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Local Consistency Does Not Guarantee Global Conservation: Auditing Zero-Shot Composition of Airway Flow Operators

Nichula Sathmith Wasalathilaka, Navodya Heshan Samarasinghe Dhanujaya Suraweera, Kevin Dawson, Chinthaka Jacob Mervyn Parakrama Bandara Ekanayake, Roshan Godaliyadda

arXiv:2610.02056v1eess.SYcs.SY

TL;DR

The paper asks whether independently learned local operators can be composed into a consistent unseen system, a question complicated by interface and accumulated errors in branching flows. It audits frozen, single-pass composition of airway primitives and finds that local regularization improves primitive and assembled-local diagnostics without ensuring better global fields or conservation.

  • Problem

    It is unclear whether independently learned operators for reusable primitives can be composed into an unseen system with mutually compatible interfaces and global consistency.

    Source: §1 · p. 2

  • Method

    The study trains separate DeepONets for airway primitives and freezes their predictions for single-pass tree assembly without iterative coupling or flow correction.

    Source: §1 · p. 2, §2 · pp. 3–4

  • Results

    Full reduces primitive composite scores by 26.7% for Y2 and 26.4% for Y3, yet increases mean tree vector RelL2 by 7.5% and external residual from 17.49 ± 4.38% for Data to 26.77 ± 3.65%.

    Source: §3 · pp. 4–5

  • Takeaways & Limitations

    Improved primitive and assembled-local diagnostics do not ensure improved global fields or external conservation.

    Source: §5 · p. 5, §3 · p. 5

    The audit uses steady laminar flow in idealized 2D rigid-wall airways and tests one topology with prescribed inputs and terminal flows, not arbitrary topologies or physiological redistribution.

    Source: §4 · p. 5

Abstract

from arXiv · show

Neural operators approximate PDE solutions within a geometry family, but independently learned local operators need not form a consistent global simulator. We study frozen, single-pass composition for steady incompressible flow in idealized two-dimensional airway trees. Separate Tube, bifurcation, and trifurcation DeepONets are trained on 4,872 primitive CFD cases using field supervision and auxiliary divergence, port-flux, component-balance, and port-pressure penalties. The validation-selected deployment is frozen before whole-tree CFD fields are inspected and assembled without tree training, iterative coupling, flux correction, or CFD-informed adjustment. It retains major flow patterns and controlled pathology responses with 0.204-0.215 s CPU inference, but has a 22.68% prescribed-inlet-normalized external residual. A post-hoc sensitivity protocol, frozen before new training and evaluation, repeats Data, Div, and Full models across three seeds with Tube fixed. Relative to Data, Full reduces primitive composite scores by 26.7% for Y2 and 26.4% for Y3 and reduces assembled component-residual and interface-mismatch RMS by 7.2% and 16.6%, respectively. Nevertheless, mean tree velocity error increases by 7.5%, while external residual increases from 17.49 +/- 4.38% to 26.77 +/- 3.65%. Local regularization can therefore improve primitive and assembled local diagnostics without ensuring accurate global fields or conservation.

1 Introduction

The paper asks whether independently learned operators for reusable physical components can be frozen and composed into an unseen system. It audits this question in idealized airway trees, where local accuracy does not ensure global consistency.

1 Introduction

  • Full-domain geometry generalization differs from composing independently learned local operators, which additionally requires compatible predictions at component interfaces.Existing learned local-to-global approaches described here retain iterative coupling or joint generation.Source: §1 · p. 2
  • Low field and physics errors on isolated primitives do not guarantee consistency after assembly, where flux mismatches, incompatible pressure gauges, and accumulated errors can arise.These system-level requirements differ from constraints imposed within individual component domains.Source: §1 · p. 2
  • The study uses idealized two-dimensional Tube, bifurcation (Y2), and trifurcation (Y3) airway components, with prescribed terminal-flow allocation rather than patient-specific physiology.It does not claim unrestricted topology generalization or physiological terminal-flow redistribution.Source: §1 · p. 2
  • The audit tests frozen, single-pass composition of independently trained operators without tree training, iterative exchange, flux correction, or global projection.The primary deployment is archived before whole-tree CFD fields are inspected.Source: §1 · p. 2

2 Method

Independent DeepONets are trained for airway primitives, then their frozen predictions are placed and audited in a tree without modifying the predicted flow. The learning objectives add physical and port-related terms to field supervision.

