LDDMM stochastic interpolants: an application to domain uncertainty quantification in hemodynamics

Sarah Katz; Francesco Romor; Jia-Jie Zhu; Alfonso Caiazzo

stat.MLMar 30, 2026

LDDMM stochastic interpolants: an application to domain uncertainty quantification in hemodynamics

Authors: Sarah Katz, Francesco Romor, Jia-Jie Zhu, Alfonso Caiazzo

Organizations: WIAS, Weierstrass Institute for Applied Analysis and Stochastics, Anton-Wilhelm-Amo-Str. 39, 10117 Berlin, Germany. · KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.

Abstract

We introduce a novel conditional stochastic interpolant framework for generative modeling of three-dimensional shapes. The method builds on a recent LDDMM-based registration approach to learn the conditional drift between geometries. By leveraging the resulting pull-back and push-forward operators, we extend this formulation beyond standard Cartesian grids to complex shapes and random variables defined on distinct domains. We present an application in the context of cardiovascular simulations, where aortic shapes are generated from an initial cohort of patients. The conditioning variable is a latent geometric representation defined by a set of centerline points and the radii of the corresponding inscribed spheres. This methodology facilitates both data augmentation for three-dimensional biomedical shapes, and the generation of random perturbations of controlled magnitude for a given shape. These capabilities are essential for quantifying the impact of domain uncertainties arising from medical image segmentation on the estimation of relevant biomarkers.

Figures & tables

Figure 1. Overview of the proposed method and application in forward domain uncertainty quantification for aortic blood flows.

Figure 2. Left: On a two-dimensional grid, the prescribed velocity field $u^{\text{cond}}_{t}$ ( 3 ) can be defined by computing the difference between the realizations of the random variables $X_{0}$ and $X_{1}$ . Right: In the case of a three-dimensional shape (described, e.g., as a surface mesh), a definition of $u^{\text{cond}}_{t}$ based on the difference ( 3 ) is no longer possible, since there is no direct correspondence between points.

Figure 3. Example of application of the multilevel ResNet-LDDMM [ 44 ] for the registration of computational meshes of aortic shapes. In this example, meshes are refined after 3,000, 4,000, and 5,000 epochs.

Figure 4. Graphical overview of the proposed conditional LDDMM stochastic interpolant method. The map defining the conditional drift $b^{\theta}$ as a function of the conditioning variable $C$ is learned registering the available shapes onto a common reference ( $\mathbf{x}_{0}$ ), in order to find a stochastic interpolant between the two distributions.

Figure 5. The conditional drift

u_{t}^{\text{cond}}(\mathbf{x}_{0},\mathbf{x}_{i})

is defined based on the LDDMM registration. When

\mathbf{x}_{0}

and

\mathbf{x}_{i}

represent domain coordinates (left), the conditional drift is computed from the registration field

{\phi}^{0,i}

between the two realizations. When

\mathbf{x}_{0}

and

\mathbf{x}_{i}

represent physical fields/solutions of PDEs (right), the conditional drift is computed, at each intermediate step

t\in[0,1]

, considering the push-forward of

{\phi}^{0,i}

(registration of

\mathbf{x}_{0}

onto

\mathbf{x}_{i}

) and the pull-back of

{\psi}^{0,i}

(registration of

\mathbf{x}_{i}

onto

\mathbf{x}_{0}

Figure 6. Dataset of aortic geometries. Left: template geometry, and example of centerline encoding with inscribed radii. Right: t-SNE clustering of the dataset based on the similarity matrix of the LDDMM registrations from the template geometry.

Figure 7. Distributions of selected quantities of interest of the registered dataset over the $1239$ training and test geometries.

Figure 8. Left: sketch of the computational domain for the computational blood flow model ( 18 )-( 19 ). Right: Time–dependent inflow profile.

Figure 9. Transport of nested hexahedral mesh hierarchies needed for geometric multigrid preconditioners.

Figure 10. Left: Example of shape generated from a convex interpolation on the latent space of four test geometries. Right: Limitations of the method with respect to local parametric deformations. We impose a variation of the inscribed radii by a factor of $\alpha_{r}\in\{0.2,0.7,1.3,1.8\}$ , limited to the centerline points close to a selected location (red circles). This results in perturbed shapes with global radius variations.

Figure 11. Shape generation with conditional LDDMM stochastic interpolant. The three different rows correspond to a different factor applied to the reference inscribed radii: $0.7$ , $1.0$ , and $1.3$ ; in each row, we show the four generated samples, varying only the Gaussian random field employed to perturb the control points coordinates, reported on the left. The displacement magnitude of the associated registration maps from the template to the generated test shape is also shown.

