WeirNet: A Large-Scale 3D CFD Benchmark for Geometric Surrogate Modeling of Piano Key Weirs

Lisa Lüddecke; Michael Hohmann; Sebastian Eilermann; Jan Tillmann-Mumm; Pezhman Pourabdollah; Mario Oertel; Oliver Niggemann

cs.LGFeb 24, 2026

WeirNet: A Large-Scale 3D CFD Benchmark for Geometric Surrogate Modeling of Piano Key Weirs

Authors: Lisa Lüddecke, Michael Hohmann, Sebastian Eilermann, Jan Tillmann-Mumm, Pezhman Pourabdollah, Mario Oertel, Oliver Niggemann

Organizations: Hydraulic Engineering Section, Civil Engineering, Helmut-Schmidt-University, Hamburg, Germany · Institute for Artificial Intelligence, Helmut-Schmidt-University, Hamburg, Germany · Professorship of Computer Science in Mechanical Engineering, Helmut-Schmidt-University, Hamburg, Germany · Federal Waterways Engineering and Research Institute, Karlsruhe, Germany ·

Abstract

Reliable prediction of hydraulic performance is challenging for Piano Key Weir (PKW) design because discharge capacity depends on three-dimensional geometry and operating conditions. Surrogate models can accelerate hydraulic-structure design, but progress is limited by scarce large, well-documented datasets that jointly capture geometric variation, operating conditions, and functional performance. This study presents WeirNet, a large 3D CFD benchmark dataset for geometric surrogate modeling of PKWs. WeirNet contains 3,794 parametric, feasibility-constrained rectangular and trapezoidal PKW geometries, each scheduled at 19 discharge conditions using a consistent free-surface OpenFOAM workflow, resulting in 71,387 completed simulations that form the benchmark and with complete discharge coefficient labels. The dataset is released as multiple modalities compact parametric descriptors, watertight surface meshes and high-resolution point clouds together with standardized tasks and in-distribution and out-of-distribution splits. Representative surrogate families are benchmarked for discharge coefficient prediction. Tree-based regressors on parametric descriptors achieve the best overall accuracy, while point- and mesh-based models remain competitive and offer parameterization-agnostic inference. All surrogates evaluate in milliseconds per sample, providing orders-of-magnitude speedups over CFD runtimes. Out-of-distribution results identify geometry shift as the dominant failure mode compared to unseen discharge values, and data-efficiency experiments show diminishing returns beyond roughly 60% of the training data. By publicly releasing the dataset together with simulation setups and evaluation pipelines, WeirNet establishes a reproducible framework for data-driven hydraulic modeling and enables faster exploration of PKW designs during the early stages of hydraulic planning.

Figures & tables

Figure 1: Main geometric parameters of a rectangular PKW in plan view (left) and side view (right) defined by Pralong et al. [ 56 ] with $Q$ indicating the direction of flow.

Parameters	Definition
$B$	Upstream-downstream length of the PKW (total weir length)
$B_{o}$	Upstream (outlet key) overhang crest length
$B_{i}$	Downstream (inlet key) overhang crest length
$B_{b}$	Base length
$P_{i}$	Height of the inlet entrance measured from the PKW crest (including possible parapet walls)
$P_{o}$	Height of the outlet entrance measured from the PKW crest (including possible parapet walls)

Table 1: Nomenclature of the parameters of the PKW geometry by Pralong et al. [ 56 ] .

Figure 2: Exemplary results of a CFD simulation of water over a PKW geometry.

Study	Size	$N_{\text{geom}}$	SpD	Dim.	Struct. type	Rep.	Outputs	Open
	[sims/cases]	[–]	[–]			(P/M/PC)
Studies on PKW discharge coefficients
Bansal et al. [ 4 ]	162	9	18	3D	PKW key width ratio	✓/ – / –	$c_{D}$ , Energy dissipation (exp.)	–
Iqbal & Ghani. [ 30 ]	88	11	8	3D	PKW key width ratio & slope ratio	✓/ – / –	$c_{D}$ (exp.)	–
Studies on Weir geometry datasets
Eilermann et al. [ 14 ]	15	15	1	2D	Weir geometries	✓/ – / –	$h(x)$ , free surface, $c_{D}$	✓

Table 2: Comparison of existing studies and datasets on hydraulic and multiphase flow processes with geometric variation. Size counts CFD simulations or experimental cases,

N_{\text{geom}}

denotes the number of distinct geometries where reported and SpD is the approximate number of simulated operating points (e.g., discharges or boundary conditions) per geometry. P/M/PC indicate availability of parametric descriptors (P), surface or volume meshes (M), and point clouds (PC). The Outputs column summarizes the primary simulated or measured quantities used for surrogate modeling.