2 Method

  • Separate DeepONets map geometry, inlet-speed scale, outlet resistances, and normalized coordinates to velocity and pressure for Tube, Y2, and Y3 components.The dataset contains 4,872 cases: 1,000 Tube, 2,162 Y2, and 1,710 Y3; original-template groups are disjoint across train, validation, and test splits.Source: §2 · p. 3
  • Data uses field loss, Div adds divergence, and Full uses the closure-aware objective; port-pressure supervision is limited to pilot Y2/Y3 cases.The field term is equal-weight MSE over standardized velocity components and pressure.Source: §2 · p. 3
  • Each component is predicted in its canonical frame and placed in the tree using its orientation and translation, while pressure receives only an additive gauge alignment.Gauge alignment uses neural interface pressures; it does not change velocity or pressure gradients.Source: §2 · pp. 3–4
  • The assembler reports conservation errors without modifying predicted flow, using no flux correction, target-flow forcing, or CFD-informed adjustment.The operators and predictions are frozen before monolithic velocity and pressure CFD fields are inspected.Source: §2 · p. 4

3 Results

The sensitivity audit finds that Full improves primitive composite scores and assembled local diagnostics relative to Data, but global field error and external conservation do not improve. The frozen Selected deployment retains major flow patterns while showing a substantial external residual.

3 Results

  • 26.7% for Y2 and 26.4% for Y3 are the mean composite-score reductions for Full relative to Data across three matched seeds.Divergence-MSE reductions of 90.5% for Y2 and 75.0% for Y3 principally drive the gains.Source: §3 · p. 4, Table 2 · p. 5
  • 7.5% higher mean vector RelL2 and external residual of 26.77 ± 3.65% for Full versus 17.49 ± 4.38% for Data accompany 7.2% and 16.6% mean reductions in component-residual and interface-mismatch RMS.The external residual is higher for Full in every seed.Source: §3 · pp. 4–5, Table 3 · p. 5
  • 22.676% is the Selected deployment's external boundary residual, versus 4.5×10^-7% for CFD; its fields retain axial-flow organization and stenosis acceleration.CPU inference is 0.204–0.215 s.Source: §3 · p. 4, Table 3 · p. 5
  • The external residual measures the external-port budget by telescoping and cannot localize internal error.It contains no pathology signal when external-port vectors do not change across conditions.Source: §3 · p. 5

4 Limitations

The audit is limited to idealized steady laminar flow in rigid 2D airways and one topology with prescribed inputs and terminal flows; its three-seed variability is descriptive, not inferential.

4 Limitations

  • The study covers idealized steady laminar flow in rigid 2D airways, excluding patient anatomy, unsteadiness, wall motion, and turbulence.It tests one topology with prescribed inputs and terminal flows, not arbitrary topologies or physiological redistribution.Source: §4 · p. 5
  • The three-seed Data/Div/Full protocol was frozen after original primitive-test and whole-tree CFD unblinding, so n = 3 variability is descriptive, not inferential.Pathology conditions are not independent replicates.Source: §4 · p. 5
  • The external residual telescopes to the external-port budget and cannot localize error; the assembler enforces neither interface-flux continuity nor global balance.Unmatched hardware also precludes a normalized speedup claim.Source: §4 · p. 5

5 Conclusion

Auxiliary objectives improve primitive scores and divergence, and Full reduces assembled component and interface discrepancies across three seeds, but these gains do not ensure better global fields or conservation.

5 Conclusion

  • Across three seeds, auxiliary objectives improve primitive composite scores and divergence, while Full reduces assembled component-residual and interface-mismatch RMS.These local improvements do not ensure improved global fields or conservation.Source: §5 · p. 5
  • Full raises mean external residual from 17.49% to 26.77%, despite reducing assembled component-residual and interface-mismatch RMS.The conclusion cautions that gains in these local diagnostics do not ensure lower whole-tree field error or imbalance.Source: §5 · p. 5
  • Frozen primitive operators are useful diagnostics, but reliable composition requires conservative interface coupling, global projection, or iterative correction rather than local regularization alone.Source: §5 · p. 5

A Additional material

Reusable flow surrogates could reduce exploratory simulation costs, but this model is not clinically validated and its errors could make unverified clinical or design uses unsafe.

A Additional material

  • Faster reusable flow surrogates could reduce the computational cost of exploratory scientific simulation.Source: A Additional material · p. 8
  • The present model is not clinically validated.Its pressure and conservation errors could lead to unsafe conclusions in diagnosis, treatment planning, or device design without independent CFD or experimental verification.Source: A Additional material · p. 8
  • Independent CFD or experimental verification is needed before using the model for diagnosis, treatment planning, or device design.Source: A Additional material · p. 8

A.1 Primitive dataset construction and splits

The primitive dataset contains geometry–operating-point cases grouped and split by original template, with fixed field sampling and training-only normalization; port supervision is limited to pilot branch cases.

