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

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

arXiv:2609.33857v1cs.LGcs.AI

TL;DR

DCE-MRI measures tracer concentrations over time, but the velocity and source mechanisms driving their propagation remain hidden. PI-NOMT infers continuous transport fields through physics-constrained recursive rollout, recovering known synthetic dynamics and consistent transport characteristics across rat sequences.

  • Problem

    DCE-MRI observes tracer distributions over time, but the underlying transport mechanisms remain hidden, making recovery from sparse spatiotemporal observations a challenging inverse problem.

    Source: §1 · p. 2

  • Method

    PI-NOMT learns continuous velocity and source fields by recursively generating a continuous density teacher through differentiable transport dynamics under optimal-transport and governing-equation constraints.

    Source: §1 · p. 2, §3 · p. 4

  • Results

    PI-NOMT recovers known synthetic advective dynamics and achieves mean local endpoint error below 1% across nine control-rat sequences.

    Source: §5 · p. 22

  • Takeaways & Limitations

    PI-NOMT offers a framework for recovering physically constrained latent transport mechanisms from observed density evolution, while inferred fields remain model-based estimates.

    Source: §5 · p. 23, §6 · p. 24

    Latent transport from tracer density alone is inherently non-identifiable, so rat-brain fields are model-consistent explanations rather than direct physiological measurements.

    Source: §5 · p. 23

Abstract

from arXiv · show

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.

1 Introduction

The paper frames tracer transport recovery as inference of hidden, physically constrained transport fields from observed density evolution. PI-NOMT represents those fields continuously and tests them through recursive physical rollout.

1 Introduction

  • DCE-MRI observes tracer distributions over time, but the underlying mechanisms governing their evolution remain hidden.Recovering those processes from sparse spatiotemporal observations is presented as a challenging scientific problem.Source: §1 · p. 2
  • PI-NOMT treats continuous velocity and source fields as a latent transport state inferred from observed tracer-density evolution under physical constraints.The target is a physically meaningful mechanism explaining the dynamics, not simply a prediction of future observations.Source: §1 · pp. 2–3
  • PI-NOMT combines a continuous neural density teacher, neural transport fields, unbalanced optimal-transport regularization, and governing-equation supervision.The density teacher supplies interframe supervision and derivatives; velocity and source fields are inferred through the combined constraints.Source: §1 · pp. 2–3
  • Unlike pairwise OMT formulations that re-optimize interval-specific variables, PI-NOMT learns continuous transport fields shared across the observed interval.The fields are constrained through recursive physical evolution.Source: §1 · p. 2
  • Recursive differentiable advection–diffusion–source rollout tests whether inferred fields can generate the observed tracer evolution.Rollout couples latent-state estimation directly to physical dynamics rather than serving only as a prediction mechanism.Source: §1 · pp. 2–3
  • The evaluation covers synthetic transport with known ground truth and DCE-MRI sequences from nine control rats.Ablations examine recursive rollout, physics supervision, and velocity-field parameterization.Source: §1 · p. 3

2 Related Work

Related work spans optimal-transport formulations, physics-informed inverse modeling, and latent continuous-time dynamics. PI-NOMT brings these perspectives together to infer continuous velocity and source fields from tracer-density evolution under physical constraints.

2 Related Work

  • Classical and dynamic OMT, unbalanced transport, rOMT, and urOMT provide formulations for transport, mass variation, diffusion, and source or sink terms.PI-NOMT instead learns shared continuous neural velocity and source fields, testing them through recursive physical evolution.Source: §2 · p. 3
  • Physics-informed learning and differentiable simulation recover hidden physical quantities from indirect observations, including latent velocity fields from time-resolved scalar data.PI-NOMT combines continuous latent transport fields, unbalanced OMT regularization, and recursive differentiable advection–diffusion–source rollout.Source: §2 · p. 3
  • DCE-MRI transport studies have estimated velocity, pressure, permeability, or mechanistic-model parameters under physical assumptions, alongside OMT-based glymphatic transport inference.PI-NOMT represents spatially and temporally varying velocity and source fields as a shared latent state tested by recursive transport dynamics.Source: §2 · pp. 3–4
  • Latent continuous-time and state-space models address hidden dynamics, while PI-NOMT targets continuous velocity–source recovery from tracer densities under unbalanced transport physics.The paper presents its framework as combining latent-state inference with rollout, optimal-transport regularization, and governing-equation supervision.Source: §2 · p. 4

3 Methodology

PI-NOMT infers continuous velocity and source fields as the latent mechanisms generating observed tracer-density evolution, using a frozen density teacher and physical constraints.