Figure 12. The 30 test geometries used to generate the different batches.

Panel of Figure 13 — Figure 13. Examples from batches 0, 10, 20 and 29 (from left to right). Top: perturbations with radius reduced by 30% (factor $\alpha_{r}=0.7$ ). Middle: perturbations with similar radius (factor $\alpha_{r}=1$ ). Bottom: perturbations with radius increased by 30% (factor $\alpha_{r}=1.3$ ).

	Incomplete batches
$\alpha_{r}$	0	2	7	16	17	18	21	23	24	25	total
$0.7$	6	4	0	1	10	8	5	10	8	10	262
$1.0$	10	10	10	10	9	10	10	5	9	9	292
$1.3$	10	10	9	10	10	10	10	10	6	10	295

Table 1. Valid geometries for each batch (numbered from

0

29

). Each cell contains the number of full simulations available for each batch and for the corresponding value of

\alpha_{r}

. For the batches not listed here, all 10 perturbations, for all three values of

\alpha

, were available. The last row provides the total number of cases for each

\alpha_{r}

Panel of Figure 14 — Figure 14. Meshes generated from average coordinates for the batches 0, 10, 20, and 29. The surface color displays the sample standard deviation.

Figure 15. Shape variability within each batch. Top: For each batch index (0 to 29), the bars show mean and sample standard deviation of all pairwise Chamfer distances of the shapes within the batch. Bottom: For each batch index (0 to 29), the bars show mean and sample standard deviation of vertex coordinate standard deviations.

Figure 16. Velocity mean and standard deviation over $10$ sampled perturbations of one selected test geometry (the base geometry for batch 1) with $\alpha_{r}=1.0$ , and for $t\in\left\{0.2\text{s},0.4\text{s},0.6\text{s},0.8\text{s}\right\}$ .

Figure 17. Pressure mean and standard deviation over $10$ sampled perturbations of one selected test geometry (the base geometry for batch 1), with $\alpha_{r}=1.0$ , and for $t\in\left\{0.2\text{s},0.4\text{s},0.6\text{s},0.8\text{s}\right\}$

Figure 18. Left: selected geometry (from batch 20) showing the cross–sections used for computing averages of quantities of interests. The outer rings are in-plane circles matching an estimate of the outer radius of each cross–section. The inner rings show the estimated hydraulic radius $r_{H}$ .

Panel of Figure 19 — Figure 19. Top row: Pressure difference between descending aorta and inlet versus mean cross-sectional normal velocity magnitude (averaged over the length of the domain) over one cardiac cycle (time goes counterclockwise). Bottom row: Pressure difference between descending aorta and inlet versus outflow at the descending aorta versus pressure difference between descending aorta and inlet over one cardiac cycle. From left to right: batches 0, 10, 20, and 29. Black: $\alpha_{r}=1.3$ (30% radius increase), blue: $\alpha_{r}=1$ , red: $\alpha_{r}=0.7$ (30% radius decrease).

Panel of Figure 20 — Figure 20. Top row: Pressure difference between descending aorta and inlet versus outflow at the descending aorta for all simulations. Bottom row: Pressure difference between descending aorta and inlet versus mean cross–sectional velocity magnitude (averaged over the length of the domain). From left to right: times $t\in\{2,2.16,2.3,2.7\}$ , corresponding to the end of diastole, peak flow, end of systole, and middle diastole. Black: $\alpha_{r}=1.3$ (30% radius increase), blue: $\alpha_{r}=1$ , red: $\alpha_{r}=0.7$ (30% radius decrease).

Panel of Figure 21 — Figure 21. Mean wall shear stress (top) and mean velocity magnitude (bottom) versus mean pressure (left) and mean pressure drop (right), for all simulations, averaged over time. Note the clear batch separation in the absolute mean pressure compared to the pressure drop. Black: $\alpha_{r}=1.3$ (30% radius increase), blue: $\alpha_{r}=1$ , red: $\alpha_{r}=0.7$ (30% radius decrease).

Panel of Figure 22 — Figure 22. Top row: Cross-sectional tangential velocity versus normal velocity over one cardiac cycle. The time runs counterclockwise around these loops. During systole the flow accelerates and the normal flow dominates, while, during diastole, the presence of dissipating eddies might result in larger tangential velocity. Middle row: SFD (averaged over the length of the domain) over one cardiac cycle. Bottom row: SFD (maximum over the length of the domain) over one cardiac cycle. From left to right: batches 0, 10, 20, and 29. Black: $\alpha_{r}=1.3$ (30% radius increase), blue: $\alpha_{r}=1$ , red: $\alpha_{r}=0.7$ (30% radius decrease).

