DragSolver: A Multi-Scale Transformer for Real-World  Automotive Drag Coefficient Estimation
Authors: Ye Liu
Yuntian Chen
Venue: ICML 2025 poster (2025)
Keywords: Aerodynamic Drag Prediction
Computational Fluid Dynamics
Transformer
Real-World Automobiles
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We introduce a multi-scale Transformer approach for real-time automotive drag coefficient estimation, handling diverse 3D car meshes,  and cutting error by 58.7% over prior methods.

Abstract: Automotive drag coefficient ($C_d$) is pivotal to energy efficiency, fuel consumption, and aerodynamic performance. However, costly computational fluid dynamics (CFD) simulations and wind tunnel tests struggle to meet the rapid-iteration demands of automotive design. We present DragSolver, a Transformer-based framework for rapid and accurate $C_d$ estimation from large-scale, diverse 3D vehicle models.
DragSolver tackles four key real-world  challenges: 
(1) multi-scale feature extraction to capture both global shape and fine local geometry; 
(2) heterogeneous scale normalization to handle meshes with varying sizes and densities;
(3) surface-guided gating to suppress internal structures irrelevant to external aerodynamics;
and (4) epistemic uncertainty estimation via Monte Carlo dropout for risk-aware design. 
Extensive evaluations on three industrial-scale datasets (DrivaerNet, DrivaerNet++, and DrivaerML) show that DragSolver outperforms existing approaches in accuracy and generalization, achieving an average reduction of relative $L_2$ error by 58.7% across real-world datasets. Crucially, DragSolver is the first to achieve reliable, real-time $C_d$ inference on production-level automotive geometries.

DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Ye Liu 12 Yuntian Chen
Abstract
200
Automotive drag coefficient (Ca) is pivotal to en- ergy efficiency, fuel consumption, and aerody- namic performance. However, costly computa- tional fluid dynamics (CFD) simulations and wind tunnel tests struggle to meet the rapid-iteration demands automotive design. We present Drag- Solver, Transformer- based framework for rapid and accurate Cd estimation from large-scale, di- verse 3D vehicle models. DragSolver tackles four key real-world challenges: (1) multi-scale feature extraction to capture both global shape and fine local geometry; (2) heterogeneous scale normal- ization to handle meshes with varying sizes and densities; (3) surface-guided gating to suppress internal structures irrelevant external aerody- namics; and (4) epistemic uncertainty estimation via Monte Carlo dropout for risk-aware design. Extensive evaluations on three industrial-scake datasets (DrivAerNer Driv and Dri- vaerML) show that DragSolver outperforms ex- isting approaches in accuracy and generalization, achieving an average reduction relative L2 error by 58.7% across real-world datasets. Crucially, DragSolver is the first to achieve reliable, real- time Cd inference on production- levell automotive geometries.
200
(a)
-3.5e102
(b)
(c)
Figure 1. Key challenges in learning-based C2 prediction. (a) Multi-scale influence: Global shape influences Ca by determining large-scale acrodynamic flow patterss, including pressure distri- bution and wake formation. (b) Local and irrelevant structures: Small details (e.g.. mirrors, wheels) induce turbulence, while the inset highlights internal components (e.g., seats, dashboard) that are irrelevant to external aerodynamics but appear Geometric and mesh variability: Differences in vehicle underbedy structures, mesh resolution, and discretization highlight the chal- lenge of inconsistent representations across datasets, necessitating robust normalization.
scans. (c)
tion, and vehicle range (Sudin al., 2014). In the early design stages, rapid and accurate C. estimation is essential for evaluating shape variants under tight development time- lines. Traditional numerical methods, such as computational fluid dynamics (CFD) simulations and wind tunnel exper- iments, provide reliable estimations (Rodi, 1997; Baker Brockie, 1991) but are computationally expensive, often requiring hours even days per iteration for complex 3D models (Moin & Mahesh, 1998). This bottleneck signif- icantly hinders the iterative workflows crucial to modern automotive engineering.
1. Introduction
The aerodynamic drag coefficient (C₂) is critical metric in automotive design, shaping energy efficiency, fuel consump-
Shhangaai Jiao Tong University, Shanghai 200240. PR China Zhejiang Key Laboratory of Industrial Intelligence and Digital Twin, Eastern Institute of Technology, Ningho, Zhejiang 315200, P.R. China 'Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo, Zhejiang 315200, P.R. China 'Ningbo Key Laboratory of Advanced Manufacturing Simulation, Eastern Institute Technology, Ningbo, Zhejiang 315200, P.R. China. Correspondence to: Yuntian Chen <ychen citech.edu.cn>.
From physics standpoint, the drag force Fp on a vehicle is
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Fp p0² Ca A,
where p air density, is velocity, and is frontal area.

