AneumoBench: A Source-Linked Benchmark for Synthetic-Geometry Transfer in Aneurysm CFD

Xigui Li; Yuanye Zhou; Feiyang Xiao; Xin Guo; Chen Jiang; Tan Pan; Xingmeng Zhang; Cenyu Liu

eess.IVMay 19, 2025

AneumoBench: A Source-Linked Benchmark for Synthetic-Geometry Transfer in Aneurysm CFD

Authors: Xigui Li, Yuanye Zhou, Feiyang Xiao, Xin Guo, Chen Jiang, Tan Pan, Xingmeng Zhang, Cenyu Liu, +11 more

Organizations: Artificial Intelligence Innovation and Incubation Institute, Fudan University, Shanghai, China · Shanghai Academy of Artificial Intelligence for Science, Shanghai, China · Human Phenome Institute, Fudan University, Shanghai, China · Huashan Hospital, Fudan University, Shanghai, China

Abstract

Scientific machine learning uses simulation data to train surrogate models for fast physical-field prediction across geometries. Local shape editing can expand limited geometry collections, but whether its variants improve prediction on unseen geometries, and how to allocate them across sources, require controlled evaluation. We introduce AneumoBench, a dataset and benchmark linking 401 source aneurysm geometries to 9,693 locally edited descendant records, with computational fluid dynamics (CFD) fields computed on both. It contains 80,752 steady velocity-pressure cases across eight inlet conditions and 9,715 transient sequences of velocity, pressure, and wall shear stress (WSS). Each sequence contains 100 frames sampled at 0.01-s intervals from a 1-s cardiac cycle. Mesh, point, and voxel interfaces support steady field prediction and WSS forecasting from four observed frames. With family-disjoint splits, we compare source-only training, descendant training, and descendant pretraining followed by source fine-tuning across nine architectures on 79 held-out sources. Under the reported schedules, two-stage training lowers steady-field and reset-window WSS errors relative to source-only training. With the number of sampled fields and training updates fixed within each comparison, GraphSAGE benefits from descendant training and from distributing a fixed number of descendants across more sources. For WSS, reset-window gains do not consistently persist through 96-step rollout, and lower trajectory error need not improve cycle-level shear metrics or hotspot localization. These data and protocols enable researchers to compare descendant selection and training strategies on the same unseen source geometries.

Figures & tables

Figure 1: AneumoBench corpus and benchmark. (A) Family-disjoint splits contain 401 sources and 9,693 descendants (280/42/79 sources). (B) Steady cases provide velocity and pressure at eight inlet conditions; transient sequences provide 100 frames of velocity, pressure, and WSS. Mesh, point, and voxel interfaces are available. (C) R1/R2/R3 compare source, descendant, and descendant-to-source training; GraphSAGE controls fix sampling and update counts. (D) Source-balanced WSS evaluation uses $4\!\to\!4$ reset and $4\!\to\!96$ rollout; the airfoil extension uses 47 held-out sources.

Resource	Data structure	Evaluation setting	Time dependence
AneuG-Flow	Generated aneurysm geometries	Geometry-to-field learning	Steady and pulsatile
AirfRANS	Airfoils and operating conditions	Data scarcity and extrapolation	Steady
APEBench	PDE trajectories	Training and rollout	Time-dependent
RealPDEBench	Simulation–measurement pairs	Simulation-to-measurement transfer	Time-dependent
AneumoBench	Source-linked geometry families	Unseen-source prediction and training-pool comparisons	Steady and pulsatile

Table 1: Data relationships and evaluation settings in selected scientific-ML resources. Appendix Table 9 details aneurysm-resource coverage and protocol records.

Figure 2: Geometry changes relative to each source. Each family contributes one median residual per descriptor. (a) Distributions of these residuals: positive values indicate larger descriptor values in descendants. Residuals use source-IQR scaling and a sign-log transform. (b) Spearman correlations of these family summaries; stars mark FDR-adjusted significance. (c) The residuals standardized across 401 families for each descriptor, showing departures from the mean family response. Color and bubble size encode signed and absolute z-scores. Families are ordered by mean z-score; Appendix A.5 gives the definitions.

	Steady velocity			Stored-gauge pressure			Reset-window WSS
Model	R1	R2	R3	R1	R2	R3	R1	R2	R3
Unstructured sampled points
GraphSAGE	0.8269	0.6812	0.6521	0.7131	0.5480	0.5525	0.3571	0.2561	0.2235
Transolver	1.0031	0.7329	0.6979	0.8672	0.5808	0.5783	0.1253	0.1092	0.1084
Galerkin Transformer	1.0033	0.8477	0.8150	0.8682	0.6417	0.6667	0.1210	0.1094	0.1090
PointNet	0.9859	0.9371	0.8899	0.8489	0.7732	0.7718	0.1889	0.1481	0.1484

Table 2: RelL2 for R1–R3 on 79 held-out sources under the reported training schedules. Each score uses one validation-selected checkpoint. Steady scores use sampled vertices from the original export arrays or the constructed fluid-voxel masks; Appendix H.4 separately reevaluates six GraphSAGE allocation checkpoints on cell-connected queries. Source-bootstrap intervals describe held-out-source variation. Lower is better; bold and underlined values mark the best and second-best recipes for each model–endpoint pair.

Panel of Figure 3 — Figure 3: GraphSAGE pool-size and allocation controls on 79 held-out sources. Small dots show individual seeds; connected markers and error bars give means and sample SD across seeds 42–44. Both panels use the same absolute RelL2 scale (lower is better); pressure uses the stored gauge. (a) Accessible-pool expansion. (b) Allocation of 280 descendants, followed by source fine-tuning. Within each panel, runs use equal numbers of sampled fields and training updates. Zero-velocity prediction has RelL2 approximately one. Complete values are in Appendix Tables 20 and 22 .