3.1 Latent Transport Formulation

  • PI-NOMT defines the latent transport state as velocity and relative source or sink rate, represented by shared continuous fields across the observed temporal domain.Inference couples these fields to density evolution through recursive physical rollout and the governing equation.Source: §3.1 · p. 4

3.2 Continuous Neural Density and Transport Fields

  • A frozen continuous density teacher supplies interframe targets and physics derivatives, while continuous neural fields represent source and velocity, with velocity split into magnitude and direction.The magnitude branch models localized speed variation; the direction branch encodes smoother spatial variation.Source: §3.2 · p. 5

3.3 Differentiable Advection–Diffusion–Source Rollout

  • The rollout recursively applies source, semi-Lagrangian advection, and backward-Euler diffusion, using approximate incompressibility to align characteristic and conservative advection.Density is initialized at the start of a rollout window and not reset between substeps; the default uses 10 substeps per sampled interval.Source: §3.3 · pp. 5–6

3.4 Physics Supervision

  • Physics supervision evaluates a continuous residual using derivatives of the frozen teacher and updates the velocity and source networks without modifying the teacher.The residual is evaluated under the rollout's approximately incompressible formulation.Source: §3.4 · p. 6

3.5 Training Objective

  • Training initializes random temporal windows from the frozen teacher and optimizes the learned fields with reconstruction, optimal-transport, source, and physics objectives.Endpoint and intermediate reconstruction losses share one recursive rollout, so their gradients pass through the learned transport sequence; rolled density weights the regularizers without receiving their gradients.Source: §3.5 · pp. 6–8

4 Experimental Evaluation

The experimental evaluation assesses PI-NOMT’s latent transport recovery on a synthetic benchmark with known dynamics and rat DCE-MRI sequences, fitting and freezing a continuous density teacher before learning transport fields through recursive physical rollout. Across the rat sequences, it examines endpoint error, speed scales, inferred field patterns, and how physics supervision and modeling choices shape physical consistency, reconstruction, and rollout stability.

4 Experimental Evaluation; 4.1.1 Synthetic Gaussian-Sphere Benchmark; 4.1.2 Rat-Brain DCE-MRI Experiment; 4.10 Ablation Tradeoffs

  • Experiments used five synthetic training volumes with 36 held-out interframes and nine rat sequences with 29 retained frames; the ablations compare structural variants with PDE and incompressibility losses on and off.Rat sequences were preprocessed percentage-signal-change tracer surrogates, globally scaled to [0, 100]; C1217 was used for detailed analyses and ablations, with the other sequences used for cross-subject evaluation.Source: §4 · p. 8, §4.1.1 · p. 8, §4.1.2 · p. 9

4.2 Continuous Density Teacher Quality; 4.3 Continuous Gaussian Density Interpolation

  • Rat-teacher mean NMSE was 0.535% across 29 frames and 0.867% on held-out frames; Gaussian-teacher NMSE was 0.0153% on training frames and 0.0202% on interframes.The Gaussian teacher’s held-out centroid error averaged 0.104 voxels and its mean absolute held-out mass error was 1.291%; the rat teacher had 26 of 29 frames below 1% NMSE.Source: §4.2 · p. 11, §4.3 · p. 11

4.4 Recovery of the Prescribed Advective Field

  • Velocity EPE and magnitude error were 0.01021 and 0.00605 voxels per numerical time unit; mean angular error was 0.293°, cosine similarity 0.999987, and pathline EPE 0.130 voxels.Across four acquisition intervals, learned trajectories traveled 22.590 voxels on average versus 22.632 prescribed voxels, a relative difference of −0.187%.Source: §4.4 · pp. 11–12, Figure 2 · p. 12, Table 1 · p. 13

4.5 Synthetic Source–Transport Decomposition and Rollout

  • The synthetic inferred mean advective speed was 1.412 voxels per numerical time unit, with a mean drift-to-advection ratio of 0.046%, while the source followed the prescribed gain-then-loss sequence.Recursive rollout endpoint NMSE rose from 0.133% at t = 1 to 0.941% at t = 4, averaging 0.503%; relative mass error increased from 5.29% to 13.56%.Source: §4.5 · p. 13