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Under standardized test conditions, p and remain fixed, leaving Ca the primary shape-dependent variable. If one could directly map vehicle's 's 3D geometry to Ca. engineers could swiftly assess design alternatives without incurring the high costs of CFD or wind tunnel experiments.
internal structures, focusing on exterior flow-dominant sur- faces. (4) uncertainty estimation via lightweight Monte Carlo dropout scheme, enabling risk-aware design deci- sions in industrial workflows. We extensively evaluate Drag- Solver on three real-world datasets ADrivAerNet, ArivAer- Net++, and DrivaerML). Results show an average 58.7% relative L2 improvement over existing baselines. Further- more, DragSolver achieves near real- time inference (~0.9- 5 per shape) on single GPU, significantly undercutting hours-long CFD pipelines. To foster broader research and real-world adoption, we will release our code upon accep- tance. Overall, our contributions are threefold:
Despite this appeal, developing a real-world practi- cal geometry-to-Ca predictor faces four critical chal- lenges (Sudin et al., 2014): (1) Multi-scale interactions: Local details (e.g., spoilers, side mirrors) and global shape factors (e.g., roof curvature) both heavily influence drag. (2) Geometric Diversity: Vehicles differ widely in size, pro- portion, and mesh density, requiring robust normalization across heterogeneous data. (3) Irrelevant Internal Struc- tures: Real-world test samples may include interior com- ponents ge.g. seats, steering wheel) absent in the training set of exterior surfaces, adding noise and hindering focus on flow-dominant geometry; (4) Predictive uncertainty: In- dustrial pipelines need confidence estimates for risk-aware design, but most existing ML methods treat Ca prediction as a "black box." " Figure 1 illustrates these issues in detail.
We present the first end-to-end multi-scale transformer- based neural solution for automotive drag forecasting on large, unstructured industrial datasets.
We design surface-guided gating mechanism with het- erogeneous normalization and uncertainty estimation, addressing real-world geometric complexities.
We validate on three high-fidelity, diverse datasets, con- sistently surpassing prior methods in accuracy, speed, and ditto--distribution generalization.
Conventional CFD and wind tunnel experiments, though accurate, remain prohibitively expensive for repeated design iterations (Moin& & Mahesh, 1998; Heinz, 2020; He et al., 2021). Recent deep-learning surrogates (Rios et al., 2021; Abbas et al., 2022) can accelerate aerodynamic analysis but still exhibit key limitations: (1) existing Ca prediction methods (Song et al., 2023; Elrefaic et al., 2024a) often rely on 2D views or dimeeniiiioal shape parameters, failing to capture real-world 3D geometry; (2) Transformer-based 3D networks (Vaswani et al., 2017; 7; Pang et al., 2022; Han et al., 2024; Chen et al., 2024; Wu et al., 2024b; 2022; Wang, atten- tion but typically assume uniform point densities and lack robust mesh normalization or interior filtering; and (3) neu- ral PDE solvers (Li et al., 2021; 2023c; Hao et al., 2023; Wu et al., 2024a) emphasize local flow states rather than a global scalar like Ca. requiring cumbersome flow-field integration for large, unstructured car meshes. Thus, while Transform- ers inherently excel at capturing distant geometric cues (e.g., mirror-to-tail or underbody interactions) that strongly in- fluence aerodynamics, a practical geometry-to-Ce pipeline still needs dedicated modules to handle heterogeneous mesh scales, interior structures, and predictive uncertainty.
2. Related Work
Automotive Aerodynamic Analysis. Accurate drag coef- ficient (Ca) estimation traditionally relies on computational fluid dynamics (CFD) and wind tunnel testing (Menter et al., 2003; Scardovelli & Zaleski, 1999; Fröhlich & Von Terzi, 2008). Though these methods yield high-fidelity aerody- namic data, they are slow and costly, limiting rapid design it- eration. While recent learning- based approaches (Song et al., 2023; Elrefaie et al., 2024a;b) reduce CFD costs, most rely on low-dimensional shape parameters (c.g.. a few geometric descriptors) or 2D projections (e.g., silhouette views). Such simplifications ignore the intricate 3D structure that sig- nificantly impacts aerodynamic flow. Consequently, these methods struggle to capture local flow phenomena, limiting their applicability in real-world design loops.
2023) excel at modeling global context via
Neural PDE Solvers. A related direction involves neural PDE solvers (Karniadakis et al., 2021; Wang et al., 2023), serving as fast surrogates for PDE-governed phenomena such as fluid flow or stress fields. Neural operators meth- ods (Li et al., 2021; 2023d; Wu et al., 2023; Li et al., 2023b) and Transformer-baser PDE solvers (Liu et al., 2022: Li et al., 2023c; Hao et al., 2023; Wu et al., 2024a) achieve strong results in predicting local flow states. However, de- riving a global scalar like Ca typically requires further in- tegration of the entire flow field or explicit knowledge of pressure/velocity distributions, making them less direct and often inaccurate for global drag prediction. In contrast, DragSolver bypasses explicit PDE modeling by learning a
In response, we introduce DragSolver, a practical deep learn ing system that delivers reliable, real-time Ct estimations for diverse, large-scale automotive models. Specifically, Specifically, DragSolver integrates: (1) Multi-scale feature extraction to capture both global shape contexts and lo- cal geometric cues crucial for aerodynamics. (2) Hetero- geneous scale normalization to handle drastically varying mesh densities and vehicle sizes consistently. (3) Surface- guided gating to automatically suppress or ignore irrelevant
2

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
geometry-to-Ca mapping end-to-end, focusing on capturing the global shape cues relevant to drag without solving local fields and better suits large-scale automotive meshes with partial interior data.
and velocity v, cf. Eq. 1). Our goal is to approximate Ca accurately and efficiently, avoiding the high cost of full- fledged CFD or wind tunnel experiments.
Input Geometry. Let G = [ denote either N sur- face vertices or an N-poiint cloud of the vehicle. In prac- tice, G may include partial interior geometry (e.g. seats, steering wheel) that does not affect external aerodynam- ics. We address this via surface-guid guided gating (Sec. 3.4) to deemphasize irrelevant internal points, focusing on the flow-dominant exterior.
Geometric Deep Learning Deep learning on 3D data has progressed through point-based, voxel-based, mesh-based, and more recently trunsformer-bused paradigms. Point- based networks (Qi et al., 2017a;b; Zhao et al., 2021; Pang et al., 2022; Han et al., 2024; Chen et al., 2024) directly process unstructured point clouds but can struggle with very dense or uneven distributions. Voxel-based methods transform point clouds into regular voxel grids to facilitate 3D convolution operations (Wu et al., 2024b; 2022; Wang, 2023). Mesh-based approaches (Pfaff et al., 2021b; Wang et al., 2019) exploit edge convolutions, though they can be sensitive to mesh topology changes and not trivially adopt multi-scale feature fusion. However, none of these main- stream methods directly address (1) heterogeneous mesh normalization, (2) gating out irrelevant internal geometry, or (3) automotive-specific C. regression with uncertainty. Our proposed DragSolver extends the seafformer-baseed pipeline, integrating multi-scale hierarchical encoding (Wu et al., 2024b) and additional modules tailored to real-world automotive drag estimation.
Learning Objective. Denote by C. = [o(GG the pre- dicted drag coefficient from a network with parameters 0. We adopt standard regression loss:
M M j=1
1
-C
Lreg
(2)
where M is the mini-batch size. Additional terms for gating or regularization may be included (Sec. 3.4). Our goal is to make Ca closely match the true C2 while remaining robust to geometric variations and interior noise.
3.2. Multi-Scale Architecture
Motivation. Automotive drag depends on both global shape (e.g.. roof curvature, body length) and local details (e.g., spoilers, side mirrors). A single-scale network may miss fine geometry or lose broad context. Hence, we adopt a multi-stage encoder design, inspired by Transformer based point processing (Wu et al., 2024b), capturing both large and small-scale patterns.
Predictive Uncertainty Estimation Uncertainty-aware ML helps practitioners gauge the reliability of estimations. Bayesian neural networks (Jospin et al., 2022) are a classic approach, but often complicated to implement and scale. MC Dropout (Gal& & Ghahramani, 2016) offers a simpler variational approximation: applying dropout at inference time for multiple forward passes. Ensemble methods (Lak- shminarayanan et al., 2017) also provide robust uncertainty but at higher memory cost. In our setting, epistemic un- certainty guides risk-aware decisions when deploying Ca estimations in industrial design pipelines.
Stage- Wise Forward. As illustrated in Fig. 2, we begin with an embedding block that transforms raw points/voxela into initial feature space. We then stack L encoding stages fenc[0)....... enc 1]}, each combining sparse or point-based convolutions with attention layers. Let (()) be the output of the embedding block. For k = I,...,L. L:
3. Methodology
(()) = enc [k] (F((-
(3)
Deeper stages expand receptive fields to capture global shape context, while earlier features preserve local detail.
We present DragSolver, a novel Transformer-based ap- proach for real-time drag coefficient estimation from large- scale 3D automotive geometries. Our design addresses four key challenges in aerodynamic optimization: multi-scale interactions, heterogeneous geometry, irrelevant internal structures, and predictive uncertainty.
Global Pooling at Each Stage. Unlike single-resolutie pipelines, we collect multi-scale global features by applying global pooling operator (GPool) at each stage output:
g (b) = GPool (F k = L.
(4)
3.1. Problem Formulation
Where (*) E RC encodes the entire shape at scale k.
Task Definition. We aim to learn a function f(G) Ca mapping an automotive geometry G to its drag coefficient Ca. Concretely, each sample (G, Ca) provides: (I) 3D representation of the vehicle (e.g., triangular mesh point cloud, or voxel grid), and (2) a scalar Ca measured or com- puted under standardized conditions g., fixed air density/ p
Concatenating these pooled features
G scle = ig (0)
g (1)
g
(5)
yields a comprehensive representation capturing both shal- low local cues and deep global context. We feed Gmulti-scale into final regression MLP to predict Ca.
3