Figure 3: Exemplary geometries from the dataset showing variations in key widths, overhang lengths and wall thickness.

Parameters	Definition
$W_{i,u}$	Inlet key width (from sidewall to sidewall) on the upstream side *1
$W_{i,d}$	Inlet key width (from sidewall to sidewall) on the downstream side *2
$W_{o,u}$	Outlet key width (from sidewall to sidewall) on the upstream side *2
$W_{o,d}$	Outlet key width (from sidewall to sidewall) on the downstream side *1
$R_{B,i}$	Ratio of the downstream (inlet key) overhang crest length to base length
$R_{B,o}$	Ratio of the upstream (outlet key) overhang crest length to base length

Table 3: Differentiation of the key widths for trapezoidal PKWs and extended nomenclature of the sidewall thickness for trapezoidal PKWs.

Figure 4: Detailed presentation of geometric relationships between inlet and outlet key widths for TPKWs with $Q$ indicating the direction of flow.

Figure 5: Additional defined parameters at the PKW sidewall. Single parameters are shown on the left, while the corresponding reference locations are indicated on the right.

Parameters	Min	Max	Constant
$W$			1000 mm
$P$			330 mm
$B_{b}$	$0.33\cdot P$	$1.67\cdot P$
$R_{B,i}$	0.25	1
$N_{u}$			3
$T_{s}$	$0.015\cdot P$	$0.18\cdot P$

Table 4: Limitations for the generation of PKW geometries via parametric design.

Parameter	Computational Relationship
$W_{u}$	$W/N_{u}$
$B_{i}$	$R_{B,i}\cdot B_{b}$
$B_{o}$	$R_{B,o}\cdot B_{b}$
$B$	$B_{b}+B_{i}+B_{o}$
$\alpha$	$\tan^{-1}\left(\frac{W_{i,u}-W_{i,d}}{2\cdot(B-T_{s})}\right)$
$T_{s,2}$	$T_{s}/\cos(\alpha)$

Table 5: Derived parameters in the parametric model

$Q$ [l/s]
50	80	120	160	200
55	90	130	170	225
60	100	140	180	250
70	110	150	190

Table 6: Simulated discharges

Q

for all geometries.

Figure 6: Example of flow behavior over an exemplary PKW geometry in a scaled physical model test with flow direction from left to right.

Figure 7: Calculated discharge coefficients from CFD simulations of all investigated geometries for a representative exemplary selection of discharges simulated for the dataset.

Figure 8: Relative deviation between numerical and experimental results of upstream flow depths used for model validation.

Figure 9: Pearson correlation of selected weir parameters.

Figure 10: Pairwise correlation between the features and discharge coefficient.

Model	MSE (↓)	$\mathbf{R^{2}}$ (↑)	MAE (↓)	Max AE (↓)	Time/epoch (↓)	Inference Time (↓)	#Params
PointNet [ 57 ]	15.30	98.96	9.06	1.67	10 s	0.49 ms	5,435,009
Mesh-GCN [ 36 ]	241.60	64.33	36.88	2.99	24 s	0.90 ms	100,481
RegDGCNN [ 16 ]	97.40	83.11	22.01	1.90	1054 s	53.15 ms	4,482,657

Table 7: In-distribution surrogate modeling results for predicting

c_{D}

at given

Q

using geometric representation data (Task 1). In addition, the Inference Time for batch size 1 as well as the total number of parameters (#Params) per model are reported. Reported errors are scaled:

