PI-NOMT: Physics-Informed Neural Optimal Mass Transport for Brain Fluid Dynamics

Mehmet Emin Acar; Vahit Bugra Yesilkaynak; Helene Benveniste; Gozde Unal

cs.LGSep 27, 2026

PI-NOMT: Physics-Informed Neural Optimal Mass Transport for Brain Fluid Dynamics

Authors: Mehmet Emin Acar, Vahit Bugra Yesilkaynak, Helene Benveniste, Gozde Unal

Organizations: Istanbul Technical University, Faculty of Computer and Informatics Engineering, AI and Data Engineering Department, Istanbul, Turkiye · Yale University, School of Medicine, New Haven, CT, USA

Abstract

Recovering hidden transport mechanisms from sparse spatiotemporal observations is a fundamental inverse problem in scientific machine learning. In brain tracer imaging, dynamic contrast-enhanced MRI (DCE-MRI) provides time-resolved measurements of tracer concentration, while the underlying velocity and source mechanisms governing tracer propagation remain unobserved. We formulate this problem as physics-informed latent-state inference, in which the transport field itself is the primary object of inference rather than an auxiliary variable used only to reconstruct observed densities. We propose Physics-Informed Neural Optimal Mass Transport (PI-NOMT), a framework that represents density, velocity, and source as continuous neural fields and combines a continuous neural density teacher, recursive differentiable advection--diffusion--source rollout, unbalanced optimal-transport regularization, and governing-equation supervision. Physical laws act as structural priors that constrain the space of admissible transport mechanisms, while observed tracer dynamics provide evidence for estimating the latent transport state. We evaluate PI-NOMT on a synthetic benchmark with known ground-truth transport and on DCE-MRI sequences from nine control rats. On the synthetic benchmark, PI-NOMT accurately recovers the prescribed velocity field, including its magnitude, direction, and integrated trajectories, rather than merely reconstructing endpoint densities. Across the nine rat datasets, the framework yields sub-percent local endpoint error, consistent physical speed scales, and low post-training PDE and incompressibility residuals. These results support physics-informed latent-state inference as a general framework for recovering hidden transport mechanisms from observed dynamic scalar fields.

Figures & tables

Figure 1. PI-NOMT training pipeline. The frozen density teacher provides rollout initialization, continuous reconstruction targets, and derivatives for physics supervision. The learned source and velocity fields generate a recursive differentiable rollout, with gradients supplied by reconstruction, optimal-transport, source, PDE, and incompressibility objectives. PI-NOMT training diagram showing a frozen continuous density teacher, trainable source and magnitude–direction velocity fields, and a recursive source–advection–diffusion rollout.

Figure 2. Recovery of the prescribed Gaussian advective field. The top row compares the prescribed and learned velocity fields on a representative central plane. The lower row compares learned pathlines with analytic trajectories initialized from identical seeds and shows magnitude and directional errors across the rollout interval. Comparison of the prescribed and learned Gaussian velocity fields, analytic and learned trajectories from identical seeds, and velocity magnitude and directional errors over the rollout interval.

Method	Velocity EPE $\downarrow$	Velocity magnitude error $\downarrow$	Velocity angular error ( ∘ ) $\downarrow$	Velocity cosine similarity $\uparrow$	Pathline EPE (voxels) $\downarrow$
PI-NOMT-Cart	0.01021	0.00605	0.293	0.999987	0.130

Table 1. Recovery of the prescribed Gaussian advective field. Field metrics are averaged over the prescribed Gaussian evaluation support

\mathcal{S}(t)

defined in Section 4.1.1 and over 40 substep times. Velocity EPE and magnitude error are reported in voxels per numerical time unit, angular error in degrees, and cosine similarity is dimensionless. Pathline EPE denotes the mean Euclidean distance between the final endpoints of analytic and learned trajectories initialized from identical seeds and is reported in voxels.

Figure 3. PI-NOMT rat-brain transport summary. Columns show density snapshots, relative-source maps, speed maps, pathlines, flux vectors, speed-lines, and Péclet-lines. Density frames span acquisition frames 6–30. Density, relative-source, and Eulerian speed fields are shown as translucent 3D slice renderings, using density- and scalar-dependent opacity and one shared scale per column. The Eulerian speed maps and speed-lines use the shared physical range 0–0.04 mm/min; values above this range are clipped to the upper color limit. Flux-vector color denotes endpoint displacement in mm. The Lagrangian panels show two complementary orientations of the inferred field; the black square in the pathline panel is connected to an enlarged detail inset between the upper and lower panel groups. Seven density frames and their time-local source and speed fields show tracer evolution from frame 6 to frame 30. Two orientations of pathlines, endpoint flux vectors, speed-colored trajectories, and Péclet-colored trajectories summarize the inferred field. A marked square in the side pathline view connects to a magnified region of densely organized curves.