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DragSolver: A Multi-Scale Transformer for Heal-World Automotive Drag Coefficient Estimation
Encoder x
Diverse Meshes
Normalized Point
Block N
Depti Filteiing
MLP
MLP
MLP with MC Dropout
Embedding
Positional
Encoding
LayerNomm
Attention
LayerNorm
MLP
NSH
Voxel
Global
Figure 2. Overview of DragSolver. The framework consists of four main components: (1) Heterogemeous Scale Normalization (HSN), which processes diverse vehicle meshes into a unified representation (Sec. 3.3). (2) Surface-Guided Gating, which filters out irrelevant internal structures to ensure a focus on aerodynamic surfaces (Sec. 3.4). (3) Multi-Scale Architecture, an encoder leveraging voxel pooling, positional encoding, and attention to capture both global and local aerodynamic features (Sec. 3.2). (4) Epistemic Uncertainty Estimation, utilizing Monte Carlo Dropout (MC Dropout) to provide confidence estimates in Ca predictions (Sec. 3.5).
Vehicle Models
Underbody (Mesh)
Body (Point cloud)
Comparison of Interior Structures
Estateback
With Interior
Fastback
Notchback
Figure 3. Variability in vehicle configurations, mesh resolution, and structural representations. The first column illustrates three distinct vehicle types: Estateback (top). Fastback (middle), and Notchback (bottom). highlighting variations in shape distribution. The second column presents underbody meshes, revealing significant differences structural complexity and mesh resolution across datasets. The third column visualizes point cloud representations, showing heterogeneous surface discretization levels. The fourth column compares vehicles with and without interior structures, emphasizing distributional shifts where real-world data includes internal components irrelevant externl acrodynamies. These differences underscore the challenges posed by dataset heterogencity in learning-based Ca prediction.
3.3. Heterogeneous Scale Normalization
overr/uudde--eepeeentttiioo Consequently, different vehicles become more comparable, aiding stable multi-scale feature extraction.
Challenge. Real-world vehicles differ drastically phys- ical scale (e.g., compact cars large SUVs) and mesh densities (hundreds of thousands up millions of poly- gons), causing inconsistent feature distributions if ingested naively.
3.4. Surface-Guided Gating
Motivation. In practice, real-world scans or CAD files can include internal components (c.g., seats, steering wheel) that do not affect external aerodynamic flow. Training datasets often provide only exterior surfaces, whereas test models might unexpectedly contain such interior geometry, causing noise and performance degradation. Manual preprocessing is generally impractical for large-scale automotive designs. Therefore, we combine depth- filtering step with multi-stage gating mechanism to robustly suppress imrele- vant internal structures.
Automotive Geometric Normalization. We employ two-stage approach: (1) Wheelbase Alignment & Global Scaling. Since wheelbase is relatively standardized, we first rescale each shape so that the wheelbase matches a fixed length, ensuring consistent front/rear alignment. We then normalize all data by scaling the bounding box, en- suring consistency and reducing large-scale discrepancies. (2) Hybrid Resolution Sampling. Even after global seal- ing, some vehicles remain overly dense, others sparse. We mix mesh-based sampling (capturing high-curvature detail) with uniform sampling (mitigating density imbalance). This yields robust coverage of local geometry while preventing
Depth-Base Basth-Based Filtering (Preprocessing). First, we per- form a coarse elimination of clearly internal voxels. We estimate the depth of each voxel relative to the outer sur-

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
face using a may-marching approach along the principal axes (e-g., d.directiin Specifically, we cast rays outward from reference plane and track the sequence of voxel cross- ings. For each voxel, its depth is defined as the number of intersections encountered before reaching the outer shell. Voxels that are clearly distant from the surface-beyone a predefined threshold-are classified as internal and re- moved. This method efficiently climinates large chunks of internal geometry while maintaining conservativeness to avoid mistakenly discarding subtle external cavities or enclosed aerodynamic features.
Table 1. Summary of the datasets used for evaluation.
Dataset
Categories
#Car
#MMesh
Estuteback
DrivAerNet++
Fastback
7673
420k-2.2M
Notchback
DrivAerNet
Fastback
3760
420k
DrivaerMI.
Notchback
483
750k
4. Experiments
We conduct extensive experiments to evaluate DragSolver under both in-distribution (ID) and out-of-distribution (OOD) settings, reflecting real-world automotive design scenarios. Our primary goals are to (1) assess accuracy on known distributions, (2) measure generalization to unseen shapes, datasets, or interior structures, and (3) validate the effectiveness of each core module via ablation.
Multi-Stage Gating Mechanism. After filtering out obvi- ous internal voxels, we assign an initial gating weight Q 1
to each remaining voxel or point. An MLP then refines 9
at select stages of our multi-scale encoder. Specifically: (1)
High-resolution input layer: Each voxel i has c; (lait) 1,
which is updated by gating MLP to account for uncertain regions near the exterior. Hence c: c((mit) + (MLL(())) letting the network softly downweight borderline internal geometry. (2) Mid-resolution layers: No gating is applied, allowing the model to freely attend to intermediate features without further constraints. (3) Final gating: Right before global pooling, we recompute c (fiual) from the final-layer features. This second pass ensures fine-graines re-check so that truly flow-relevant surface voxels remain prominent, while any residual internal points are minimized. If c, x O, that voxel's 'ss'i feature p.d is effectively nulled for subsequent aggregation. In the end, we compute
We use three real-world datasets: (DrivAer- Net++, DrivAerNet, and DrivaerMI..coverinn diverse body shapes (estateback, fastback, notchback), mesh resolutions (420k to 2.2M polygons), and potential interior geometry. Table 1 summarizes key statistics. Each dataset reserves 20% for testing. For cross- or cross-shape evalua- tions, we carefully partition training sets in one subset while testing on different subsets or entirely different datasets. More detailed descriptions of the datasets can be found in Appendix B.
Datasets.
Evaluation Metrics. For accuracy, we report: (1) Rela- tive &2 Error: highlighting percentage error relative to the ground-truth drag coefficient. much variance in C3 explained by the model,
Σ. (final) (ffaall) (ffinal)
gllobal
Σ, 4
(Anal)
(6)
(2) R² Score: capturing how
focusing attention on truly external, aerodynamically rel- evant regions. During training, we insert random "ghost"" voxels inside the bounding box to simulate unexpected inte- rior structures.
E (Ca,; - -Ca;-Ca..)
R² 1
(8)
M EM (Ca,i CCa) 2'
where C. is the mean of {Ca,s} over the test set. A perfect model yields R² 1, while R²2 < indicates no explana- tory power relative to the mean.
3.5. Epistemic Uncertainty Estimation
Rationale. In industrial design loops, single Čd estima- tion may be insufficient for critical decisions. Epistemic uncertainty indicates model ignorance, helping engineers weigh risk in choosing final shapes.
Baselines. We compare DragSolver with more than 10 baselines: including typical 3D geometric deep learning methods: PointNet (Qi al., 2017a), PointNet++ (Qi et al., 2017b), PointTransformer (Zhao et al., 2021), Point- GPT (Chen et al., 2024), Mamba3D (Han et al., 2024), Point- Transformer V3 (Wu et al., 2024b), MeshGraphNet (Pfaff et al., a0) RegDGCNN (Elrefaie et al., 2024b). Includ- ing state-of-the-arr neural operators: GNOT (Hao et al., 2023), GINO (Li et al., 2023a), Transolver (Wu et al., 2024a)
MC Dropout. We adopt Monte Carlo Dropout (Gal & Ghahramani, 2016), activating dropout at inference. Let fe. (G) be the k-th forward pass with distinct dropout masks. Then
K (fa (
C. R Σ (G),
R k=
1=1
(7)
Implementations. For fairness, we implement a fivee multi- encoder, each stage halving the spatial resolu- tion (stride= 22) to progressively expand the receptive field.
Hence, 2Z, approximates epistemic uncertainty. In practice, K 10 yields stable estimates with minimal overhead.
5