Model	Source-only	$280\!\times\!1$	$140\!\times\!2$	$70\!\times\!4$	$35\!\times\!8$
GraphSAGE	$0.0482$ $\pm 0.0005$	$\mathbf{0.0450}$ $\mathbf{\pm 0.0001}$	$0.0463$ $\pm 0.0012$	$0.0453$ $\pm 0.0017$	$0.0454$ $\pm 0.0009$
Transolver	$0.0585$ $\pm 0.0115$	$0.0623$ $\pm 0.0028$	$0.0541$ $\pm 0.0129$	$0.0581$ $\pm 0.0089$	$\mathbf{0.0516}$ $\mathbf{\pm 0.0086}$

Table 3: Airfoil pressure-coefficient RelL2 on 47 sources: mean

\pm

sample SD across three full-pipeline seeds. The four 280-descendant allocations share a training budget; Source-only is a separately trained reference recipe. Bold marks the lowest mean within each model.

Model	Reset window			96-step rollout			Pattern
Model	R1	R3	$\Delta$	R1	R3	$\Delta$	Pattern
Transolver	0.128	0.109	$-14.9\%$	1.279	0.966	$-24.5\%$	Both improve
Voxel U-Net	0.247	0.208	$-15.6\%$	0.954	0.956	$+0.2\%$	Reset gain; rollout mixed
PointwiseMLP	0.146	–	–	0.735	–	–	Local-history reference
Persistence	0.258	–	–	0.881	–	–	Training-free

Table 4: Mean reset-window and recursive-rollout RelL2 across three training seeds.

Endpoint	R1 initialization	R3 initialization
96-step rollout RelL2	$0.46840\pm 0.01369$	$0.37076\pm 0.00390$
TAWSS relative error	$0.32239\pm 0.02310$	$0.21591\pm 0.00458$
OSI absolute error	$0.02747\pm 0.00025$	$0.02836\pm 0.00013$

Table 5: WSS endpoints after identical eight-epoch

F=16

feedback adaptation of fixed Transolver R1/R3 checkpoints. Entries are means

\pm

sample SD over three adaptation seeds on 79 sources. Lower is better. Appendix Table 28 reports all endpoints.

Appendix figures & tables49 assets

Supplementary material from the paper’s appendix.

Appendix

Release unit	Train	Validation	Test	Total	Evaluation treatment
Source geometries	280 (69.8%)	42 (10.5%)	79 (19.7%)	401	Split unit
Descendant records, eligible	6,687 (69.0%)	1,055 (10.9%)	1,951 (20.1%)	9,693	Inherit source split
Distinct wall-coordinate arrays	6,638	1,052	1,943	9,633	Coordinate-array equality
Descendant transient, available	6,438 (69.1%)	1,001 (10.7%)	1,878 (20.2%)	9,317	Readable 101-entry HDF5
Descendant transient, unavailable	249	54	73	376	Tracked in manifest; excluded from transient use
Source transient, archived/readable	280	42	79	401	Three train-only exclusions below

Appendix

Table 6: Cohort ledger. “Eligible” records belong to the construction manifest and have a retained source; “available” records have a readable modality that passes structural validation. Counts refer to the unit named in each row; source geometries are counted in the first row. Parentheses give each split’s share of the row total.

Population	Operational transition	Removed	Retained
Source acquisition inventory	Locally archived source surfaces	–	427
Sources	$<100$ transient outputs (25) or no complete eight-flow steady set (1)	26	401
Raw descendant inventory	All records; every record has a source mapping	–	10,660
Descendants	Incomplete transient index (330) or missing eight-flow steady set (7), overlap 2	335	10,325
Descendants	Otherwise complete but linked to one of the 26 excluded sources	632	9,693
Descendants	Missing conversion source files	26	9,667

Appendix

Table 7: Release-cohort accounting for the archived 427-surface acquisition inventory. The archive lacks a record-level mapping to the complete 668-record AneuX collection. Counts in overlapping exclusion categories are unioned before subtraction.

Figure 4: Descendant coverage for the 401-source/9,693-descendant manifest. (a) Eligible descendant counts and (b) transient availability within each lineage, grouped by inherited split. Each point represents a source lineage; boxes show the interquartile range and median, and labels give split medians. The manifest retains the source identifiers of the 376 unavailable descendant sequences.

Figure 5: Geometry differences between the 376 descendants without a transient HDF5 file and the 9,317 available descendants. Standardized mean differences and Kolmogorov–Smirnov (KS) statistics compare eight rotation-invariant descriptors and source–descendant Chamfer distance. A class-balanced logistic model, evaluated by five-fold GroupKFold over lineages, achieves an out-of-fold area under the receiver operating characteristic curve (AUC) of 0.563. The recorded descriptors therefore predict availability only weakly.

Audit quantity	Population / comparison	Result
Surface variation	Source–descendant ( $n=9,693$ )	0.0519 [0.0421–0.0625]
Surface variation	Random source–source pairs ( $n=20,000$ )	0.0613 [0.0533–0.0743]
Descriptor proximity	Nearest sibling	0.2056 [0.1481–0.2777]
Descriptor proximity	Nearest non-sibling	0.2614 [0.2045–0.3343]
Exact coordinate arrays	Groups / affected / excess records	47 / 107 / 60
Distinct coordinate arrays	One canonical record per exact group	9,633

Appendix

Table 8: Geometric coverage and redundancy audit for the release cohort. Surface distances use 512-point deterministic samples after PCA canonicalization and are normalized by the mean wall bounding-box diagonal. Descriptor distances use median–IQR scaling fitted only on the 401 sources. Values are median [interquartile range]. Duplicate groups use coordinate-array equality; the near-neighbor rows report descriptor distances.