4.6 Inferred Rat-Brain Transport Field

  • Rat-brain mean learned Eulerian speed was 0.00846 mm/min, and pathline-sampled mean advective speed was 0.00839 mm/min; the inferred transport and source fields were spatially structured.The source maps shifted from widespread positive tracer gain early to spatially separated gain and loss later; inferred fields are model-consistent explanations, not direct measurements of physiological fluid transport.Source: §4.6 · p. 13, Figure 3 · p. 14

4.7 Recursive Rollout Quality

  • Continuous rat-brain rollout NMSE was 26.58%, compared with mean reference-start endpoint NMSE of 0.680%; the former accumulates error across frame 6–34, while the latter reinitializes every two frames.The recursive rollout preserved dominant tracer-distribution structure, although discrepancies became more pronounced over the full rollout.Source: §4.7 · p. 14

4.8 Effect of Source Regularization

  • As source regularization increased, continuous-rollout NMSE rose from 9.28% to 20.18%, while unweighted mean Lsrc fell from 3.239 × 10^-3 to 4.18 × 10^-4 over frames 6–22.Stronger source suppression also increased speed P99, advection-induced mass change, PDE residual, and incompressibility, consistent with source–velocity compensation; similar short-interval errors can mask different decompositions.Source: §4.8 · p. 15

4.10 Ablation Tradeoffs

  • Physics supervision lowered PI-NOMT’s mean PDE residual from 2.0644 to 0.0742, incompressibility from 0.01666 to 0.00701, and speed P99 from 0.0376 to 0.0265 mm/min.It also reduced final continuous NMSE from 28.75% to 26.58% and mean reference-start NMSE from 0.713% to 0.680%, while mean absolute advection mass change increased from 0.524% to 0.592%.Source: §4.10 · pp. 16–17
  • Removing the source branch raised final and reference-start NMSE from 26.58% and 0.680% to 31.97% and 1.150%, respectively.PI-NOMT-NoSrc also increased speed P99 from 0.0265 to 0.0352 mm/min and mean absolute advection mass change from 0.592% to 0.834%; physics supervision still lowered its PDE residual from 0.7699 to 0.1734 and speed P99 from 0.0527 to 0.0352 mm/min, although final NMSE increased from 26.97% to 31.97%.Source: §4.10 · p. 17
  • The Cartesian velocity model had higher incompressibility error (0.01301 versus 0.00701) and speed P99 (0.0358 versus 0.0265 mm/min) than PI-NOMT.Its direction varied more rapidly and its pathlines appeared less coherent; physics supervision lowered its PDE residual from 2.4808 to 0.0943 and incompressibility error from 0.13618 to 0.01301, but final continuous NMSE increased from 23.76% to 89.00%. The smaller 1.048% reference-start error coincided with unstable long-horizon rollout; the authors suggest magnitude–direction factorization supplies an inductive bias for stability.Source: §4.10 · p. 17
  • Using two rather than 10 transport cycles raised final NMSE from 26.58% to 39.19%, reference-start NMSE from 0.680% to 0.973%, and mean absolute advection mass change from 0.592% to 2.771%.Each numerical step was five times larger; reconstruction degraded despite lower PDE residual, incompressibility, and speed, consistent with reduced temporal resolution and larger per-step advection mass change.Source: §4.10 · p. 17
  • PI-NOMT-OneStep’s final continuous NMSE reached 65.05% and reference-start NMSE reached 2.565%, despite low speed and incompressibility values.Unlike recursive supervision, each OneStep update starts from teacher density, so later reconstruction losses do not propagate through earlier field decisions; recursive training instead exposes fields to repeated composition.Source: §4.10 · p. 17

4.11 Cross-Subject Consistency

  • Across the cohort, mean Eulerian speed was 0.00738 ± 0.00111 mm/min and pathline-sampled mean advective speed was 0.00711±0.00108 mm/min.The authors describe these similar cohort-level summaries as consistent in physical scale with rOMT-based rat DCE-MRI transport speeds.Source: §4.11 · p. 18, Table 5 · p. 19
  • Across nine rats, mean reference-start endpoint NMSE was 0.801%±0.167% (range 0.532–1.090%), while mean frame-6–34 rollout same-time NMSE was 11.773% ± 2.606%.The authors report accurate local reconstruction alongside moderate subject-to-subject variation in accumulated full-rollout error.Source: §4.11 · p. 18, Table 5 · p. 19
  • Across the nine subjects, inferred fields showed localized high-speed structure and organized positive- and negative-relative-source regions, with subject-specific extent and intensity.Figure 4 presents shared-scale fields; the results support consistency within this control cohort, not anatomically identical fields or direct measurements of physiological fluid velocity.Source: §4.11 · p. 19, Figure 4 · p. 20

5 Discussion

PI-NOMT recovers latent transport fields in synthetic and rat-brain experiments, while ablations show that stable inference depends on recursive rollout, physics supervision, and velocity-field representation. The inferred fields remain model-consistent explanations rather than direct physiological measurements.