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Table 2. Performance comparison for in in-distribution evaluation on two real-world datasets, where we record Relative Le and R². A smaller Relative is better, while a larger R is better. For clarity, the best result is bold, and the second-best is underlined. "Promotion" refers to the relative error reduction w.r.t the second-best model, i.e. The best Note: DragSolver applies MC Dropout uncertainty (see $3.5), yielding mean 1 std. for each metric. Other baselines do not estimate model uncertainty and therefore report single values.
Our erre
DrivAerNet
DrivAerNet++
Model
Relative L2 4
R 1
Relative L2
R
PointNet (Qi et al., 2017a)
0.0062
0.6184
0.0079
0,5170
PointNet++ (Qi al., 2017b)
0.0046
0.7538
0.0064
0.6241
PointTransformer (Zhao al., 2021)
0.0024
0.8844
0.0031
0.8015
PointGPT (Chen et al., 2024)
0.0023
0.8910
0,0028
0.8210
Mamba3D (Han et al., 2024)
0.0015
0.9173
0.0014
0.9192
PointTransformer V3 (Wu et al., 2024b)
0.0021
0.9051
0.0022
0.8901
MeshGraphNet (Pfaff' et al., 2021a)
0.0230
-0.2334
0.0238
-0.5858
RegDGCNN (Elrefaic et al., 2024b)
0.0030
0.8418
0.0059
0.7064
GINO (Li et al., 2023a)
0.0286
-0.4361
0.0316
-0.5463
Transolver (Wu et al., 2024a)
0.0180
-0.1646
0.0223
-0.3931
GNOT (Hao et al., 2023)
0.0036
0.7926
0.0028
0.8312
DragSolver (Ours)
0.0007= 0.0001
0.9582 = 0.0016
0.0005= 0.0001
0.9712 0.0011
Relative Promotion
53.3%
4.2%
64.2%
7.05%
Channel dimensions grow from 32 to 256, and we employ robust patch-based attention, which ensures that our model models
DrivAerNer+++ yet remain behind DragSolver. PDE-based or neural operator methods often focus on local flow states and require additional integration for a global Ca. leading to inconsistent or even negative R² on complex real-world shapes. Our DragSolver substantially outperforms all base- lines both datasets, reducing relative L2 error to 0.0007 (Driv AerNet) and 0.0005 (DrivAerNet++) It achieves It to 0.9712, indicating an excellent fit with ground-truth labels. Compared to the second best method, DragSolver yields a 53.3% and 64.2% promotion error reduction, re- spectively, demonstrating its ability to consistently capture global aerodynamic interactions in purely data-driven man- ner. Overall, these in-distribution results confirm that multi-scale network, aided by robust normalization excels at learning complex geometry-to-C, mappings.
parameter is comparable to other Transformer-baat such as Point Transformer V3 (Wu et al., 2024b). This hier- archical design balances local and global aerodynamic cues while efficiently handling diverse vehicle meshes. For Het- erogeneous Scale Normalization (Sec. 3.3), the wheelbase is fixed at 2.8 m. and we apply a hybrid sampling strategy that yields around 600 points per vehicle. We enable Monte Carlo Dropout at the final regression layers with dropout rate p 0.1. At inference, we draw K 10 forward passes, yielding a set of predicted C, values (C(R) Ik=1- We then report mean (Ca) and standard deviation (# We optimize with AdamW (Loshchilov, 2019) at base learning rate of 10-3 batch size 24, and train for 50 epochs, adopting cosine LR decay. All experiments run on NVIDIA A800 GPUs unless stated otherwise. Unless otherwise stated, these hyperparameters remain fixed across all experiments. See Appendix C for comprehensive de- scriptions of implementations.
4.2. Results of
Although ID accuracy is important, practical deployment often faces unseen shape configurations or distinct mesh distributions. We thus evaluate Dut-offDisttriuun (OOD) settings where training and testing data differ shape type, dataset origin, or internal structures. Figure illustrates the diversity real-world datasets, encompassing variations in vehicle shape, underbody complexity, and point cloud resolution. The presence of both high-density and low- density mesh structures, along with the inclusion of real- world interior components in some cases, creates significant distribution shifts. These differences pose challenges for learning-bas based prediction.
4.1. Results of In-Distribution
inDiitribbutinn Performance. We first evaluate each model in -ame-distribution scenario, where training and test sets share the same shape/mesh statisties. Specifically, we independently train and test DrivAerNe and Dri- vAcrNet++ (Tab. 2). Point-based networks generally benefit from learning unstructured data but still suffer with fine geometry. PointGPT and Mamba3D attain higher R² on
6