	Cohort size		CFD coverage		Benchmark protocol
Resource	Source units	Synthetic records	Steady	Pulsatile cases (frames)	Linked lineage	Inherited split	Matched exposure	Recursive rollout
Aneurisk	103	–	–	–	n/a	n/a	n/a	–
CMHA	143	–	–	–	n/a	n/a	n/a	–
AneuG-Flow	–	14,000	14,000	730 (80)	–	–	–	–
AneumoBench	401	9,693	80,752	9,715 (100)	Yes	Yes	Yes	Yes

Appendix

Table 9: Comparison with aneurysm resources providing hemodynamic annotations.

Figure 6: CFD domain construction. (a) Inlet, outlet, and wall boundaries on a representative surface mesh. (b) Cross-section of the internal volume mesh and its wall-adjacent boundary-layer cells.

Panel of Figure 7 — Figure 7: Mesh-resolution comparison and steady-solver residuals. (a) Velocity profiles for one case at five mesh resolutions, used to inform the 0.15-mm production setting. (b) Velocity-component and pressure residuals over iterations for a representative steady case.

Quantity	Stored solver form	Released/evaluated form	Domain
Coordinates	Geometry source units standardized to metres	$x,y,z$ in m; optional unit-bbox copy for intervention	Volume or wall vertices
Velocity	$\mathbf{u}$	$\mathrm{m\,s^{-1}}$	Internal volume mesh
Pressure	$p_{k}=p/\rho$	$p=\rho p_{k}$ in Pa	Internal volume mesh
Wall shear stress	Kinematic wallShearStress vector	$\bm{\tau}_{w}$ in Pa	Wall surface mesh
TAWSS	Not stored as a target	Cycle mean of $\\|\bm{\tau}_{w}(t)\\|_{2}$ in Pa	Wall surface vertices
OSI	Not stored as a target	Dimensionless vector-cycle index in $[0,0.5]$	Wall surface vertices

Appendix

Table 10: Field and coordinate conventions used by the release and evaluator.

Diagnostic	Source geometries	$n$	Descendants	$n$
Mean Reynolds number	608 [479, 748]	398	602 [477, 743]	9,317
Womersley $\alpha$	2.01 [1.63, 2.55]	398	2.03 [1.65, 2.56]	9,317
Peak velocity ( $\mathrm{m\,s^{-1}}$ )	3.53 [2.7, 5.02]	398	3.71 [2.75, 5.26]	9,317
WSS p99 (Pa)	146 [83.1, 249]	398	164 [93.4, 290]	9,317
Mean TAWSS (Pa)	17 [10.3, 27.8]	398	21.8 [13.2, 37.4]	9,317
Mean OSI	0.0224 [0.0144, 0.0356]	398	0.0182 [0.0125, 0.0274]	9,317

Appendix

Table 11: Transient CFD screening summaries. RRT denotes relative residence time. Values are median [interquartile range] across included records with a finite summary;

n

gives the case count for each diagnostic. Three divergent source sequences are excluded.

Figure 8: Empirical distributions for the audited transient cohort, including dimensionless flow descriptors, field magnitudes, and solver balance. The source curve contains 398 geometries and the descendant curve 9,317 available variants. Positive heavy-tailed quantities use logarithmic horizontal axes.

Figure 9: Ground-truth WSS for eight source geometries selected at approximately even intervals from the official test list, shown at four cardiac phases. All 32 panels use the released polygonal wall meshes and a shared color scale. Sources are arranged in two groups of four, with the same phase in corresponding rows. The gallery shows how the location and persistence of high-shear regions vary across source geometries.

Figure 10: Ground-truth steady pressure and speed for eight approximately evenly spaced held-out source geometries at $0.003\,\mathrm{kg\,s^{-1}}$ . Each column uses a deterministic geometry-centered PCA plane and the original VTU triangulation. Shared row colorbars show variation in the fields across geometries.

Data form	Principal contents	Benchmark role	Audited accounting
STL surface	Vascular lumen boundary	Geometry/provenance	401 sources + 9,693 eligible descendant records
NIfTI mask	Binary voxelized lumen	Auxiliary binary geometry mask	Linked by geometry manifest
CFD mesh / VTU/VTK	Volume grid, boundary and field exports	Mesh-resolved CFD outputs and provenance	Availability recorded per modality
Steady HDF5	Coordinates, boundary/flow features, $\mathbf{u}$ , $p$	Eight-flow field prediction	3,208 source + 77,544 descendant files
Transient HDF5	Volume mesh, $\mathbf{u}$ , $p$ , boundary meshes, WSS, time	WSS 4 $\to$ 4 and 4 $\to$ 96 forecasting; volume fields retained	401 archived source (398 usable) + 9,317 descendants

Appendix

Table 12: Release representation schema. Counts distinguish geometries from flow-condition files and transient sequences.

Task	Data form	Model input	Target	Within-case sampling	Primary endpoint	Extended diagnostics
Steady field	Point / unstructured	$xyz$ , boundary type, wall distance, mass flow	$\mathbf{u},p$	20,000 sampled volume points; fixed diagnostic subset	Source RelL2	MSE, RMSE, MAE, $R^{2}$ , direction
Steady field	Voxel $64^{3}$	Occupancy, query coordinates, mass flow	$\mathbf{u},p$	Complete constructed fluid mask	Source RelL2	Same physical-unit metrics
Transient reset	Point / surface	4 WSS frames, wall query, phase	Next 4 WSS frames	4,096 sampled wall points; fixed diagnostic subset; stride 5	Source-window RelL2	RMSE, magnitude and angle
Transient reset	Voxel $64^{3}$	Occupancy grid, wall queries, 4 WSS frames, phase	Next 4 WSS frames	Same fixed wall subset	Source-window RelL2	Same physical-unit metrics
Transient rollout	Point or voxel	Geometry, phase, and first 4 WSS frames	Next 96 frames	No ground-truth reset	Mean $E_{i}^{\mathrm{roll}}$	Mean cumulative error, TAWSS, OSI, top-10%, peak time/magnitude

Appendix

Table 13: Benchmark task card. The evaluation unit remains the source geometry even when a task contains multiple flow conditions, windows, frames, or points. Extended diagnostics include mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE).