Density teacher

  • The continuous density teacher reproduces held-out tracer dynamics and mass evolution, but errors in its interpolation or derivatives can propagate into rollout targets and physics residuals.It provides a fixed continuous reference for transport learning and derivatives needed by governing equations.Source: §5 · p. 21

Rollout ablations

  • Teacher-reset one-step training and reducing rollout resolution from ten to two substeps both degrade long-horizon reconstruction, indicating that local fitting or pointwise consistency alone is insufficient for stable recursive inference.Recursive supervision requires learned fields to generate intermediate teacher states across multiple substeps.Source: §5 · p. 21

Physics and representation ablations

  • Physics supervision reduces PDE and incompressibility residuals and contracts high-speed tails, but its effect on reconstruction depends on velocity representation; PI-NOMT-Cart gains physical consistency while reconstruction markedly degrades.PI-NOMT is the only evaluated family where improved physical consistency also improves local reconstruction and long-horizon prediction, consistent with its magnitude–direction factorization.Source: §5 · pp. 21–22

Interpretation and scope

  • Different source penalties can yield comparable local density fits but substantially different recursive behavior, while inferred rat-brain source fields shift from early positive values to later negative values consistent with tracer gain then loss.The decomposition is conditional on fixed diffusion coefficient D; the fields are physically constrained explanations, not direct measurements, and biological validation covers nine control-rat sequences under one protocol.Source: §5 · pp. 22–23

Cross-subject evaluation

  • Across nine control-rat sequences, mean local endpoint error is below 1%, with consistent physical speed scales and low post-training PDE and incompressibility residuals.Eulerian fields and trajectories show recurring organization alongside subject-specific variation.Source: §5 · p. 22

Synthetic validation

  • The synthetic benchmark recovers the prescribed velocity field, with low velocity, angular, magnitude, and pathline endpoint errors, rather than merely matching endpoint densities.The continuous density teacher also reconstructs the unseen interframe sphere trajectory, enabling continuous evaluation of the learned transport field.Source: §5 · p. 22

6 Conclusion

PI-NOMT infers continuous velocity and source fields from sparse scalar observations using recursive differentiable transport and governing-equation supervision. Synthetic and rat-brain experiments support recovery of physically constrained transport quantities, with inferred fields interpreted as model-based estimates.

Framework

  • PI-NOMT replaces interval-specific voxelwise optimization with continuous neural transport fields trained through recursive differentiable transport and governing-equation supervision.The framework represents recovery from sparse time-resolved scalar observations as physics-informed latent-state inference.Source: §6 · p. 23

Evidence

  • Synthetic experiments show direct recovery of known advective dynamics, while rat-brain DCE-MRI experiments demonstrate accurate short-interval reconstruction and consistent transport characteristics across subjects.Ablations indicate that governing-equation supervision works best alongside an appropriate latent-field representation and recursive transport formulation.Source: §6 · pp. 23–24

Scope

  • The inferred fields are physically interpretable model-based estimates, not direct physiological measurements, and the framework is presented for inverse transport problems involving observable evolving scalar fields.The paper identifies this as a broader latent-state perspective for hidden dynamics.Source: §6 · p. 24

Dedication

The appendix situates PI-NOMT within optimal transport and scientific machine learning, distinguishing its latent-field inference task from related frameworks. Its dedication honors Allen Tannenbaum's influence on the work.

Dedication

  • The work is dedicated to Allen Tannenbaum in recognition of his contributions to optimal transport, medical imaging, and mathematical data science.The dedication also acknowledges his mentorship, scientific vision, and intellectual influence.Source: Dedication · p. 24

A.1 Dynamic Optimal Mass Transport

  • Dynamic OMT describes transport through time-varying density and velocity fields satisfying a continuity equation, linking optimal transport with fluid mechanics.This formulation provides a basis for recovering velocity fields from evolving density observations.Source: §A.1 · p. 26