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DragSolver: A Multi-Scale Transformer for Automotive Drag Coefficient Estimation
Interior Structure Variation Results. Industrial pipelines often differ in interior geometry. We model this mismatch by artificially adding random "ghost" interior points to both training and testing data in DrivAerNet++. The training set initially contains only exterior surfaces, but we insert a limited number of synthetic interior points to teach the network that such geometry should be downweighted. At test time, we similarly append more extensive ghost interiors mimic real scans that include seats or dashboards. Table 6 shows that, without gating, Relative L2 error degrades to 0.078; with gating, the error is reduced to 0.065. Baselines that lack a dedicated filtering mechanism degrade more severely, underscoring the value of surface-guides gating in practical scenarios where training and testing distributions differ in interior geometry.
Table Cross-shape generalization within DrivAerNet++. Train columns list body types used for training. Test indicates the held- out shape (estateback E, fastback F, notchback N).
Model
Train
Test
Relative L2 L
R²1 f
Mamba3D
F4N
E
0.0097
0.4912
PointGPT
F+N
0.0117
0,3616
RegDGCNN
F+N
E
0.0124
0.3245
DragSolver
F+N
E
0.0031 0.0003
0.8054z 0.0021
Mamba3D
F+E
N
0.0054
0.6179
PointGPT
F+R
N
0.0094
0.4984
RegDGCNN
F+F
N
0.0120
0.3838
DragSolver
F+E
0.0034x 0.0003
0.7842x 0.0019
Table 4. Performance comparison for cross-dataset generaliza- tion. DN+ DrivAerNet++, DML: DrivaerML. Models are trained DN++ and tested on DML to evaluate generalization to unseen high-resolution meshes.
Scalability analysis. As shown in Table 8, DragSolver is already the most accurate model with only 5 % of the training data, achieving = where the closest base- line (Mamba3D) is still negatively correlated with ground truth. Accuracy improves smoothly as the set grows, yet the gain from 50 % 100 % data is marginal (A Rel. L2 0.06 10-3), indicating good cumple-efficienge The base- lines, in contrast, require the full dataset to stabilise and still exhibit 2-6 x higher error at 100 % data. These trends confirm that our architecture scales favourably in data and compute, delivering state-of-the-an accuracy under limited supervision while adding negligible inference overhead as the dataset expands.
Model
Train
Test
Relative Lz
R²a 1
Mamba3D
DN+
DML
0.0197
-0.0684
PoinGGPT
DN+
DML
0.0447
-0.5447
RegDGCNN
DN+
DML
0.0561
-0.6145
DragSolver
DN+ DML
0.0091: 0.0001
0.2732+ 0.0004
Cross-Shape within DrivAerNet++. The DrivAerNet++ series includes Estateback, Fastback, and Notchback de- signs. Table 3 shows results where we train only on two body types (F, N) and test on the third (E) (E)-absent from training, and vice versa. When testing on estateback (E) after training on fastback & notchback (F+N), DragSolver achieves the lowest relative L2 (0.0031) and highest I² (0.8054), outperforming Mamba3D by an over 3x reduc- tion in error. Similarly, for the F+E-N split, DragSolver again surpasses competing methods with an error of 0.0034 and R² = 0.7842. The substantial gap underscores the model's ' strong capacity to handle shape distribution shifts and capture exterior geometry cues that impact Ca.
Noise robustness. Table 5 shows that DragSolver is far less sensitive to real-worid-style noise (dropout, pose jät- ter, sensor error) than the three baselines. Even at the strongest perturbation it keeps a positive nt² whereas PTV3, Mamba3D, and PointGPT all become negatively correlated with CFD ground truth. This confirms the superior resilience of our physics-aware architecture for noisy, in-situ data.
4.3. Ablation Study
We now ablate the main modules in three settings: (I) in- distribution, (2) cross- cross-shape,, and (3) interior mismatch. The variants tested are: Single-Scale Only: no multi-scale fea- ture fusion.+ Multi-scale (no gating, no normalization). + Multi-scale + Normalization (no gating). DragSolver Full (Multi-scale + Normalization + Gating). Table 7 shows that each module contributes noticeably. Multi-scale is crucial. Comparing "Single-Scale Only" (0.0049 in ID), we see nearly a 47% relative error reduction in the in- distribution scenario. Similar improvements appear in cross- shape (0.0081 1 vs. 0.0065) and interior mismatch (0.0096 vs. 0.0079), indicating that capturing both global and local geo- metric cues is essential. Normalization tackles large-scale variation. Adding heterogeneous normalization (*+ MS + Norm.") ") yields a further jump in accuracy, e.g. from 0.0026 to 0.0007 in the ID setting, showing that unifying vehicle
-atts-D Generalization Results. We next exam- ine a cross- datssed scenario by training on Driv DritAerNet++ (DN++) and testing on DrivaerMI. (DML), where vehicles differ more substantially in shape design, mesh resolution, and potential internal complexities. Table 4 shows that Drag- Solver attains a much lower relative L2 error (0,0091) and a positive R² of 0.2732, indicating it effectively transfers across dataset discrepancies. In contrast, Mamba3D yields a higher error (0.0197) and a negative R² (-0.0684), while PointGPT further degrades to 0.0447 error and -0.5447 R². A negative R² suggests these methods underperform even a naive "constant mean" predictor in this cross-dataset context. By contrast, DragSolver's 's robust normalization, multi-scale architecture enable it to retain stable perfor- mance despite the substantial shift from DN++ to DML.
7