Stage	Input	Output	Recorded checks
Surface repair	Public-source triangular surface	Watertight repaired source surface	Topology defects, source identifier
Local deformation	Editing surface after sac removal and selected wall patch	Synthetic descendant surface	Source–descendant mapping and fixed output surface
Voxelization	Descendant or source STL	Binary NIfTI lumen mask	Geometry identifier, foreground/background convention
CFD preparation	Repaired STL	Named volume mesh and boundary patches	ANSYS 2026 R1; inlet/outlet/wall labels and mesh settings
Steady solve	Mesh and one mass-flow condition	$\mathbf{u},p$ volume fields	OpenFOAM v2312 icoFoam ; PISO; residual criterion and complete case configuration archived
Transient solve	Mesh and periodic waveform	101-entry raw field sequence	OpenFOAM v2312 pimpleFoam ; $\Delta t=10^{-4}$ s; PIMPLE/convergence logs and complete case configuration archived

Appendix

Table 14: Stage-wise provenance for the released representations. Source and geometry identifiers are preserved across every stage.

Channel	Mean	Standard deviation
Steady source
2,240 training flow cases; 514,573,832 points
$u_{x}$	$0.00069$	$0.17669$
$u_{y}$	$0.01249$	$0.16057$
$u_{z}$	$0.01157$	$0.17094$
$p$	$169.49$	$266.63$

Appendix

Table 15: Output normalization statistics in channel order:

u_{x},u_{y},u_{z},p

for the steady task and the three WSS components for the transient task. Velocity is measured in

\mathrm{m\,s^{-1}}

; pressure and WSS are in Pa.

Model	Steady	Transient
Unstructured sampled points
GraphSAGE	0.37M	0.37M
Transolver	3.86M	3.87M
Galerkin Transformer	1.10M	1.10M
PointNet	0.23M	0.23M
Regular voxel geometry ( $64^{3}$ )

Appendix

Table 16: Trainable parameter counts for completed R3 runs in each task, as recorded in the training logs. M denotes millions.

Model	Width	Depth	Heads	Batch (steady / transient)	Geometry/operator setting
PointwiseMLP (aux.)	128	4	–	– / 32	Shared residual multilayer perceptron; no point interaction
PointNet	16	8	–	32 / 64	Point-set multilayer perceptron
GraphSAGE	128	8	–	32 / 32	16-nearest-neighbor graph
Transolver	256	8	8	16 / 16	32 physics-attention slices
Galerkin Transformer	128	8	8	16 / 32	Galerkin attention
U-Net	64	4	–	32 / 16	$64^{3}$ voxel grid

Appendix

Table 17: Architecture and batch configurations. PointwiseMLP is an auxiliary transient diagnostic; the remaining rows form the nine-architecture reference suite. Width and depth follow the released implementations.

Regime	Training data	Initial weights	Steady epochs	Transient epochs	Learning rate	Selection / evaluation
R1	Source geometries	Random	200	200	$10^{-3}$	Minimum validation endpoint; held-out sources
R2	Synthetic descendants of training sources	Random	200	50	$10^{-3}$	Source-validation selection; common source test
R3	Source training geometries	Best R2 checkpoint	100	100	$10^{-4}$	Minimum validation endpoint; same test cohort

Appendix

Table 18: Optimization schedules for the core R1–R3 comparison. Table 17 lists batch sizes, widths, and key operator settings; the released configurations and logs give the full settings.

Figure 11: Paired comparisons of the three training settings across nine reference architectures. (a) Cells report relative error changes for R2 versus R1, R3 versus R2, and R3 versus R1; negative values denote lower error. Bold R3–R1 values have paired-source 95% bootstrap intervals excluding zero. The upper four rows are point/unstructured architectures and the lower five are voxel architectures. (b) Cells give the percentage of the same 79 source geometries improved by R3 relative to R1; darker fill denotes broader consistency. Figure 3 presents the GraphSAGE pool-size and allocation controls.

Model	Representation	Field	RelL2	Physical-unit RMSE
PointNet	Point set	Velocity	0.891 $\pm$ 0.001	0.12606 $\pm$ 0.00009
		Pressure	0.773 $\pm$ 0.003	130.7 $\pm$ 0.2
GraphSAGE	16-NN graph	Velocity	0.654 $\pm$ 0.001	0.09486 $\pm$ 0.00001
		Pressure	0.560 $\pm$ 0.006	96.4 $\pm$ 0.3
U-Net	$64^{3}$ voxel	Velocity	0.704 $\pm$ 0.002	0.09795 $\pm$ 0.00010
		Pressure	0.795 $\pm$ 0.012	125.4 $\pm$ 0.2

Appendix

Table 19: Steady fine-tuning variability for three representative architectures. Each architecture uses one fixed seed-42 descendant-pretraining checkpoint and source fine-tuning seeds 42–44. Entries are held-out-source means

\pm

sample SD across these seeds, conditional on the shared pretrained weights. RMSE is computed from decoded fields and reported in

\mathrm{m\,s^{-1}}

for velocity and Pa for pressure. Lower is better.

Figure 12: Source-level R3 steady relative-

\ell_{2}

distributions for the nine reference architectures. Blue denotes unstructured sampled-point architectures and orange voxel architectures. Each source contributes once after averaging its eight mass-flow conditions; boxes show quartiles and whiskers the 5th–95th percentiles.

Figure 13: Steady-field comparison on a common plane through the 410,202-vertex, 137,745-cell internal VTU for held-out real_0072 at $0.003\,\mathrm{kg\,s^{-1}}$ . Rows show pressure in Pa, absolute pressure error, speed, and absolute speed error. The mostly-polyhedral mesh is tetrahedralized before slicing; all panels use the same plane triangulation and share a colorbar within each row. Point-model outputs are evaluated directly at 20,000 deterministic vertices and nearest-neighbor interpolated elsewhere for display. Voxel outputs are sampled trilinearly from the $64^{3}$ grid. Cohort-level metrics appear in Tables 35 and 36 .