A.2 Unbalanced and Regularized Optimal Transport

  • Unbalanced OMT extends transport to mass variation, accommodating diffusion, production, clearance, and exchange; PI-NOMT draws on this transport–source setting without computing an exact unbalanced-transport geodesic.The Wasserstein–Fisher–Rao construction combines transport and mass change.Source: §A.2 · p. 26

A.3 Physics-Informed and Operator Learning

  • Unlike neural operators and related scientific machine-learning approaches, PI-NOMT targets latent transport fields whose recursive physical evolution explains an observed density sequence.The related approaches are described as learning mappings between function spaces for surrogate modeling and simulation acceleration.Source: §A.3 · pp. 26–27

A.4 Neural-Field Architectures and Density-Teacher Training

The appendix describes separate neural representations for density, source, and velocity, then specifies a differentiable recursive rollout that splits source, advection, and diffusion. It also states the rollout's incompressibility assumption and fixed-diffusion treatment.

A.4 Neural-Field Architectures and Density-Teacher Training

  • The density teacher is a residual fully connected network with four residual blocks, width 256, tanh activations, and a Chebyshev feature map for spatiotemporal coordinates.Its supervised loss uses observed tracer samples, with MAE replacing MSE for the final 1,000 epochs; the trained density network is then frozen during transport inference.Source: §A.4 · p. 27
  • Separate residual networks represent source, velocity magnitude, and direction; the magnitude branch uses Chebyshev features while source and direction use spatiotemporal coordinates directly.This encodes localized speed variation separately from smoother directional structure; PI-NOMT-Cart instead predicts three Cartesian velocity components directly.Source: §A.4 · p. 27

A.5 Differentiable Rollout: Numerical Formulation and Implementation

  • Each rollout interval is divided into substeps that split source, advection, and diffusion, combining an explicit relative-source step, semi-Lagrangian advection, and implicit diffusion.The source update is first-order and remains nonnegative for nonnegative input density when its stated condition holds; source regularization and small substeps promote that condition.Source: §A.5 · pp. 27–28
  • The rollout uses an approximately incompressible formulation, controlling divergence through a penalty rather than an explicit Jacobian correction to the semi-Lagrangian update.The same assumption is used in the continuous physics residual to keep rollout and supervision consistent.Source: §A.5 · p. 28
  • Diffusion uses a backward-Euler solve on the rollout grid, and each diffused density feeds the next source–advection–diffusion cycle without resetting to the teacher between substeps.Conjugate gradients solve the diffusion system; differentiation uses the corresponding adjoint system, and gradients do not propagate through fixed D.Source: §A.5 · pp. 28–29

A.6 Random-Window Training and Gradient Implementation

Randomly sampled temporal windows train transport fields using recursive rollout reconstruction alongside local physics supervision. A frozen continuous density teacher supplies targets between measured frames, while rollout gradients connect transport estimates across substeps.

A.6 Random-Window Training and Gradient Implementation

  • Random windows are sampled continuously over the valid interval, with nominal rollouts of 10 substeps and truncation at temporal boundaries.The teacher supplies initialization and reconstruction targets even at interframe times without direct measurements.Source: §A.6 · pp. 29–30
  • Each window produces a recursive sequence without resetting to the teacher, and endpoint and intermediate losses supervise states on that same trajectory.For a one-substep window, no intermediate state exists and the intermediate loss is zero.Source: §A.6 · p. 30
  • OMT kinetic and source regularizers weight activity along the predicted density trajectory, but stop-gradient prevents their density weights from changing earlier rollout states indirectly.Gradients from those terms act directly on the transport fields.Source: §A.6 · p. 30
  • Reconstruction gradients propagate through the full source–advection–diffusion chain, while physics and incompressibility losses use teacher density derivatives and train the transport fields.The density teacher remains frozen; physics-loss gradients are accumulated at active substeps separately for memory efficiency without changing the objective.Source: §A.6 · p. 30

B Density-Teacher Validation

  • The density-teacher validation figures compare teacher predictions with measured or analytic densities and report frame-wise errors, mass, and centroid diagnostics.The Gaussian benchmark figure includes held-out interframe times, while the rat validation figure marks acquisition frames excluded from teacher fitting.Source: Figure S1 · p. 31, Figure S2 · p. 32

C Additional Synthetic Results

The supplementary synthetic results visualize learned transport fields and assess recursive rollout against analytic ground truth.