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Table 5. Comparison under limited training data (30% split of DrivAerNet+++) with increasing augmentation intensity.
Model
# #Train Samples
Time/ epoch (s)
1x Inf.
Drop- out
Rot.
Flip
Trans.
Noise
Rel. L2!
R²²
Time (s)
(%)
(%)
(x10-*)
1638
52.09
16.82
20%
~"
50%
0.01
5%
2.9420
0.8583
DragSolver (Ours)
1638
52.02
16.81
30%
~
50%
0.02
5.7857
0.7460
1638
53.46
17.51
40%
-40
50%
0.03
9%
16,9077
0.2344
1638
42.51
14.86
20%
~
50%
0.01
8,6164
0,4669
PTV3
1638
43.14
14.90
30%
~9
50%
0.02
7%
11.4341
0.2501
1638
43.97
15.04
40%
44
50%
0.03
9%
26.5797
-0.5991
1638
152.39
21.86
20%
~
50%
0.01
5%
13.7013
0.0853
Mamba3D
1638
153.51
22.90
30%
~3
50%
0.02
7%
19,8485
-0.0894
1638
154.54
23.04
40%
~4
50%
0.03
9%
37.4115
1.5153
1638
157.23
23.15
20%
~0
50%
0.01
5%
18.3948
-0.1941
PointGPT
1638
158.32
23.45
30%
~9
50%
0.02
7%
36,6032
-1.4599
1638
159.15
24.64
40%
1 19
50%
0.03
95.2137
-4.8912
Table 6. Impact of interior structure mismatch: training data a small portion of synthetic interior points, but test samples include more extensive interior additions. Baseline methods de- grade severely when confronted with these unseen interiors, while DragSolver's gating mechanism preserves robust performance.
Table 8. Scalability analysis on DrivAerNet++ under different
training-set sizes. Lower relative Le better; higher R² better.
Model
#Train Sumples
Train Ratio
Time/ epoch (s)
Inf.
ReL. L (x10
Time (s)
268
5%
5.40
11.98
3.5119
0.8243
536
10%
12.62
12.04
1,3929
0,9255
Method
Relative L2
R22 f
DragSolver
30%
0.7026
0.9620
2680
50%
60.73
12.68
0.5927
0,9673
PointTransffrmer
0.0096
0.4343
0.5351
0,9710
PoineTransffrmer V3
0.0078
0.5455
268
5%
4.48
11.54
68. 1518
- -3.0564
Mamba3D
0.0084
0.4604
PTV3
1638
30%
29.07
11.41
6.4185
0.6150
PointGPT
0.0097
0,4241
2680
3.2661
0.8088
5361
100%
98.77
12.25
1,7319
0.9024
DragSolver (No Gating)
0.0057 0.0001
0.65382 0.0005
DragSolver (Full)
0.0024x 0.0001
0.8739 0.0094
536
10%
35.12
22.15
32.8624
1.1419
1.6104
Mamba3D
1638
30%
103.54 54
22.30
9,2257
0.3926
2680
50%
178.41
22.30
2.8901
0.8362
5361
100%
343.65
23.01
1.4811
0,9191
Table Ablation of DragSolver components DrivAerNet++ under (a) in- in-distribution, (b) cross-shape, and (c) interior mismatch
268
5%
18.21
23.54
70,8708
-3.234
536
10%
37.02
23.64
62.9200
-3.29419
PoinGGPT
1638
30%
106.32
23.41
13:2661
0.1315
scenarios, Metries: Relative (lower better).
2680
50%
185.15
23.56
6.2985
0.6180
5361
100%
369.01
23.67
3.1240
0,8207
Method Variant
(a) In-Dist
(b) Cross-Shape
(c) Interior
Single-Scale Only
0.0049
0.0081
0.0096
+ Multi-scale
0.0026
0.0065
0.0079
Multi-scale + Norm.
0.0007
0.0034
0.0057
Table 9, Impact of Multi-Scale Depth L on DrivAerNet++.
DragSolver (Full)
0.0IM7= 0.0001
0.0034: I.0002
0.0024x 0.0MIZ
Depth
L=1
L=3
L=5
I.=7
Relative L2 4 0.0020
0.0010
0.0008
0.0009
sizes/densities helps the model converge more effectively. Gating is pivotal for interior mismatch. In the final column, adding gating drives error from 0.0057 to 0.0024, 58% re- duction, clearly demonstrating how surface-guin guided filtering helps when test data includes unexpected internal structures.
Multi-Scale Depth (L). As discussed in Sec. 3.2, we vary the encoder depth L E 3, 7 to assess how capturing multiple scales influences accuracy. Table indicates that L 5 achieves the lowest relative L2 error (0.0008). When L 1, the network struggles to encode global shape context (error 0.0020), and L 7 brings diminishing returns with higher overhead. Hence, L = efficiency.
4.4. Model Analysis
We further analyze core hyperparameters and design deci- sions in DragSolver, focusing on DrivAerNet++ under the in-distribution (ID) scenario unless stated otherwise. Our goal is to demonstrate how multi-scale depth, pooling strat- egy, voxel granularity, sampling approaches, and runtime efficiency collectively shape the final performance.
balances accuracy and
Global Pooling: Max vs. Average. After each en- coder stage (Sec. 3.2), we apply a global pooling operator (GPool). As shown in Table 10, max pooling consistently
8

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Table 10. Impact of Global Pooling.
Table 13. Performance evaluation of DragSolver under varying numbers of sampled points.
Global Pooling Strategics
Mcan
Max
#Train Samples
#Sammped Points
Sampled Pts.
Rel. Lz (x10
R² f
Relative L2 |
0.0009
0.0007
Ratio
1638
60 000
2.72 14.38%
0.7026
0.9620
Table 11.
Impact of voxel size 5 (lower 8 finer granularity.
1638
50000
2.27
0.7442
0,9606
1638
40000
1.18-9.599
0.7766
0,9572
Voxel Size
0.01
0.05
0.1
1638
30000
1.36-
0.8844
0,9534
1638
20000
0.59 4.79%
1.0284
0,9460
Relative L2
0.0009
0.0016
0.0029
1638
10000
0.30 2.40%
4.1502
0.7718
outperforms average pooling, likely due to its ability to bet- ter retain salient geometric features (e.g., spoilers, ridges), which are crucial for aerodynamic prediction.
combined with FlashAttention (Dao, 2024), significantly re- ducing computational complexity by avoiding costly KNN operations. Table 14 shows that DragSolver achieves 3x faster training time compared to Point-based methods. Our method processes normalized point in under 0.9 with L 5, 6 0.01. Monte Carlo Dropout (T = 10) raises inference to #4.8 s/shape, yet remains far below CFD times (hours to days), thereby retaining viability for rapid design.
Voxel Size. When using sparse voxel backend, we eval-
uate voxel sizes E {0.01, 0.05, 0.10) in normalized coor- dinates. As Table 11 suggests, smaller 5 = 0.01 captures more fine-grained geometry, improving accuracy but slightly increasing memory/time cost. Larger voxels (8 = 0.10) lose local detail, hurting Ca prediction. We thus adopt ō = 0.01. Hybrid Sampling. To handle heterogeneous vehicle meshes, we compare three sampling strategies under an OOD setting on DrivAerNet++. Specifically, the test distri- bution includes shapes and resolutions not seen at training. Mesh-based sampling alone can overfit to the training mesh topology, while uniform sampling may underrepresent high- curvature regions. Table 12 shows that hybrid approach achieves the best accuracy, effectively balancing local geo- metric detail against distribution shifts.
Table 14. Training/ifference time (per epoch, per sample).
Method
PointGP
Mamba3D
PTV3
Ours
Train (slepoch)
1680
1600
490
530
Infer (s/sample)
0.97
0.95
0.79
0.93
Point Downsampling. Table 13 quantifies the impact of progressively reducing the number of surface points fed to the network. A gentle reduction from 60 k to 40 k points increases the relative L2 by only ~0.07!x!10-3 and keeps R² above 0.95, indicating that the model is robust to moderate sparsity. Below 30 k points, however, errors rise rapidly- most notably a six-fold jump in Ta and 19- point drop in R² at 10 k shontt--howwng that excessive decimation removes critical boundary-layer information.
5. Conclusions and Feature Work
We presented DragSolver, a Transformer-base framework for real-time drag coefficient estimation from 31) automo- tive geometries. By integrating multi-scale features, het- erogeneous normalization, surface-guided gating, and MC Dropout uncertainty, DragSolver robustly handles diverse mesh densities, interior structures, and shape variations. Experiments across three real-world datasets show state- of-the-art results and near real-time inference, confirming each module's contribution and the method's potential for practical automotive design. Future work will explore par- tial CFD constraints, expanded uncertainty calibration, and broader domains (e.g. aerospace).
Table 12. Comparing sampling strategics.
Strategies
mesh-based uniform hybrid
Relative L2 +
0.0019
0.0016
0.0012
Training and Inferenee Efficiency. Finally, we com- pare DragSolver's efficiency against ttate-of-the-art 3D point-based methods under the same hardware environ- ment. As highlighted in PTV3 (Wu et al., 2024b), KNN- based approaches (e.g., PointGPT (Chen et al., 2024), Mamba3D (Han et al., 2024)) incur high computational costs due to repeated neighbor graph construction and at- tention computation. Inspired by this, we adopt a SpConv- based sparse convolutional backbone (Contributors, 2022)
Acknowledgements
This work was supported by the National Key Research and Development Program of China 2024YFF15006M and the Innovation Capability Support Program of Shaanxi(Program We also thank Shenzhen TenFong Technology Co. Ltd. for valuable engineering support throughout the project.