$w_{\mathrm{src}}$	Final NMSE (%)	Ref.-start NMSE (%)	Mean PDE	Mean incomp.	Mean $\mathcal{L}_{\mathrm{src}}$	Speed P99 (mm/min)	Mean $\|\Delta M_{\mathrm{adv}}\|$ (%)
1,000	9.279	0.687	0.0967	0.00709	0.003239	0.0360	0.467
5,000	11.247	0.670	0.1084	0.00937	0.001222	0.0399	0.568
10,000	13.843	0.776	0.1243	0.01014	0.000710	0.0413	0.696
20,000	20.185	1.146	0.1699	0.01163	0.000418	0.0433	0.714

Table 2. Source-regularization sensitivity over frames 6–22. Reference-start NMSE is averaged over the eight two-frame chunk endpoints. Mean PDE and incompressibility are evaluated at the same 80 post-step times from 6.2 through 22.0. Mean source regularization is the unweighted

\mathcal{L}_{\mathrm{src}}

from Eq. ( 13 ); speed P99 is reported in mm/min, and mean absolute advection mass change is computed over the rollout steps.

Method	Mean endpoint NMSE (%) $\downarrow$
urOMT ( $\alpha=1{,}000$ )	$0.51\pm 0.22$
urOMT ( $\alpha=3{,}000$ )	$0.98\pm 0.39$
urOMT ( $\alpha=6{,}000$ )	$1.74\pm 0.79$
urOMT ( $\alpha=10{,}000$ )	$2.58\pm 1.28$
urOMT ( $\alpha=20{,}000$ )	$4.92\pm 2.24$
urOMT ( $\alpha=50{,}000$ )	$10.94\pm 3.92$

Table 3. Rat-brain short-interval endpoint reconstruction. Representative urOMT values are the means and standard deviations reported in the legend of Figure 6b in Chen et al. ( Chen et al., 2024 ) over its chained, interval-specific optimizations. The PI-NOMT value is the mean and sample standard deviation over 14 teacher-initialized, 10-substep reference-start endpoint NMSE values. The comparison provides short-interval reconstruction context under the respective inference procedures rather than a matched optimization protocol.

Variant	Final NMSE (%)	Ref.-start NMSE (%)	Mean PDE	Mean incomp.	Speed P99 (mm/min)	Mean $\|\Delta M_{\mathrm{adv}}\|$ (%)
PI-NOMT †	26.58	0.680	0.0742	0.00701	0.0265	0.592
PI-NOMT	28.75	0.713	2.0644	0.01666	0.0376	0.524
PI-NOMT-NoSrc †	31.97	1.150	0.1734	0.01243	0.0352	0.834
PI-NOMT-NoSrc	26.97	0.764	0.7699	0.04490	0.0527	0.918
PI-NOMT-Cart †	89.00	1.048	0.0943	0.01301	0.0358	0.579
PI-NOMT-Cart	23.76	0.505	2.4808	0.13618	0.0405	0.729

Table 4. Paired ablation of physics supervision and transport-model structure. Continuous NMSE is measured after rolling from frame 6 to frame 34. Reference-start NMSE is the mean two-frame chunk endpoint error when each chunk starts from the teacher density. PDE and incompressibility are the space–time means in Eq. ( 23 ), evaluated on the same 140 times and all masked points for every model without density rollout. Speed P99 is reported in mm/min, and mean absolute advection mass change is computed over the rollout steps. Lower values indicate better reconstruction or lower physical/numerical residuals for the corresponding error metrics; Speed P99 is reported as a characteristic of the inferred velocity field. A superscript

\dagger

denotes physics supervision; unmarked rows omit the PDE and incompressibility losses. The selected PI-NOMT formulation is marked in bold.

Metric	Mean $\pm$ SD	Subject range
Reference-start endpoint NMSE (%)	$0.801\pm 0.167$	$0.532$ – $1.090$
Continuous-rollout NMSE (%)	$11.773\pm 2.606$	$6.555$ – $15.298$
Mean Eulerian speed (mm/min)	$0.00738\pm 0.00111$	$0.00533$ – $0.00930$
Pathline-sampled mean advective speed (mm/min)	$0.00711\pm 0.00108$	$0.00508$ – $0.00876$
Speed P99 (mm/min)	$0.02968\pm 0.00592$	$0.02235$ – $0.03989$
PDE residual MSE	$0.07010\pm 0.02977$	$0.02997$ – $0.13078$

Table 5. Cross-subject consistency over nine control-rat DCE-MRI sequences. Values are the mean

\pm

sample standard deviation across subjects. The continuous-rollout entry first averages same-time teacher NMSE over frames 6–34 within each subject. PDE and incompressibility entries are post-training space–time means obtained by directly querying the neural fields. Eulerian speed summarizes each subject’s full-interval time-averaged velocity field, whereas pathline-sampled mean advective speed pools the local advective magnitude over all mask-inside trajectory-time samples of the corresponding integrated trajectories.