Figure 23. Sample means and standard deviations for different quantities over the whole dataset of geometries, divided according to the corresponding batch (from 0 to 29, in the

x

-axis). First row: maximum velocity magnitude at peak systole (

t=2.16

s); second row: largest pressure difference between inlet and descending aorta over the cardiac cycle; third row: mean wall shear stress magnitude over the whole blood vessel wall and over the cardiac cycle; Fourth row: mean oscillatory shear index over a reference patch on the underside of the descending aorta. Black:

\alpha_{r}=1.3

(30% radius increase), blue:

\alpha_{r}=1

, red:

\alpha_{r}=0.7

(30% radius decrease).

Panel of Figure 24 — Figure 24. Top row: Distribution (mean and standard deviation) of mean pressure in the aorta (left) and pressure difference between descending aorta and inlet (right) over time. Black: larger batch of $100$ simulations; red: original smaller batch. Bottom row: Joint mean of the separate distributions and Wasserstein–1 distance between them.

Panel of Figure 25 — Figure 25. Top row: Distribution (mean and standard deviation) of mean cross–sectional velocity magnitude (left) and overall maximum velocity magnitude (right) over time. Black: larger batch of $100$ simulations; red: original smaller batch. Bottom row: Joint mean of the separate distributions and Wasserstein–1 distance between them.

Panel of Figure 26 — Figure 26. Left: Mean pressure in the aorta versus mean cross–sectional velocity magnitude, averaged over the length of the domain (top) and outflow through the descending aorta (bottom), over one cardiac cycle. Right: Pressure difference between descending aorta and inlet versus mean cross–sectional velocity magnitude, averaged over the length of the domain (top) and outflow through the descending aorta (bottom), over one cardiac cycle. Black: larger batch of $100$ simulations; red: original smaller batch.

Panel of Figure 27 — Figure 27. Pressure difference between descending aorta and inlet versus mean cross–sectional velocity magnitude (averaged over the length of the domain) for all simulations at times $t\in\{2,2.16,2.3,2.7\}$ , corresponding to the end of diastole (top-left), peak flow (top-right), end of systole (bottom-left), and middle diastole (bottom-right). Black: larger batch of $100$ simulations; red: original smaller batch. Crosses and ellipses indicate sample means and standard deviations.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 28.

L^{2}

-relative errors on the drift during training (blue curve) and fine-tuning (red curve) phases, evaluated only on the test set of

30

shapes.

Figure 29. Left: Drift $L^{2}$ -relative error on the test set during training and fine-tuning for different sampled time instances. Right: Chamfer distance between the target shapes and the generated shapes with the learned drift on the test set during training and fine-tuning.

$\alpha_{r}=0.7$	$\overline{d}_{c}$	$\overline{\sigma}$	$\alpha_{r}=1$	$\overline{d}_{c}$	$\overline{\sigma}$	$\alpha_{r}=1.3$	$\overline{d}_{c}$	$\overline{\sigma}$
0	1.419	1.79	0	1.923	2.902	0	2.043	2.875
1	1.561	2.389	1	1.871	2.901	1	1.712	2.207
2	1.223	1.694	2	1.359	1.938	2	1.735	2.791
3	1.302	1.848	3	1.483	1.949	3	1.602	1.966
4	1.42	2.031	4	1.432	1.943	4	1.631	2.674
5	1.444	2.019	5	1.904	3.128	5	1.571	2.12

Appendix

Table 2. Mesh variation statistics. Here

\overline{d}_{c}

is the mean pairwise chamfer distance over each batch, whereas

\overline{\sigma}

is the root mean square sample standard deviation of the vertex coordinates. Both are in units of millimeters.

$\alpha_{r}=0.7$	$\mu$	$\sigma$	$\alpha_{r}=1.0$	$\mu$	$\sigma$	$\alpha_{r}=1.3$	$\mu$	$\sigma$
0	$4.108$	$0.696$	0	$1.750$	$0.177$	0	$1.368$	$0.109$
1	$7.032$	$0.828$	1	$4.886$	$0.800$	1	$3.059$	$0.250$
2	$5.395$	$0.418$	2	$3.850$	$0.393$	2	$2.815$	$0.313$
3	$4.098$	$0.142$	3	$2.398$	$0.076$	3	$1.778$	$0.072$
4	$5.198$	$0.490$	4	$3.903$	$0.568$	4	$2.590$	$0.423$
5	$3.774$	$0.260$	5	$2.402$	$0.465$	5	$1.992$	$0.346$

Appendix

Table 3. Maximum velocity magnitude (in m/s) per batch at

t=2.16

(peak systole), sample means and standard deviations.

