias. This comparison bounded the discretization error of the reported sampler. The direct reference for the target itself was the analytic sampler of Eq. ( 3 ), introduced below.

Metric	Value
Mean novelty score, $1-\max_{k}\cos(\hat{\xi},\hat{m}_{k})$	0.50
Median synth–real nearest-neighbor distance (std. units)	6.14
Median real–real nearest-neighbor distance (std. units)	6.87
Distance ratio (synth–real / real–real)	0.89

Table S9 : Proximity to stored profiles. Two measures compared synthetic and stored profiles. Novelty is the angular distance between a synthetic profile and its nearest stored pattern at

\beta=\beta^{*}

. Nearest-neighbor distances are computed in the per-visit standardized 72-dimensional feature space.

Condition	Feature	MRE	Frac. $p>0.05$
Uncomplicated ( $n{=}18$ )	FVIII	0.057	0.952
Uncomplicated ( $n{=}18$ )	vWF	0.022	0.999
Uncomplicated ( $n{=}18$ )	FIX	0.018	0.998
Uncomplicated ( $n{=}18$ )	FV	0.006	0.996
Uncomplicated ( $n{=}18$ )	$\alpha_{2}$ -AP	0.019	1.000
Uncomplicated ( $n{=}18$ )	FVII	0.057	0.941

Table S10 : Descriptive bootstrap Mann–Whitney diagnostic per feature and condition. For each of 24 feature–condition pairs, we subsampled the synthetic cohort to match the real sample size, applied a Mann–Whitney U test, and repeated 1,000 times. Frac.

p>0.05

is the fraction of replicates in which the test detected no difference at the available sample size. This power-limited diagnostic is not an equivalence test. Twenty of 24 pairs had

p>0.05

\geq 90\%

of replicates; the four below-threshold pairs (italicized) involved high-variance features in the smallest subgroups.

Condition	TGA Feature	Cloud overlap	KS $D$	KS $p$
TF-only	Lagtime	0.92	0.112	0.48
TF-only	Peak	0.91	0.081	0.86
TF-only	T.Peak	0.92	0.102	0.61
TF-only	Max Rate	0.91	0.083	0.84
TF-only	ETP	0.83	0.123	0.36
TF+TM	Lagtime	0.92	0.127	0.32

Table S11 : Mechanistic validation diagnostics per feature and condition. Cloud overlap is the fraction of synthetic ODE-predicted/measured ratios falling within the 5th–95th percentile of the corresponding real distribution. KS

D

and KS

p

are from a two-sample Kolmogorov–Smirnov test on the same ratio distributions. Real

n{=}69

(23 patients

\times

3 visits); synthetic

n{=}300

(100 patients

\times

3 visits).

TGA Feature	Real-cal MRE	Synth-cal MRE	Synth/Real
Lagtime	0.165	0.162	0.98 $\times$
Peak	0.097	0.087	0.90 $\times$
T.Peak	0.330	0.306	0.93 $\times$
Max Rate	0.286	0.259	0.91 $\times$
ETP	0.197	0.183	0.93 $\times$
Overall median	0.201	0.189	0.94 $\times$

Table S12 : Downstream utility: per-feature held-out V2/V3 performance. Median relative error of the BZ2012 ODE-predicted TGA values against the held-out real V2 and V3 measurements, for the model calibrated on real V1 patients (

K{=}23

) and the model calibrated on synthetic V1 patients (

N{=}100

). The synth-calibrated model achieves slightly lower error on every feature. Synth/Real ratio

<1

favors the synth-calibrated model.

Method	MRE (%)	Cross-visit Frob.	Mech. overlap	Median novelty	Frac. novel
SA	1.24	0.641	0.902	0.514	1.00
PCA+GMM	1.13	0.422	0.918	0.497	0.94
PCA+KDE	1.94	0.337	0.823	0.249	0.65
PCA+kNN	1.63	0.587	0.938	0.051	0.05
PCA+Copula	1.34	0.417	0.934	0.462	1.00

Table S13 : PCA-space ablation. Five generators used the same

d{=}18

PCA subspace and evaluation. PCA+GMM is the fitted two-component diagonal-covariance baseline, not the exact stochastic-attention mixture. MRE measures marginal fidelity. Mechanistic overlap is the fraction of pooled synthetic predicted-to-recorded TGA ratios within the real 5th–95th percentile range. Median novelty is

1-\max_{k}\cos(\hat{\boldsymbol{\xi}},\hat{\mathbf{m}}_{k})

; Frac. novel is the fraction above

0.2

. Cross-visit Frobenius error measures deviation from the real correlation matrix. Lower error indicates closer in-sample reproduction and can reward memorization.