A.1 Primitive dataset construction and splits

  • The 4,872 dataset entries are geometry–operating-point cases, not necessarily distinct geometries, because operating speeds can share an original-template group.Table 4 reports family, source, split, and port-supervision counts as case or original-template-group counts.Source: §A.1 · p. 8, Table 4 · p. 8
  • All cases sharing an original legacy template identifier are assigned to one split, with empty train, validation, and test group intersections for Tube, Y2, and Y3.Pilot assignments were propagated to matching legacy cases, requiring 32 Y2 and 8 Y3 legacy split-label overrides.Source: §A.1 · p. 8
  • Each prepared case stores 1,024 field points; training randomly samples 256 field points and 64 divergence points per case per epoch.Frozen primitive-test evaluation uses all 1,024 field points and 256 divergence points per case.Source: §A.1 · p. 8
  • Branch-feature and target-channel normalization statistics are fitted on training cases only and applied without refitting to validation and test cases.Target statistics use all 1,024 spatial samples of training cases.Source: §A.1 · p. 8
  • Only pilot branch cases provide port supervision: 180 Y2 cases and 144 Y3 cases, while Tube has none.Legacy Y2/Y3 port losses are masked because exact port geometry is unavailable.Source: §A.1 · p. 8
  • Primitive port supervision uses fixed-order arclength midpoint quadrature on mesh-realized CFD caps, while tree inference uses Gauss–Legendre quadrature on exact analytic interfaces.Each procedure uses 32-point quadrature.Source: §A.1 · p. 8

A.2 Loss, score, and metric definitions

The section defines field and auxiliary training losses, a common primitive audit score, and metrics for primitive and assembled-tree evaluation.

A.2 Loss, score, and metric definitions

  • The field loss gives standardized ux, uy, and pressure channels equal pointwise weight, with 256 field points and independently sampled divergence points per training case.Frozen primitive testing uses 1,024 field points and 256 divergence points per case.Source: §A.2 · pp. 8, 10
  • The Full objective combines standardized field loss with divergence, flux, component-balance, and port-pressure terms, ramping auxiliary weights to full strength by epoch 60.Data uses only field loss, Div adds divergence, and Full uses all four auxiliary terms; port losses apply only to pilot Y2/Y3 cases.Source: §A.2 · p. 9
  • The port-pressure loss compares learned outlet-mean pressure with stored primitive-CFD targets, without resistance, inlet-pressure supervision, or resistance-flow decoding.Port losses are restricted to pilot Y2/Y3 cases.Source: §A.2 · p. 9
  • The common primitive audit score weights field and divergence errors and adds flux, balance, and resistance-flow-decoder terms for Y2 and Y3.It is a post-training audit quantity, distinct from the variant-specific validation objectives used to select checkpoints.Source: §A.2 · p. 9
  • Primitive confidence intervals are 95% geometry-group bootstrap intervals that quantify held-out group variability for fixed checkpoints, not training-seed uncertainty or paired-model significance.They use 2,000 source-stratified nonparametric replicates.Source: §A.2 · p. 10
  • Tree pressure metrics use one scalar offset fitted across the complete tree; velocity is neither rescaled nor corrected, and component-wise pressure shifts are prohibited.The evaluation-only resistance-flow decoder is not used for pressure alignment.Source: §A.2 · p. 10

A.3 Whole-tree CFD reference simulations

The whole-tree CFD references solve steady, two-dimensional incompressible Newtonian flow with prescribed inlet and terminal velocities. Mesh refinement and boundary integration provide numerical checks, while the recorded timings and provenance have limits.

A.3 Whole-tree CFD reference simulations

  • The references solve steady two-dimensional incompressible Newtonian Navier–Stokes flow with retained convection and no physical body force, using Taylor–Hood P2/P1 discretization on triangular meshes.Velocity is exported as P1 nodal data; boundary fluxes are integrated from the original finite-element velocity.Source: §A.3 · pp. 10, 12
  • The inlet and all seven terminals use prescribed smooth parabolic velocity profiles, with terminal target flows listed for the three final CFD references.The inlet maximum speed is 0.110 m s^-1, and the prescribed inlet flow is 3.74 × 10^-4 m^2 s^-1.Source: §A.3 · p. 11, Table 6 · p. 11
  • The CFD pressure null space is fixed by setting the P1 pressure degree of freedom nearest the inlet centre to 0 Pa, without a post-solution shift.Tree-wide pressure alignment for scoring is separate; component-wise CFD-fitted shifts are not permitted.Source: §A.3 · p. 11
  • The Picard solve starts from zero, uses under-relaxation α = 0.7, and stops at the stated relative-change criterion or after 100 iterations.The linear configuration uses PETSc preonly with LU factorization, requesting MUMPS when available and otherwise SuperLU_DIST.Source: §A.3 · pp. 11–12
  • L40-to-L45 maximum pressure-drop changes were 0.0001882% for normal, 0.0003814% for stenosis, and 0.0002722% for dilation, below the predefined 1% mesh-independence threshold.This assesses mesh sensitivity, not agreement with an analytical solution.Source: §A.3 · p. 12
  • Finite-element boundary integration gave mass imbalances of 4.50655518 × 10^-7% for normal and 4.50655503 × 10^-7% for both stenosis and dilation.The values come from boundary integration rather than only the analytical target-flow sum.Source: §A.3 · p. 12
  • The three Picard-loop times were 498.52 s, 512.42 s, and 588.23 s for normal, stenosis, and dilation, respectively, but exclude setup and post-processing.Because hardware is unmatched, these timings do not support a hardware-normalized speedup comparison with neural inference.Source: §A.3 · p. 12
  • The records do not retain the CFD CPU model or core count, RAM or peak memory, exact LU backend, or exact integrated per-port final flows.Other unretained items include exact pressure-gauge coordinates, nonlinear iteration counts, total-script times, and solver or mesh hashes.Source: §A.3 · p. 12