Figure 4. Cross-subject Eulerian transport comparison across all nine control-rat sequences. Each subject is shown with its time-averaged speed and relative-source fields over frames 6–34. Speed and source panels use shared scales of 0–0.04 mm/min and $[-0.02,0.02]$ , respectively. A three-by-three arrangement shows time-averaged Eulerian speed and relative-source maps for all nine control-rat subjects under shared color and opacity scales.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1. Continuous density teacher validation. The top grid compares measured target density, neural teacher density, and absolute error at five representative held-out validation frames using a shared normalized density/error scale. The lower panels report frame-wise NMSE, MSE, relative mass error, measured and teacher total mass, and the voxel-density distributions across all frames. Open diamond markers identify acquisition frames excluded from teacher fitting. Five held-out density volumes are reconstructed with faint spatial errors. Below them, NMSE is below one percent at 26 of 29 frames, MSE remains small in absolute units, the teacher follows the measured mass trajectory, and the measured and reconstructed voxel-density distributions nearly overlap.

Figure S2. Continuous density interpolation on the Gaussian benchmark. The upper rows compare analytic ground truth, density-teacher prediction, and absolute error at four held-out interframe times. The lower panels show dense-time NMSE, total mass, the centroid trajectory projected onto the prescribed diagonal, and centroid error. Diamonds mark the five integer times used to fit the teacher. Ground-truth and teacher Gaussian spheres nearly coincide at four held-out times. Error images are faint on the shared density scale, while the metric panels show low interpolation error, matching mass evolution, and an accurately recovered diagonal centroid trajectory.

Figure S3. Learned source and transport fields on the Gaussian benchmark. Columns show the five input densities, interval-wise relative-source and speed maps, full-interval averages, pathlines, endpoint flux vectors, speed-lines, and Péclet-lines. Trajectory geometry incorporates the diffusive drift $-D\nabla\log\rho$ , whereas trajectory speed coloring represents the magnitude of the raw advective velocity $\left\|\mathbf{v}_{\boldsymbol{\theta}}\right\|_{2}$ . Eulerian maps use tracer-dependent opacity and one common scale per quantity. The four trajectory panels include their corresponding trajectory-time, displacement, speed, and Péclet color scales. The Gaussian sphere moves diagonally while its inferred source is positive in the first two intervals and negative in the final two. Velocity maps and trajectory renderings show a smooth, coherent, nearly constant transport field.

Figure S4. Recursive Gaussian rollout against analytic ground truth. The model is initialized once at $t=0$ and recursively evolved for 40 substeps. Analytic density, rollout density, and absolute error share the single 0–30 color scale shown beneath all three rows. NMSE values at each integer endpoint are reported below the corresponding error maps. Analytic and recursively predicted Gaussian spheres remain spatially aligned through time. Absolute-error maps remain faint on the shared density scale, while NMSE rises gradually from 0.13 to 0.94 percent.

Figure S5. Spatial rollout reconstruction for PI-NOMT. The rollout is initialized at frame 6 and recursively evolved to the selected future frames. Density and absolute-error panels use the same colormap and numerical scale, allowing error magnitude to be interpreted directly relative to the density signal. Grid showing measured density, density-teacher projections, PI-NOMT rollout projections, and absolute rollout–teacher error at selected acquisition frames.

Figure S6. Effect of source regularization over frames 6–22. (a) Continuous rollout NMSE from a single initialization at frame 6. (b) Reference-start two-frame endpoint NMSE, with an inset enlarging frames 10–22. (c–d) Spatial mean PDE residual and incompressibility evaluated on the common 80-time grid. (e) Full-interval relative-source and Eulerian-speed maps followed by top and side views of speed-lines for four source weights. Source maps share the range $[-0.02,0.02]$ , while Eulerian speed and speed-lines share the physical range $[0,0.04]$ mm/min, with values above this range clipped to the upper color limit. The Eulerian opacity depends jointly on tracer density and the displayed scalar. Four temporal plots and a four-by-four visual comparison show the effect of increasing source regularization. Higher source weight weakens the relative-source field, increases continuous rollout error, and produces stronger high-speed transport together with larger PDE and incompressibility residuals.