Hull geometry of the real cohort
Quantity	Synthetic ( $n{=}100$ )	Real, leave-one-out ( $n{=}23$ )
Fraction inside hull	$0.00$	$0.00$
Median distance to hull	$11.0$	$11.9$
Median radial overshoot $1/t^{*}$	$10.5$	$15.2$
Median patient radius	$13.8$
Median hull extent, generic direction	$2.0$

Table S14 : Convex-hull geometry and plausibility at

K{=}23

d{=}18

. We compared

100

synthetic profiles with leave-one-out real profiles using hulls of

22

real points. For each synthetic profile, we omitted its nearest real profile. The radial overshoot

1/t^{*}

used the point where a ray from the hull centroid exited the hull. All profiles lay outside their comparison hulls. Median hull distance was

11.0

for synthetic profiles and

11.9

for real profiles (

p{=}0.07

); radial overshoot was smaller for synthetic profiles (

p<10^{-4}

). These descriptive tests treated the synthetic profiles as independent, so their significance was optimistic. Hull membership had little resolution because every real profile formed an outer corner, and a unit-normalized generated direction could remain inside only by matching a stored direction. The biological check required nonnegative values and limited coagulation-factor and TGA values to five real-cohort standard deviations from the mean. Fifty-four synthetic profiles passed. The other

46

contained negative estradiol or progesterone values: estradiol was negative in

19

profiles and progesterone in

37

, with

10

overlapping; the minima were

-2109.8

and

-75.8

. No variable was constrained to be positive, and linear decoding had unbounded support. Hormones were closer to zero relative to their observed variation than the coagulation factors, which explained why only hormones crossed zero in these draws but did not guarantee positive factors in other draws. All nine factor inputs to BZ2012 were positive, so all

100

profiles entered the mechanistic check. The unconstrained MVN baseline produced negative hormones in

43

profiles, compared with

46

for stochastic attention.

Median nearest-neighbor distance
Quantity	Value
Memory-matrix member to nearest synthetic	$10.7$
Held-out non-member to nearest synthetic	$15.4$
Real to nearest real, leave-one-out	$15.9$
Synthetic to nearest real	$13.8$
Membership-inference AUC over $40$ random splits

Table S15 : Membership inference recovered memory membership, not unseen patients. Median nearest-neighbor distances in the standardized

216

-dimensional space, and the test’s discrimination. For each split, we constructed the memory matrix from

15

patients and scored each real record by its distance to the resulting synthetic set, treating the

15

included patients as members and the

8

held-out patients as non-members. Every quantity came from the same protocol. We drew

40

partitions at random with the generation seed held fixed, so that only the partition varied, and pooled the distances across all partitions. Non-members sat at the same distance from the synthetic cohort as unrelated real patients sat from one another. The synthetic cohort carried no proximity signal about patients the generator never saw, and the discrimination came entirely from members sitting closer. The area under the receiver operating characteristic curve (AUC), calculated using the equivalent Mann–Whitney rank statistic, measured how often the test assigned stronger membership evidence to a randomly chosen member than to a randomly chosen non-member;

0.5

indicated no discrimination and

1.0

indicated perfect discrimination.

Figure S1 : Cross-visit correlation structure (amplified residual scale). Same layout as the main-text correlation figure, but with the bottom-row residual panels plotted on a $\pm 0.5$ color scale (vs. $\pm 1.0$ in the main text) to reveal fine structure. Frobenius norms of each residual matrix are shown in the panel titles.

Figure S2 : Covariance eigenvalue spectrum. Log-scale eigenvalues of the

216\times 216

sample covariance for real (

K{=}23

, rank 22), SA (

N{=}100

, rank 18), and MVN (

N{=}100

) populations. Dashed vertical lines mark the SA PCA truncation point (component 18) and the data rank limit (component 22). Ledoit–Wolf regularization for MVN inflates eigenvalues beyond component 22 (green shaded region), introducing spurious variance in 194 dimensions where the data contain no signal.

Figure S3 : Conditioned generation: PCA projection (pooled). Large markers: real patients (blue = Uncomplicated,

n{=}18

; orange = PCOS,

n{=}3

; red = Developed PE,

n{=}5

). Small markers: synthetic patients (

N{=}100

per condition). With only 3 PCOS and 5 Developed PE patients, the subgroups do not form clearly separable clusters in the first two principal components, but the synthetic cohorts concentrate around their respective real counterparts.

Figure S4 : Conditioned generation: PCA projection by visit. Same as Fig. S3 but separated by visit (V1, V2, V3). Condition-specific clustering is maintained within each individual visit.

Figure S5 : Mechanistic validation (TF+TM condition). Same layout as main-text Fig. 6 but under TF+TM conditions. The model systematically underpredicts peak and maximum rate (

\approx 0.5\times

measured) for both real and synthetic patients, reflecting overcorrection of the protein C anticoagulant pathway. Cloud overlap was 88–92%, showing similar model behavior for real and synthetic profiles despite imperfect calibration.

Figure S6 : Cross-visit calibration generalization. Boxplots of the ODE-predicted/dataset ratio by visit for real patients. The model was calibrated on Visit 1 only. Top: under TF-only conditions, ratios remained near 1.0 at Visits 2 and 3. Bottom: under TF+TM conditions, the ratios differed more across visits, particularly for peak and maximum rate.