$\alpha_{r}=0.7$	$\mu$	$\sigma$	$\alpha_{r}=1.0$	$\mu$	$\sigma$	$\alpha_{r}=1.3$	$\mu$	$\sigma$
0	$28.644$	$1.120$	0	$11.824$	$0.427$	0	$8.618$	$0.377$
1	$88.027$	$12.313$	1	$43.453$	$6.466$	1	$21.899$	$0.787$
2	$63.888$	$5.698$	2	$36.947$	$4.547$	2	$20.059$	$1.510$
3	$42.669$	$1.190$	3	$20.990$	$0.762$	3	$13.680$	$0.311$
4	$40.048$	$6.825$	4	$21.498$	$2.175$	4	$14.061$	$0.969$
5	$29.633$	$1.412$	5	$16.304$	$0.374$	5	$12.987$	$0.387$

Appendix

Table 4. Largest pressure difference between inlet and descending aorta over one cardiac cycle (in mmHg), sample means and standard deviations.

$\alpha_{r}=0.7$	$\mu$	$\sigma$	$\alpha_{r}=1.0$	$\mu$	$\sigma$	$\alpha_{r}=1.3$	$\mu$	$\sigma$
0	$3.671$	$0.273$	0	$1.532$	$0.133$	0	$1.137$	$0.118$
1	$7.392$	$0.968$	1	$4.333$	$0.682$	1	$2.409$	$0.248$
2	$7.223$	$0.659$	2	$4.291$	$0.337$	2	$2.669$	$0.342$
3	$4.276$	$0.172$	3	$2.173$	$0.132$	3	$1.327$	$0.086$
4	$4.931$	$0.604$	4	$2.533$	$0.192$	4	$1.699$	$0.190$
5	$3.113$	$0.155$	5	$1.478$	$0.123$	5	$1.097$	$0.081$

Appendix

Table 5. Mean wall shear stress magnitude over the whole blood vessel wall and one cardiac cycle (in Pa), sample means and standard deviations.

$\alpha_{r}=0.7$	$\mu$	$\sigma$	$\alpha_{r}=1.0$	$\mu$	$\sigma$	$\alpha_{r}=1.3$	$\mu$	$\sigma$
0	$0.260$	$0.019$	0	$0.209$	$0.022$	0	$0.231$	$0.014$
1	$0.119$	$0.060$	1	$0.178$	$0.057$	1	$0.250$	$0.040$
2	$0.142$	$0.020$	2	$0.081$	$0.024$	2	$0.155$	$0.016$
3	$0.307$	$0.027$	3	$0.226$	$0.039$	3	$0.197$	$0.019$
4	$0.077$	$0.045$	4	$0.044$	$0.014$	4	$0.088$	$0.028$
5	$0.114$	$0.048$	5	$0.125$	$0.017$	5	$0.239$	$0.023$

Appendix

Table 6. Mean oscillatory shear index over a reference patch on the underside of the descending aorta, sample means and standard deviations.

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High-fidelity computational models of cardiac mechanics provide mechanistic insight into the heart function but are computationally prohibitive for routine clinical use. Surrogate models can accelerate simulations, but generalization across diverse anatomies is challenging, particularly in data-scarce settings. We propose a two-step framework that decouples geometric representation from learning the physics response, to enable shape-informed surrogate modeling under data-scarce conditions. First, a shape model learns a compact latent representation of left ventricular geometries. The learned latent space effectively encodes anatomies and enables synthetic geometries generation for data augmentation. Second, a neural field-based surrogate model, conditioned on this geometric encoding, is trained to predict ventricular displacement under external loading. The proposed architecture performs positional encoding by using universal ventricular coordinates, which improves generalization across diverse anatomies. Geometric variability is encoded using two alternative strategies, which are systematically compared: a PCA-based approach suitable for working with point cloud representations of geometries, and a DeepSDF-based implicit neural representation learned directly from point clouds. Overall, our results, obtained on idealized and patient-specific datasets, show that the proposed approaches allow for accurate predictions and generalization to unseen geometries, and robustness to noisy or sparsely sampled inputs.

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