Training pool	Records	Velocity RelL2	$\Delta_{\mathrm{rel}}$	Pressure RelL2	$\Delta_{\mathrm{rel}}$	Wins
Source geometries	280	0.830 $\pm$ 0.005	–	0.731 $\pm$ 0.027	–	–
Descendant $k{=}1$	280	0.815 $\pm$ 0.002	-1.7%	0.706 $\pm$ 0.012	-3.3%	3/3
Descendant $k{=}2$	560	0.785 $\pm$ 0.002	-5.4%	0.680 $\pm$ 0.013	-6.8%	3/3
Descendant $k{=}4$	1,120	0.762 $\pm$ 0.003	-8.2%	0.663 $\pm$ 0.003	-9.2%	3/3
Descendant $k{\leq}8$	2,238	0.739 $\pm$ 0.002	-11.0%	0.630 $\pm$ 0.017	-13.8%	3/3
Descendant $k{\leq}16$	4,422	0.738 $\pm$ 0.006	-11.1%	0.635 $\pm$ 0.004	-13.0%	3/3

Appendix

Table 20: GraphSAGE pool-size comparison on 79 held-out source geometries. All branches use 280 source lineages, 2,240 fields per epoch, 28,000 updates, and identical deterministic test samples. Values are source-level means

\pm

training-seed SD across three seeds.

\Delta_{\mathrm{rel}}

is the paired relative change from source-only; “Wins” counts seeds improved on both fields. Pool membership is fixed across seeds at every size.

	Velocity RelL2			Pressure RelL2
Seed	Source only	Descendant $k{=}4$	Paired $\Delta_{\mathrm{rel}}$	Source only	Descendant $k{=}4$	Paired $\Delta_{\mathrm{rel}}$
42	0.8263	0.7649	-7.4%	0.7008	0.6607	-5.7%
43	0.8274	0.7593	-8.2%	0.7529	0.6660	-11.6%
44	0.8361	0.7616	-8.9%	0.7383	0.6615	-10.4%
Mean (SD)	0.830 (0.005)	0.762 (0.003)	-8.2 (0.7)%	0.731 (0.027)	0.663 (0.003)	-9.2 (3.1)%

Appendix

Table 21: Seed-level GraphSAGE

k{=}4

control on held-out source geometries. Both branches cover the same 280 source families and use equal numbers of sampled fields and updates, with the same normalization, validation, and deterministic test samples. The synthetic

k=4

membership is fixed across seeds. Metrics average eight flows per source; the final row reports mean (sample SD). Negative paired changes favor synthetic training.

	Velocity RelL2			Pressure RelL2
Allocation	Mean $\pm$ SD	$\Delta$	Wins	Mean $\pm$ SD	$\Delta$	Wins
$280\!\times\!1$	0.799 $\pm$ 0.019	–	–	0.717 $\pm$ 0.032	–	–
$140\!\times\!2$	0.818 $\pm$ 0.012	+0.018	0/3	0.743 $\pm$ 0.027	+0.026	0/3
$70\!\times\!4$	0.821 $\pm$ 0.014	+0.022	1/3	0.743 $\pm$ 0.029	+0.026	1/3
$35\!\times\!8$	0.848 $\pm$ 0.022	+0.049	0/3	0.786 $\pm$ 0.040	+0.069	1/3

Appendix

Table 22: GraphSAGE fixed-budget lineage breadth–depth allocation using 280 descendant records on the same 79 held-out source geometries. All branches use 2,240 stage-1 fields per epoch, 28,000 stage-1 updates, and the same 14,000-update source fine-tuning stage. Values are mean

\pm

sample SD across seeds 42–44.

\Delta

is the mean paired absolute change from

280\!\times\!1

; “wins” counts seeds with lower error than that reference. Lower is better.

Allocation	Stored-gauge RelL2	Centered-pressure RelL2	Centered-pressure RMSE (Pa)	Pressure-drop relative error
$280\!\times\!1$	0.717	1.082	96.47	0.514
$140\!\times\!2$	0.743	1.117	99.07	0.505
$70\!\times\!4$	0.743	1.127	98.60	0.546
$35\!\times\!8$	0.786	1.174	101.94	0.650

Appendix

Table 23: GraphSAGE pressure-structure diagnostics for the fixed-280-descendant allocation control. Entries average source-balanced endpoints across seeds 42–44. Each checkpoint uses the same uniformly weighted fixed point samples as its primary allocation evaluation. Centered-pressure metrics subtract each field’s spatial mean; a spatially constant pressure prediction has centered-pressure RelL2 approximately one. Lower is better.

Figure 14: Cross-domain protocol transfer on source-edited airfoils. (a) The first training lineage in the frozen manifest: source airfoil ag03 and its first four accepted CST descendants. The source surface is colored by its XFoil pressure coefficient at the declared flow condition; gray lines show descendant outlines. Separate $x/c$ and $y/c$ axis scales make the local edits visible. (b) GraphSAGE with broad descendant pretraining versus source-only training. (c) Transolver with the same 280-descendant budget allocated across 280 or 35 source lineages. Gray paths connect the same full-pipeline seed across conditions; colored diamonds and whiskers show mean $\pm$ training-seed SD for seeds 42–44. Both result panels use the same 47 held-out source airfoils; lower is better.