Figure S7 : Spearman rank correlations between ODE-predicted and dataset TGA features. Left: TF-only. Right: TF+TM. ETP is the best-predicted feature (

\rho\approx 0.6

–

0.8

for real patients under TF-only). Weak rank correlations for other features are expected given the 5-parameter population-level calibration.

Figure S8 : PCA projections by visit: SA and MVN. Each panel shows the first two principal components (PC1: 27.7% variance, PC2: 15.4%) computed from standardized per-visit real patient data (

K{=}23

patients per visit, 72 features). Dark markers indicate real patients; lighter markers indicate synthetic patients (

N{=}100

). Within each visit (column), the SA panel (top) and MVN panel (bottom) share identical axis ranges, so the relative dispersion of the two methods can be read against the same real data cloud. Top row: SA-generated patients (circles) cluster tightly around the real data cloud at all three visits. Bottom row: MVN-generated patients (diamonds) show greater dispersion, particularly at Visits 2 (first trimester) and 3 (third trimester), where MVN patients extend beyond the convex hull of the real data. This excess scatter reflects the spurious variance introduced by Ledoit–Wolf regularization of the rank-deficient covariance (Fig. S2 ). Colors encode visit: blue (V1, not pregnant), green (V2, first trimester), orange (V3, third trimester).

Rung	Sampler	MRE (%)	Cross-visit Frob.	Mech. overlap	Novelty	DCR	MIA AUC
V0	anchor, endpoint	1.31	0.65	0.89	0.51	13.97	0.97
V1	sphere init	1.53	0.57	0.89	0.50	13.96	n/a
V2	pooled ULA	2.02	0.59	0.88	0.50	14.15	n/a
V3	pooled MALA	1.89	0.62	0.92	0.52	14.33	0.96
Exact	analytic ancestral	1.68	0.60	0.91	0.50	13.87	0.98

Table S16: Sampler comparison. All procedures used the same

\beta^{*}

, memory, and decoder. We report median relative error (MRE), cross-visit correlation error, mechanistic overlap, novelty, DCR, and membership-inference AUC where evaluated. The real-to-real DCR baseline was

15.9

. The AUC used the same

40

splits for V0, V3, and the direct stochastic-attention sampler. Exact drew directly from Eq. ( 3 ) and was not a Markov-chain procedure. Across

100

direct-sampler cohorts, median MRE was

1.40\%

(5th–95th percentile

0.97

–

1.91\%

) and median DCR was

13.95

(

13.61

–

14.39

Figure S9 : Sampling correctness and the membership signal. (A) Across V0–V3, DCR rose slightly but stayed below the real-to-real baseline, while membership-inference AUC remained

0.96

–

0.97

. (B) Median relative error across the four procedures. (C) Across a

40

-fold inverse-temperature sweep around

\beta^{*}

, DCR increased as

\beta

fell but remained below the real-to-real baseline. Cross-visit covariance error increased, while mechanistic overlap changed little.

Figure S10 : Mixing diagnostic at $\beta^{*}$ . Energy traces, the between-chain $\hat{R}$ statistic, and the anchor-distance trajectory for sphere-initialized Langevin chains. The target distribution at $\beta^{*}$ was multimodal, with one component centered on each stored pattern. Metropolis acceptance near $0.999$ indicated negligible discretization bias.

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HyperNSDE: Personalized Neural SDEs for Joint Static-Longitudinal Clinical Data Generation

Perrine Chassat, Agathe Guilloux

Inria, Université Paris Cité, Inserm, HeKA F-75015 Paris, France

Stochastic Differential Equations Synthetic Data Generation Time Series Generation Longitudinal EHR Modeling Conditional Generative Modeling

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Synthetic patient data generation is a promising solution to the dual challenge of data scarcity and privacy constraints in healthcare machine learning. Realistic synthesis of patient-level clinical data requires jointly modeling heterogeneous static covariates, irregularly sampled longitudinal trajectories, and informative observation times - three tightly coupled components in practice yet rarely addressed together. We propose HyperNSDE, a continuous-time generative model that conditions a latent Neural SDE on static patient representations through a hypernetwork, allowing baseline characteristics to shape trajectory evolution beyond the initial condition without requiring a trajectory encoder, while stochastic latent dynamics capture realistic variability in generated paths. Observation times are modeled jointly through a latent-state-dependent intensity process, and training on irregular stochastic paths is stabilized via a deterministic-stochastic path decomposition with a non-adversarial signature-kernel objective. Experiments on simulated and real clinical datasets show improved observation-time fidelity and competitive performance, while matched-grid analyses reveal that forecasting and correlation metrics are affected by observation-grid regularity and trajectory smoothness.

Abstract

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HyperNSDE: Personalized Neural SDEs for Joint Static-Longitudinal Clinical Data Generation

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