\mathrm{MSE}

\times 10^{-5}

R^{2}

\times 10^{-2}

\mathrm{MAE}

\times 10^{-3}

, and

\mathrm{Max\ AE}

\times 10^{-1}

Model	MSE (↓)	$\mathbf{R^{2}}$ (↑)	MAE (↓)	Max AE (↓)
RandomForest [ 9 ]	5.20	99.67	3.94	2.90
XGBoost [ 10 ]	6.80	99.58	5.53	1.44
LightGBM [ 34 ]	7.50	99.54	6.00	1.05
GradientBoosting [ 21 ]	119.50	92.63	19.61	3.54

Table 8: In-distribution surrogate modeling results with regression models for predicting

c_{D}

at given

Q

using parametric data (Task 1). Reported errors are scaled: MSE

\times 10^{-5}

R^{2}

\times 10^{-2}

, MAE

\times 10^{-3}

, and Max MAE

\times 10^{-1}

Figure 11: Parity plot of predicted versus CFD discharge coefficients $c_{D}$ on the in-distribution test set (Task 1).

Figure 12: SHAP summary plot for predictions based on all parametric models.

Figure 13: Distribution of the discharge coefficiency $c_{D}$ (left), discharge $Q$ (middle), and inclination angle $\alpha$ (right).

Model	Split	MSE (↓)	$\mathbf{R^{2}}$ (↑)	MAE (↓)	Max MAE (↓)
RandomForest [ 9 ]	$\alpha_{\leq 2^{\circ}}$	1628.40	-19.43	94.34	5.27
XGBoost [ 10 ]	$\alpha_{\leq 2^{\circ}}$	1350.50	0.95	86.13	4.78
LightGBM [ 34 ]	$\alpha_{\leq 2^{\circ}}$	1242.40	8.88	83.35	4.80
GradientBoosting [ 21 ]	$\alpha_{\leq 2^{\circ}}$	1582.60	-16.07	94.28	5.19
RandomForest [ 9 ]	$\alpha_{3^{\circ}\text{--}5^{\circ}}$	87.20	87.59	19.11	4.93
XGBoost [ 10 ]	$\alpha_{3^{\circ}\text{--}5^{\circ}}$	37.10	94.72	14.53	1.57

Table 9: Out-of-distribution performance (

\mathrm{MSE}

R^{2}

\mathrm{MAE}

, and

\mathrm{Max\ AE}

) for all surrogate models on in-distribution (ID), OOD-Geom, and OOD-Head splits. Reported errors are scaled: MSE

\times 10^{-5}

R^{2}

\times 10^{-2}

, MAE

\times 10^{-3}

, and Max MAE

\times 10^{-1}

Figure 14: Data-efficiency of surrogate models on WeirNet.

Surrogate Modeling

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FluidFlow: a flow-matching generative model for fluid dynamics surrogates on unstructured meshes

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ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain · Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB), 07122 Palma de Mallorca (Spain) · Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain

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Computational fluid dynamics (CFD) provides high-fidelity simulations of fluid flows but remains computationally expensive for many-query applications. In recent years deep learning (DL) has been used to construct data-driven fluid-dynamic surrogate models. In this work we consider a different learning paradigm and embrace generative modelling as a framework for constructing scalable fluid-dynamics surrogate models. We introduce FluidFlow, a generative model based on conditional flow-matching, a recent alternative to diffusion models that learns deterministic transport maps between noise and data distributions. FluidFlow is specifically designed to operate directly on CFD data defined on both structured and unstructured meshes alike, without the needs to perform any mesh interpolation pre-processing and preserving geometric fidelity. We assess the capabilities of FluidFlow using two different core neural network architectures, a U-Net and diffusion transformer (DiT), and condition their learning on physically meaningful parameters. The methodology is validated on two benchmark problems of increasing complexity: prediction of pressure coefficients along an airfoil boundary across different operating conditions, and prediction of pressure and friction coefficients over a full three-dimensional aircraft geometry discretized on a large unstructured mesh. In both cases, FluidFlow outperform strong multilayer perceptron baselines, achieving significantly lower error metrics and improved generalisation across operating conditions. Notably, the transformer-based architecture enables scalable learning on large unstructured datasets while maintaining high predictive accuracy. These results demonstrate that flow-matching generative models provide an effective and flexible framework for surrogate modelling in fluid dynamics, with potential for realistic engineering and scientific applications.

Abstract

Surrogate Modeling

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