Method	Window RelL2	Rollout RelL2	Mean cumulative RelL2
Copy-last persistence	0.2580 [0.2482, 0.2679]	0.8806 [0.8785, 0.8828]	0.7389 [0.7370, 0.7407]
Linear extrapolation	0.3426 [0.3229, 0.3618]	0.9929 [0.9802, 1.0135]	0.8040 [0.7962, 0.8152]
History+phase ridge, no coordinates	0.2706 [0.2613, 0.2810]	0.9911 [0.9888, 0.9937]	0.8624 [0.8582, 0.8668]
PointwiseMLP R1, seed 42	0.1463 [0.1341, 0.1596]	0.7797 [0.7614, 0.7986]	0.6446 [0.6282, 0.6612]
PointwiseMLP R1, 3-seed mean $\pm$ SD	$0.1459\pm 0.0022$	$0.7346\pm 0.1047$	$0.6013\pm 0.0869$

Appendix

Table 24: Temporal reference baselines and an exploratory PointwiseMLP diagnostic. Persistence and linear extrapolation require no training-data fit; history+phase ridge regression is fit in closed form on source-training data. Bracketed entries give source means and 95% source-bootstrap CIs over the 79-source test set. The final row reports mean

\pm

training-seed SD over three independently trained and validation-selected PointwiseMLP pipelines.

Model	Regime	Reset RelL2	Rollout RelL2	TAWSS rel.	OSI abs.	Top-10% overlap
Transolver	R1	0.1253	1.5907	1.2793	0.0383	0.7898
	R2	0.1092	0.9810	0.8873	0.0602	0.8211
	R3	0.1084	0.9836	0.8785	0.0569	0.8240
Voxel U-Net	R1	0.2369	0.9118	0.7673	0.0461	0.6128
	R2	0.2216	0.9450	0.8296	0.0380	0.4118
	R3	0.2064	0.9824	0.8740	0.0381	0.6117

Appendix

Table 25: R1–R3 checkpoints under reset-window and 96-step WSS evaluation on the same 79 held-out sources, using seed 42. Lower is better except top-10% overlap. Panel (b) reports paired-source rollout changes; negative values favor the later regime. Table 26 reports independent pipeline repeats across training seeds.

Model	Seed	R1 reset	R3 reset	$\Delta$	R1 rollout	R3 rollout	$\Delta$
Transolver	42	0.1253	0.1084	-0.0169	1.5907	0.9836	-0.6071
	43	0.1272	0.1091	-0.0181	1.1231	0.9467	-0.1764
	44	0.1305	0.1083	-0.0222	1.1219	0.9673	-0.1546
	Mean $\pm$ SD	$0.1276$ $\pm 0.0026$	$0.1086$ $\pm 0.0004$	$-0.0190$ $\pm 0.0028$	$1.2786$ $\pm 0.2703$	$0.9659$ $\pm 0.0185$	$-0.3127$ $\pm 0.2552$
Voxel U-Net	42	0.2369	0.2064	-0.0304	0.9118	0.9824	+0.0707
	43	0.2469	0.2098	-0.0371	0.9835	0.9509	-0.0326

Appendix

Table 26: Independent full-pipeline replications for transient transfer. Each seed trains R1 from random initialization, trains its own R2 checkpoint, and initializes R3 from that checkpoint. Panel (a) reports vertex-sampled RelL2;

\Delta

is paired within seed. The final rows give mean

\pm

sample SD across three training seeds. Panel (b) reports cycle endpoints as mean

\pm

sample SD. All evaluations use the same 79 sources. R2 rollout was not repeated and remains in Table 25 as a seed-42 diagnostic.

Figure 15: Transolver rollout accuracy after 30 epochs of adaptation from one shared R3 checkpoint. Each training window predicts

F

frames, with either observed or predicted frames appended to the history. Small dots are individual adaptation seeds; markers show means with sample SD. Table 27 gives the complete metrics.

$F$	Feedback	Reset $4\!\rightarrow\!4$	96-step rollout	TAWSS rel.	OSI abs.	Top-10% overlap
4	Teacher forcing	$0.1087$ $\pm 0.0001$	$1.0387$ $\pm 0.0747$	$0.9245$ $\pm 0.0693$	$0.0561$ $\pm 0.0002$	$0.816$ $\pm 0.003$
4	Full feedback	0.1073 $\mathbf{\pm 0.0001}$	$0.9640$ $\pm 0.0064$	$0.8809$ $\pm 0.0126$	$0.0422$ $\pm 0.0005$	0.830 $\mathbf{\pm 0.003}$
8	Teacher forcing	$0.1088$ $\pm 0.0001$	$0.6096$ $\pm 0.1286$	$0.4263$ $\pm 0.1264$	0.0273 $\mathbf{\pm 0.0003}$	$0.794$ $\pm 0.007$
8	Full feedback	$0.1078$ $\pm 0.0003$	$0.3990$ $\pm 0.0035$	$0.2416$ $\pm 0.0093$	$0.0282$ $\pm 0.0000$	$0.827$ $\pm 0.001$
16	Teacher forcing	$0.1090$ $\pm 0.0001$	$0.5431$ $\pm 0.0109$	$0.3573$ $\pm 0.0148$	$0.0281$ $\pm 0.0003$	$0.802$ $\pm 0.004$
16	Full feedback	$0.1084$ $\pm 0.0001$	0.3673 $\mathbf{\pm 0.0025}$	0.2088 $\mathbf{\pm 0.0017}$	$0.0283$ $\pm 0.0001$	$0.830$ $\pm 0.001$

Appendix

Table 27: Three-seed Transolver horizon–feedback factorial from one shared R3 initialization. All settings use 30 epochs. Entries are mean

\pm

adaptation-seed sample SD. Lower is better except top-10% overlap.

Endpoint	R1 initialization	R3 initialization	Difference
Reset RelL2	$0.12811\pm 0.00060$	$0.10826\pm 0.00004$	$-0.01986$
96-step rollout RelL2	$0.46840\pm 0.01369$	$0.37076\pm 0.00390$	$-0.09764$
Mean cumulative RelL2	$0.38256\pm 0.00826$	$0.30933\pm 0.00297$	$-0.07323$
TAWSS relative error	$0.32239\pm 0.02310$	$0.21591\pm 0.00458$	$-0.10649$
OSI absolute error	$0.02747\pm 0.00025$	$0.02836\pm 0.00013$	$+0.00089$
Top-10% overlap	$0.83226\pm 0.00015$	$0.83737\pm 0.00027$	$+0.00512$

Appendix

Table 28: Transolver initialization comparison after identical eight-epoch

F=16

full-feedback adaptation. Entries are source-level means

\pm

sample SD over three adaptation seeds, conditional on one fixed upstream checkpoint per initialization. The final column is R3 minus R1. Lower is better except hotspot overlap.