C Additional Synthetic Results

  • The learned-field figure lays out input densities, interval-wise and full-interval source and speed maps, pathlines, flux vectors, speed-lines, and Péclet-lines.Trajectory geometry uses diffusion-augmented drift, while speed coloring shows raw advective velocity magnitude.Source: Figure S3 · p. 33
  • The transport visualizations use tracer-dependent opacity and shared quantity scales, with trajectory panels providing their corresponding time, displacement, speed, and Péclet scales.The displayed pathline geometry incorporates diffusive drift, not just raw advection.Source: Figure S3 · p. 33
  • The recursive Gaussian rollout starts at t=0, advances for 40 substeps, and compares analytic density, rollout density, and absolute error on a shared scale.NMSE is reported at each integer endpoint.Source: Figure S4 · p. 34

C.1 Synthetic Source and Transport Diagnostics

Synthetic diagnostics describe recovery of the prescribed source schedule and coherent transport, alongside rat rollout and source-regularization visualizations.

C.1 Synthetic Source and Transport Diagnostics

  • The learned relative source is positive over intervals 0–1 and 1–2, then negative over intervals 2–3 and 3–4, matching the benchmark’s gain-then-loss schedule.The prescribed ground-truth source lies within [−0.025, 0.025].Source: §C.1 · p. 35
  • Within the Gaussian support, mean advective speed is 1.412 voxels per numerical time unit and mean diffusion-induced drift is 6.50 × 10−4.The mean drift-to-advection ratio is 0.046%, with a 99th percentile of approximately 0.068%; trajectory geometry uses diffusion-augmented drift, but speed coloring shows raw advection.Source: §C.1 · p. 35

D Rat-Brain Rollout Diagnostics

  • The rat rollout figure shows density and absolute-error panels at selected future frames after recursive evolution initialized at frame 6.The shared colormap and scale allow error magnitude to be read relative to the density signal.Source: Figure S5 · p. 35

E Source-Regularization Diagnostics

  • The source-regularization figure compares four source weights using continuous-rollout and reference-start endpoint NMSE, physical residuals, and source and speed maps.Its maps use shared source and physical speed ranges, with speed values above 0.04 mm/min clipped.Source: Figure S6 · p. 36

F Detailed Ablation Diagnostics

The ablation figures compare physics supervision and structural variants using density-error, physical-consistency, speed, and pathline diagnostics.

F Detailed Ablation Diagnostics

  • Across five transport-model families, physics supervision improves field-consistency diagnostics in every family, while PI-NOMT alone improves both density-error metrics.The figure reports continuous-rollout NMSE, reference-start endpoint NMSE, PDE residual, incompressibility, speed P99, and single-step advection mass change.Source: Figure S7 · p. 37
  • Post-training consistency curves report spatial-mean PDE residual and incompressibility at 140 shared post-step times, querying frozen networks without rolled-out density states.All displayed variants except PI-NOMT-NoPhys use physics supervision.Source: Figure S8 · p. 37
  • The density-error figure contrasts recursive rollout from frame 6 with two-frame chunks initialized from the teacher at each chunk start.The caption identifies a large local-versus-continuous error gap for PI-NOMT-Cart as recursive error accumulation.Source: Figure S9 · p. 38
  • Physics supervision contracts the high-speed tails of PI-NOMT-NoSrc and PI-NOMT-Cart, but their P99 values remain above the PI-NOMT reference.The physical-diagnostic figure also compares speed, signed divergence, PDE residual, and pathlines across PI-NOMT and three ablations.Source: Figure S10 · p. 39, Figure S11 · p. 40

G Cross-Subject Visualizations

The cross-subject visualizations compare learned speed, relative-source, and trajectory fields across selected subjects and the full nine-subject cohort using shared scales or rendering settings.

G Cross-Subject Visualizations

  • Time-averaged Eulerian speed and relative-source fields are shown over frames 6–34 for four subjects, with common scales and density-based opacity mappings.The speed scale is 0–0.04 mm/min and the relative-source scale is [−0.02, 0.02].Source: Figure S12 · p. 41, Figure S13 · p. 41
  • For four selected subjects, side-view pathlines show progression from trajectory start to end, while speed-lines color points by local advective speed on a shared 0–0.04 mm/min scale.The pathlines use the diffusion-augmented drift and identical rendering settings.Source: Figure S14 · p. 41, Figure S15 · p. 42
  • The complete pathline and speed-line comparisons show top and side views for all nine subjects under common rendering settings, with speed-lines using a shared 0–0.04 mm/min scale.Pathline color marks progression from trajectory start to end; speed-line color marks instantaneous learned speed.Source: Figure S16 · p. 43, Figure S17 · p. 44
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