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Impact Statement
Elrefaie, M., Morar, F,, Dai, A., and Ahmed, F. Drivaer- net++: A large-scale multimodal car dataset with com- putational fluid dynamics simulations and deep learning benchmarks. NeurlPS, 2024b.
Our approach accelerates aerodynamic evaluation, poten- tially reducing prototyping costs and improving energy ef- ficiency. While we do not foresee unusual ethical issues, biased or incomplete training data could yield unexpected failure certain vehicle shapes. Proper dataset diversity and worldw validation remain critical. This paper aims to advance Machine Learning for automotive aerodynam- ics. We identify no major ethical or societal risks beyond standard MI. considerations.
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DragSolver: A Multi-Scale Transformer for Automotive Drag Coefficient Estimation
A. Visualization of Predicted Drag Coefficients
this section, we present additional qualitative insights into DragSolver's predictions on various car geometries. Specifically, we visualize each vehicle's exterior mesh (or point cloud) alongside its predicted drag coefficient (Ca) and compare it to the ground-truth reference (c.g., wind-tunnel or CFD-derived). Figure 4 illustrates the predicted and ground truth drag coefficients (Ca) across multiple vehicle geometries. The model demonstrates high accuracy, with predictions closely aligning with ground truth values. The results underscore the effectiveness of our framework in learning aerodynamic features and estimating drag coefficients in real-world automotive applications.
True 0.3298
True 0.3054
Pred. : 0.3241
Pred. : 0.3074
True 0.2702
True 0.2740
Pred. : 0.2745
Pred. : 0.2689
True 0.2941
True 0.2791 Pred. 0.2768
Pred. : 0.2912
Figure 4. Comparison between true and predicted aerodynamic drag coefficients (C.) for different vehicle geometries. Each presents a pair of vehicles with their corresponding ground truth and predicted Ca values. The top row shows vehicle designs with higher drag values (True: Ca 0.3298, 0.3054; Pred: C. 0.3241, 0.3074), indicating greater aerodynamic resistance. The middle illustrates models with moderate drag coefficients (TTrue: C, = 0.2740, 0.2702; Pred: Ca 0.2689, 0.2745). The bottom row presents the most acrodynamically optimized designs (True: C. 0.2941, 0.2791: Pred: C. 0.2912, 0.2768). results demonstrate the accuracy our model in predicting Ca. with minimal deviation from the true values. Notably, variations in acrodynamic design lead to distinct drag values, impacting energy efficiency in real-world applications.
B. Detailed Dataset Descriptions
Overview of Datasets. Figure 5 offers a comparative look at the three automotive datasets used in our experiments: DrivAerNet++, DrivAerNet, and Mr. Each varies in mesh density, inclusion of internal geometry, and parametric scope. Such differences demand robust mesh normalization and multi-scale feature extraction (as detailed in our main text) to achieve reliable Ca regression. Below, we provide concise, third-person summary each dataset, referencing the
13

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
original authors and publications.
Surface
Mesh
Point Cloud
(a) DrivAerNett+
(b) DrivAerNet
(c) DrivAerML
Figure Visual Comparison of DrivAerNet++, DrivAerNet, and DrivAerML. Each row corresponds a dataset: DrivAer- Net++ nhigh-iidliity external surfaces, parametric), (b) DrivAerNet (conventional automotive meshes capturing underbody details), (c) DrivAerML (even denser CAD scans including internal structures). For each dataset, we show the surface geometry (left), mesh representation (middle), and point-cloud view (right). The progression from simpler external shapes to highly detailed, partially interior meshes underscores the increasing complexity data richness across datasets. These contrasting resolutions, structural details, geometric variations pose challenges for generalization in Ca prediction.
B.1. DrivAerNet++ Dataset
Origin and Motivation. Elrefaie et al. (2024b) introduced DrivAerNer++ to supply a large-scale collection of watertight 3D car meshes, validated against the DrivAer model (Heft et al., 2012b;a). Unlike generic repositories (e.g. ShapeNet (Chang et al., 2015)), DrivAerNet++ aims to preserve engincering- resolution for acrodynamic simulations.
Parametric Models. Using A @ morphing, up 26 design parameters alter both global car dimensions (length, roof height) and local features (diffusor, underbody, wheels). This yields baseline configurations-ffsttuuee estateback, Brrchback-expandee to thousands of variants, each with final watertight mesh. Body shapes differ significantly in rear geometry, occupant space, matching diverse industrial needs.
CFD Simulation and Validation. Elrefaie et al. (2024b) use snappyfeeMeeh (OpenFOAM Foundation, 2023) to refine around wheels, underbody, and wake regions, often reaching 12-24 million cells. They adopt a k-w SST RANS approach at 30 m/s (Ress 107), achieving drag predictions within 2-5% of TUM's wind-tunnel data (Heft et al., 2012a). While the fidelity lower than full LES, balances mesh size vs. feasibility for large parametric sweeps.
Relevance for MI. Because Driv covers multiple rear-end shapes (fastback, estateback, notchback) moderate to high mesh resolution, it provides diverse design space for training data-driven aerodynamic surrogates. The authors note possible future expansions with multi-fidelity or hy brid RANS-LES approaches.
B.2. DrivAerNet Dataset
Background and Baseline Geometry. Elrefaie al. (2024a) present DrivAerNet as an earlier, flexible variant of the DrivAer model (Heft al., 2012b), bridging simple bodies (Ahmed (Ahmed et al., 1984) or SAE (Cogotti, 1998)) and proprietary industrial CAD. Unlike the single baseline STL, they introduced about 50 morphable parameters (roof angle, wheel/mirror geometry) to broaden shape variation.
CFD Setup and Mesh. In DrivAerNet, the authors also rely on OpenFOAM with a k-w SST scheme, refining the underbody, wheels, and wake regions via snappyHexMesh. typical velocity 30 m/s. matching the TUM reference, and the Res 9.39 105. They particularly highlight a fastback with detailed underbody + wheels mirrors (FDw WwM),
14