Method	Rollout RelL2	Mean cumulative RelL2	TAWSS rel.	OSI abs.
Persistence	0.8806	0.7389	0.7254	0.0298
Linear extrapolation	0.9929	0.8040	0.8349	0.1482
History+phase ridge	0.9911	0.8624	0.9045	0.0696
PointwiseMLP R1	0.7346	0.6013	0.5694	0.0287
U-Net R3	0.9824	0.8171	0.8740	0.0381
U-Net SS(0.5)	0.9711	0.8084	0.8670	0.0364

Appendix

Table 29: Exploratory full-cycle transient diagnostics. Lower is better except top-10% overlap; peak timing uses circular frame distance. PointwiseMLP entries average three independently trained R1 pipelines. F=16 entries average three adaptation seeds from one fixed R3 initialization: 30 epochs for Transolver and 20 for U-Net. The eight-epoch initialization comparison is in Table 28 . The remaining neural-model rows use single validation-selected checkpoints.

Figure 16: Exploratory 96-step rollout diagnostics aligned with the cardiac phase. (a) Mean ground-truth frame change peaks near predicted step 19. (b) Source-mean cumulative error for four reference methods over the same rollout, with step-96 values labeled directly. The standard neural pipelines separate from 16-step full-feedback Transolver near the rapid transition. Additional exploratory methods are reported in Table 29 .

Figure 17: WSS evolution for an illustrative held-out case at rollout steps 4, 20, 48, and 96. Rows compare CFD, standard Transolver R3, scheduled sampling, and $F=16$ full-feedback adaptation. All panels use a shared scale derived from ground truth. Values at evaluation vertices are shown directly; values elsewhere are nearest-neighbor interpolated for display. Table 29 reports cohort-level trajectory and hemodynamic metrics.

Figure 18: Exploratory source-level 96-step rollout distributions for temporal reference baselines and the R3, scheduled-sampling, and F=16 interventions. Each panel uses the metric’s natural scale; “Trajectory AUC” denotes mean cumulative RelL2. Boxes show quartiles and whiskers the 5th–95th percentiles. Top-10% overlap is higher-is-better; all other panels are errors.

Figure 19: Explicit-coordinate and registered-history interventions on point-attention reset checkpoints. (a) Changes from coordinate permutation, coordinate zeroing, and the joint permutation–undo check for the two raw-coordinate checkpoints, shown in units of

10^{-4}

. The panel also includes a unit-box Transolver sensitivity. (b) History-only permutation on the natural RelL2 scale. Breaking the WSS-history correspondence raises error by about 1.255, whereas explicit-coordinate and joint-equivariance changes remain near zero. Points are mean source-paired changes and bars are 95% source-bootstrap intervals over the same 79 held-out sources.

Model	Track	Regime	$\mathcal{E}_{u}$	$\mathcal{E}_{u}^{(V)}$	$\mathcal{E}_{p}$	$\mathcal{E}_{p}^{(V)}$	$\mathcal{E}_{p}^{\mathrm{center}}$	Drop abs. (Pa)	Drop rel.
GraphSAGE	Points	R1	0.8269	0.8011	0.7131	0.6769	1.1994	186.6	0.5317
GraphSAGE	Points	R3	0.6521	0.6395	0.5525	0.5287	0.9211	137.7	0.4176
Transolver	Points	R1	1.0031	1.0009	0.8672	0.8021	1.0228	401.6	0.9962
Transolver	Points	R3	0.6979	0.6714	0.5783	0.5062	0.8614	124.5	0.3864
U-Net	Voxel	R1	0.9867	0.9867	0.8147	0.8147	1.0174	–	–
U-Net	Voxel	R3	0.7050	0.7050	0.7930	0.7930	1.1333	–	–

Appendix

Table 30: Representative quadrature and pressure-gauge sensitivity on the 79 held-out sources. Panel (a) compares the primary, uniformly weighted vertex/mask evaluation with native-VTU lumped cell-volume weighting; uniform voxel volumes cancel exactly. Centered-pressure RelL2 (

\mathcal{E}_{p}^{\mathrm{center}}

) subtracts each field’s weighted spatial mean. The final two columns are sampled inlet–outlet pressure-drop absolute error in Pa and relative error; both are unavailable for the voxel representation. Panel (b) uses native wall-polygon area weights. Negative change favors R3.

Model	Regime	Velocity RelL2	Stored-gauge pressure RelL2	Centered-pressure RelL2
U-Net	R1	0.9803	1.5510	1.3582
	R2	0.8209	0.7442	1.0601
	R3	0.7264	0.6702	0.9786
FNO	R1	1.0093	1.3407	1.2235
	R2	0.8585	1.0012	1.2836
	R3	0.8403	0.9276	1.2037

Appendix

Table 31: Sensitivity to centered physical coordinates on 79 held-out sources. Panel (a) reports source-averaged point estimates. Panel (b) gives paired R3–R1 changes with 10,000 source-bootstrap resamples; negative values favor R3. These reruns restore the stored longest-side physical scale. The main voxel table reports the original unit-box checkpoints.