Figure S7. Effect of physics supervision within five transport-model families. Open and filled markers denote physics off and on, respectively; connected markers share the same architecture and rollout design. Panels report (a) final continuous-rollout NMSE, (b) mean reference-start endpoint NMSE, (c) common-grid PDE residual, (d) common-grid incompressibility, (e) speed P99, and (f) mean absolute single-step advection mass change. Lower values indicate better reconstruction or lower physical/numerical residuals for the corresponding error metrics; speed P99 is reported as a characteristic of the inferred velocity field. Physics supervision improves the field-consistency diagnostics in every family, whereas PI-NOMT is the only family that also improves both density-error metrics. Six paired-dot panels compare physics-off and physics-on versions of PI-NOMT, NoSrc, Cart, 2S, and OneStep. Physics lowers PDE residual, incompressibility, and speed P99 throughout, while density errors improve only for PI-NOMT.

Figure S8. Post-training physical consistency on the common evaluation grid. Curves report the spatial mean squared PDE residual and incompressibility at each of 140 shared post-step times. The evaluation directly queries the frozen density, velocity, and source networks and does not use rolled-out density states. Except for PI-NOMT-NoPhys, all displayed variants use physics supervision. Two logarithmic line plots compare six transport models over continuous evaluation time. PI-NOMT maintains low PDE residual throughout the interval. PI-NOMT-NoPhys remains substantially higher, while the physics-enabled structural and rollout ablations are regularized toward the PI-NOMT range.

Figure S9. Density-error curves for selected ablations. Panel (a) measures recursive rollout from frame 6. The reference-start endpoint curve in panel (b) measures two-frame chunks initialized from the teacher at the start of each chunk and includes the intrinsic change between consecutive measured frames. The large gap between local and continuous error for PI-NOMT-Cart reveals recursive error accumulation. Continuous-rollout and reference-start endpoint NMSE panels compare six PI-NOMT variants. PI-NOMT-Cart has moderate local endpoint error but sharply increasing long-horizon error.

Figure S10. Physical and Lagrangian diagnostics for PI-NOMT and three ablations. Columns show PI-NOMT, PI-NOMT-NoPhys, PI-NOMT-NoSrc, and PI-NOMT-Cart. Rows show time-averaged speed, signed velocity divergence, mean absolute incompressible-form PDE residual, top-view pathlines, and side-view pathlines. Speed maps use the shared physical range 0–0.04 mm/min, with larger values displayed at the upper color, and the same combined opacity as the main-paper rat-brain transport summary, $\alpha=\rho_{n}^{0.90}(0.01+0.99v_{n})$ . Five-by-four grid comparing Eulerian speed, velocity divergence, PDE residual, and two pathline orientations for PI-NOMT and three ablations.

Figure S11. Velocity-magnitude distributions for PI-NOMT and the two structural ablations. Panels compare physics-off and physics-on versions of (a) PI-NOMT-NoSrc and (b) PI-NOMT-Cart against the selected PI-NOMT reference. Physics supervision contracts both high-speed tails, but their P99 values remain above that of PI-NOMT. Two overlaid velocity-magnitude histograms compare physics-off and physics-on NoSrc and Cart models with PI-NOMT; legends report P99 values.

Figure S12. Time-averaged Eulerian speed over frames 6–34 for the detailed C1217 experiment and three additional subjects. All panels use the same 0–0.04 mm/min scale and the density–speed opacity mapping used in the main-paper rat-brain transport summary. Four side-view rat-brain maps compare time-averaged Eulerian speed for subjects C1217, C1189, C1204, and C1265 under one shared color scale.

Figure S13. Time-averaged Eulerian relative-source fields over frames 6–34 for the same four subjects. A common $[-0.02,0.02]$ scale and common density–source opacity mapping are used throughout. Four side-view rat-brain maps compare signed time-averaged relative-source fields for subjects C1217, C1189, C1204, and C1265.

Figure S14. Side-view pathlines generated using the diffusion-augmented drift for the four selected subjects. Identical rendering settings are used, and color indicates progression from trajectory start to end. Four side-view pathline renderings show trajectory progression for subjects C1217, C1189, C1204, and C1265.

Figure S15. Side-view speed-lines for the four selected subjects. Trajectory points are colored by the local advective speed $\|\mathbf{v}_{\boldsymbol{\theta}}\|_{2}$ using one shared 0–0.04 mm/min scale. Four side-view speed-line renderings compare instantaneous learned speed for subjects C1217, C1189, C1204, and C1265 under one shared scale.

Figure S16. Complete cross-subject pathline comparison. Top and side views are shown for all nine subjects under common integration and rendering settings; color indicates progression from trajectory start to end. A three-column arrangement shows stacked top and side pathline views for all nine control-rat subjects under common rendering settings.

Figure S17. Complete cross-subject speed-line comparison. Top and side views are shown for all nine subjects; trajectory points are colored by instantaneous learned speed using a shared 0–0.04 mm/min scale. A three-column arrangement shows stacked top and side speed-line views for all nine control-rat subjects under a shared speed scale.

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