A.4 Whole-tree operating-input provenance

The operating inputs were constructed from prescribed outlet and component metadata, with resistance conditioning selected from training-split rows. The support audit found marginal extrapolation only for two stenotic root-tube instances, while joint-support and arbitrary-topology generalization were not claimed.

A.4 Whole-tree operating-input provenance

  • The conditioning inflow is defined using the prescribed terminal outlets downstream of each component.Resistance entries came together from one training-split row selected using standardized geometry and inlet-speed features; Y2 and Y3 matching also used a normalized-conductance split proxy, not a terminal-flow predictor.Source: §A.4 · pp. 12–13
  • The inlet reference was constructed from boundary-condition metadata before any monolithic solution field was opened and was not imposed on predicted velocity.The second identity for the prescribed inflow follows from the two-dimensional parabolic profile and the exact operator-compatible inlet radius.Source: §A.4 · p. 12
  • The support audit found marginal extrapolation for Umax,c only at root_tube_2 and root_tube_3 under stenosis; joint-distribution support and arbitrary-topology generalization were not claimed.The target geometry and Umax,c were retained exactly.Source: §A.4 · p. 13

A.5 Detailed primitive and tree results

The detailed audit preserves the blind Selected evaluation while reporting post-hoc diagnostics, fixed-Tube conservation attribution, and signed whole-tree sums. Local discrepancy magnitudes and the external boundary budget are distinct diagnostics, and a smaller external residual does not imply uniformly smaller local errors.

A.5 Detailed primitive and tree results

  • The original Selected evaluation remained unchanged: reconstructed blends reproduced aggregate metrics and bootstrap intervals exactly, and reconstructed tree arrays were bitwise identical.The post-hoc Data, Div, and Full diagnostics did not alter checkpoints, blend weights, selection, or hyperparameters.Source: §A.5 · p. 13
  • The three-seed sensitivity audit froze seeds, configurations, validation-only selection, evaluation order, and outputs before training the additional seeds; controlled tree conditions were not statistical replicates.Table 9 intervals quantify held-out geometry-group variability for fixed checkpoints, not training-seed uncertainty or paired model-difference significance.Source: §A.5 · p. 13, Table 9 · p. 14
  • The controlled pathology figures compare Selected with CFD for stenosis-related maximum-velocity change and lesion-pressure changes under stenosis and dilation; dilation pressure-response markers are nearly coincident.Per-condition whole-tree metrics cover velocity-vector RelL2, speed RelL2, pressure RelL2, and pressure MAE for normal, stenosis, and dilation conditions.Source: Table 10 · p. 14, Figure A3 · p. 15, Table 11 · p. 15

A.6 Fixed-Tube and branching-only conservation attribution

  • The conservation decomposition separates branch-component, branch–branch interface, and external-boundary terms from fixed-Tube contributions and the condition-dependent mixed Tube–Y2 interface.Table 12 reports the decomposition; the secondary attribution uses seed-2026 arrays, whereas the three-seed robustness result uses all nine components and eight interfaces.Source: §A.6 · pp. 16–17, Table 12 · p. 17
  • The fixed Tube checkpoint makes Tube residuals identical across branching variants, and pathology-dependent signed shifts arise from the three shared Tube instances.Internal port terms cancel between component and interface sums; the external residual does not localize internal error and has no pathology signal here because external-port flux vectors do not change across conditions.Source: §A.6 · pp. 16, 18

A.7 Signed whole-tree sums and alternative normalization

  • RMS values quantify unsigned local discrepancy magnitude, while signed sums reconstruct the external residual; Data’s smaller external residual does not imply uniformly smaller local errors.The identity Scomp − Sedge = Sext holds up to numerical precision.Source: §A.7 · p. 18, Figure A7 · p. 19, Table 14 · p. 19

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