	Velocity		Stored-gauge pressure
Evaluation	$280\times 1$	$35\times 8$	$280\times 1$	$35\times 8$
Original inputs and scoring	$0.7993\pm 0.0186$	$0.8480\pm 0.0218$	$0.7167\pm 0.0324$	$0.7856\pm 0.0397$
Active scoring, original inputs	$0.7766\pm 0.0178$	$0.8265\pm 0.0205$	$0.6620\pm 0.0291$	$0.7327\pm 0.0431$
Active inputs and scoring	$0.7788\pm 0.0186$	$0.8274\pm 0.0216$	$0.6705\pm 0.0334$	$0.7363\pm 0.0443$

Appendix

Table 32: Effects of scoring and input point domains on the two GraphSAGE allocations. Values are source-mean RelL2

\pm

sample SD across three training seeds. Model weights are fixed during evaluation. The two cell-connected evaluations use the same subset of the original query points.

Model	Fixed 4,096	Fixed 8,192	Full surface
Transolver	0.108406	0.108419	0.108648
Galerkin Transformer	0.109043	0.108883	0.108975

Appendix

Table 33: Transient reset-window sensitivity to wall-point evaluation density. All columns evaluate the same R3 checkpoint on the same 79 held-out sources; lower RelL2 is better.

Track	Model	Base R3	Learning rate $5\times 10^{-4}$	Synthetic normalization
Unstructured	Galerkin Transformer	0.8150 / 0.6667	0.8227 / 0.6880	0.8143 / 0.6548
Unstructured	GraphSAGE	0.6521 / 0.5525	0.6550 / 0.5465	0.6542 / 0.5421
Unstructured	PointNet	0.8899 / 0.7718	0.8945 / 0.7598	0.9021 / 0.7687
Unstructured	Transolver	0.6979 / 0.5783	0.6997 / 0.5875	0.6997 / 0.5920
Voxel	FNO	0.8387 / 0.9126	0.8405 / 0.9278	0.8414 / 0.9106
Voxel	U-FNO	0.7568 / 0.7925	0.7603 / 0.8168	0.7622 / 0.8279

Appendix

Table 34: Targeted steady R3 sensitivity runs. Cells are velocity / pressure RelL2. The base setting uses learning rate

10^{-4}

and source-training normalization; the other columns change one choice at a time.

Model	RelL2 95% CI	Mean MSE	Mean RMSE	Mean MAE
Unstructured
GraphSAGE	0.6521 [0.6070, 0.6982]	1.342e-02	0.0948	0.0622
Transolver	0.6979 [0.6467, 0.7510]	1.677e-02	0.1033	0.0673
Galerkin Transformer	0.8150 [0.7536, 0.8769]	2.332e-02	0.1206	0.0786
PointNet	0.8899 [0.8257, 0.9593]	2.291e-02	0.1260	0.0880
Voxel

Appendix

Table 35: R3 steady velocity diagnostics on 79 held-out source geometries. Metrics are computed from decoded fields, averaged over eight flow conditions within each source, and then averaged across sources. RelL2 brackets are 95% source-bootstrap CIs. MSE has units

\mathrm{m^{2}\,s^{-2}}

; RMSE, MAE, and magnitude MAE have units

\mathrm{m\,s^{-1}}

Model	RelL2 95% CI	Mean MSE ( $\mathrm{Pa^{2}}$ )	Mean RMSE (Pa)	Mean MAE (Pa)	Mean case $R^{2}$
Unstructured
GraphSAGE	0.5525 [0.4679, 0.6570]	2.938e+04	96.151	75.164	-0.9774
Transolver	0.5783 [0.5083, 0.6681]	3.697e+04	107.712	85.817	-1.0354
Galerkin Transformer	0.6667 [0.5741, 0.7917]	4.912e+04	122.183	97.056	-1.8980
PointNet	0.7718 [0.6782, 0.8885]	4.202e+04	130.467	104.165	-2.6911
Voxel

Appendix

Table 36: R3 steady stored-gauge pressure diagnostics on the same 79 held-out sources. MSE is in

\mathrm{Pa^{2}}

; RMSE and MAE are in Pa. Columns average per-condition metrics within sources, then across sources. Per-case

R^{2}

values are averaged without clipping; negative per-case values indicate greater squared error than prediction of that case’s target mean.

Model	RelL2 95% CI	Mean MSE	Mean RMSE	Mean MAE
Unstructured
Transolver	0.1084 [0.0969, 0.1218]	9.021e+01	3.694	2.075
Galerkin Transformer	0.1090 [0.0977, 0.1221]	9.595e+01	3.747	2.091
PointNet	0.1484 [0.1373, 0.1610]	1.332e+02	4.582	2.648
GraphSAGE	0.2235 [0.2014, 0.2498]	1.633e+02	5.577	3.240
Voxel

Appendix

Table 37: R3 reset-window

4\rightarrow 4

WSS diagnostics on 79 held-out source geometries, with observed histories restored between windows. Values are source means of per-window metrics. MSE is in

\mathrm{Pa^{2}}

; RMSE, component MAE, and magnitude MAE are in Pa. Direction metrics include only locations at which both vector magnitudes exceed

10^{-8}

Pa.

Reproduction task	Materials and use	Review	Full
Cohort and schema	Source-family splits, availability records, and the compact HDF5 sample; verifies identifiers, inherited splits, array shapes, units, and time grids.	✓	✓
Metric and report	Source-level result JSON files and evaluators; regenerates reported aggregates, paired intervals, tables, and figures.	✓	✓
Loader and evaluator example	Baseline configurations, loaders, and the compact sample; exercises data loading and endpoint computation on sample fields.	✓	✓
Checkpoint inference	Fixed 79-source WSS samples and actual Transolver weights; recomputes reset, rollout, and cycle metrics.	✓	✓
Source-only WSS training and adaptation	Native-resolution WSS arrays for 277 training and 42 validation sources, with training recipes and normalization statistics.	✓	✓
Steady allocation retraining	Pool manifests, selected descendant fields, common source fine-tuning fields, and complete steady validation and test fields.	–	✓

Appendix

Table 38: Reproduction tasks supported by the anonymous review packages and the full release. For numerical provenance, “Spec.” denotes the solver specification; “Archive” denotes retained OpenFOAM case configurations.

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