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
since each adds significant drag counts. Summaries show that these morphological changes are well captured within few percent of experimental values.
Parameter Range and Mesh Resolution. Mesh resolutions vary from around 500k faces up to millions of cells, enabling more detailed geometry than prior open datasets (Li et al., 2023a; Song et al., 2023). The authors provide code and mor- phological scripts at https: //:: / thub. com/ Mohamede re an external resource for data-driven aerodynamic tasks.
/Paramet r cModel, offering
B.3. DrivAerML. Dataset
Notchback Configuration. DrivAerML around the notchback variant of the DrivAer body, originally used in the 2nd-4th AutoCFD Workshops. Wheels are static (no rotation), and cooling inlets are sealed to reduce complexity and facilitate consistent flow-field comparisons among different CFD codes.
CFD Domain and Mesh. They employ large "opee-road" domain with negligible blockage ratio (~ 0.25% The free-stream velocity is 38.889 m/s, giving Rese 7.19 10t based on wheelbase L 2.786 m. The reference frontal area A = 2.17 m² is used for force and moment coefficients. Mesh generation uses ANSA's HeXtreme, producing hexa/polyhedral grid of ~ 160 million cells (high-y* approach, 7 prism layers). This strategy, validated in the 2nd AutoCFD workshop, balances resolution and runtime for industrial-level setups, capturing underbody flow, mirrors, and wake zones.
Focus on Industrial Fidelity. Driv AerMA aims to provide near-industrial RANS mesh advanced turbulence modeling or data-driven aerodynamic surrogates. The geometry's 's sealed inlets and static wheels reduce complexity yet maintain key automotive features (e.g. underbody structures, detailed external surfaces). Full details, domain boundaries.
relevant for
Conclusion. These three datasets Deltastss-DrivAeeNNeee AerNett,, DrivAerNet, and DrivAcrML-collettve span wide spectrum of vehicle shapes, mesh densities, and structural details (exterior-only vs. partial interior). Each dataset has been validated against Driv Aer wind-tunnel data from TUM or other references, making them valuable external resources for learning-based Ca prediction and surrogate modeling of aerodynamic flows.
C. Implementation Details
This appendix provides additional information about DragSolver's implementation and the experimental protocols referenced in $4. It covers model hyperparameters, training configurations, hardware setups, and specific differences from baseline methods.
C.1. Model Configuration
Overall Architecture. As described in $3.2, DragSolver uses a five-stage encoder, each halving the spatial resolution (stride= 2) via sparse convolutions or point-based downsampling. For each stage k. the channel dimension grows from an initial 32 to up to 256 by the final stage (Table 15). We adopt multi-head attention with = 8 heads. A patch-based or voxel-based partition is used to handle large unstructured point sets. Following $3.3, we fix the wheelbase to 2.8 m, then apply a hybrid sampling (uniform mesh-based) that yields 6600 effective points per shape.
Surface-Guided Gating. We apply the depth-based filtering from $3.4 at an initial voxel resolution of A 0.01 m. Voxels beyond threshold depth (e.g. 40% layers inside) are tagged internal and removed. In high-resolution input layers, each surviving voxel has an initial gating weight = 1 and is refined by gating MLP (MLPinput). We skip gating at intermediate layers, then re-apply gating at the final layer via MLPPaaa--
Uncertainty Estimation. We use MC Dropout (Gal & Ghahramani, 2016) with dropout rate of 0.1 in the final regression MLP. At inference, each shape is passed K = 10 times with different dropout masks, yielding an ensemble (2(1) ... (11)) The mean Ca is reported as the prediction, while variance approximates epistemic uncertainty.
15

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
Table Model layer configuration for DragSolver, with stages plus final regression MLP. Notation: dm is input channels, dou output channels. After concatenating multi-scale features into 768 768-dimennae di--immnnsinnl vector, we feed it into the shown MLP structure to predict Ca.
Stage
Input Size
# Blocks
#Heads
Output Channels
Embedding
32
Stage
64
Stage 2
64
128
Stage 3
128
128
Stage 4
128
4
16
256
Stage 5
256
16
256
Concat g (0)
[|
[g)())
(5) - IR
768
Regression
Linear(668-250)
BN ReLU Dropout,
Linear (26-2r(26--128)) BN ReLU
- Linear (inear(128-1 (128-1] 1)
C.2. Training Configuration
Training Hyperparameters. Unless otherwise stated, we train for 50 epochs using AdamW (Loshchilov, 2019) with a base learning rate of x 10m3 and a cosine decay schedule. Our batch size 24 (due to memory constraints). We set weight decay x 10-5, We do not tune these hyperparameters across tasks, using a unified setting for consistency.
Losses. As described in $3.1, we define standard MSE loss on the predicted Ca. Specifically,
M -o(0)²
(00 Σ 10
Greg
M 1=1
Data Preprocessing. For DrivAerNet++ and DrivAerNet, reserve 20% for testing. We randomize the order of shapes each epoch. For cross-shape cross-dataset evaluations, we isolate the geometry categories or entire dataset for training vs. testing as described in $4.1.
C.3. Comparison with Baselines
Point-based Methods. We implement take code from offical library for standard 3D methods: PointNet, PointNet+++, Mamba3D. Each uses the recommended AdamW or Adam optimizer with identical learning rate as DragSolver. Graph-based expansions (e.g. DGCNN, MeshGraphNet) require adjacency or face definitions, which are straightforward DrivAerNet++ but require additional conversion for DrivAerML's partial interior data. We attempt to keep their hyperparameters near the official defaults, only adjusting channel size and training epochs to match our setup.
Point-based Methods (Baselines). We adopt or reference the official code repositories from the original authors aiming to remain as faithful as possible their recommended training procedures. For consistency with DragSolver's approximate 600 k-point representation (see $3.3), we also sample 600 points for each baseline method. This ensures all models operate similarly detailed point sets and fairly compare performance on high-density automotive meshes. Because many of these baselines rely on k-nearest neighbors (KNN) for local feature aggregation, we fix h: 32 capture sufficient neighborhood context in complex car surfaces. However, this large k significantly increases memory usage, nearly exhausting GPU resources in our setup. Apart from adjusting the batch size to accommodate memory constraints, we keep other hyperparameters as suggested by the original authors (c.g., learning rates, layer widths, activations).
Neural Operators/ / PDE Solvers. We adapt PDE-based surrogates (e.g. GNOT, Transolver (Wu et al., 2024a)) by letting them predict single scalar Cg. That is, instead of local field predictions, we pool features globally and feed an MLP to regress Ca-
16

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DragSolver: A Multi-Scale Transformer for Real-World Automotive Drag Coefficient Estimation
C.4. Hardware and Runtime Statistics
We conduct all experiments on up 2 NVIDIA A800 GPUs. On DrivAerNet++ at full resolution (~> 600k points), DragSolver typically processes batch of size 24. Inference single shape is under 100 ms for single forward pass, plus K 100 ms for MC Dropout. This is dwarfed by the hours days needed for high-fidelity CFD wind tunnel tests.
C.5. License and Data Availability.
DrivAerNet++ and DrivAerNet from Elrefaie et al. (2024a), while DrivAerML is from Ashton et al. (2024). Please Appendix B for direct links and usage instructions.
17


Reviews:
[{"review_id": "w7WvcGEXJw", "reviewer": "Reviewer_bnLA", "summary": "In this paper, the authors propose DragSolver—a method to effectively estimate physical properties of shapes, such as cars, without the need to run expensive CFD simulations, allowing for the design of novel shapes much faster and more effectively. The proposed method consists of four major blocks: (1) multi-scale feature extraction, which enables more accurate predictions across multiple scales, (2) input normalization, allowing the model to work with various meshes of changing size and number of nodes/edges, (3) an effective approach to ignoring irrelevant parts of the shape—such as interior objects—to produce more precise predictions, and (4) an MC-Dropout method that enables the approach to estimate uncertainties, which are of great importance in industrial workflows. The method is compared against a number of modern 3D architectures and demonstrates superior performance in terms of physical properties prediction accuracy.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s positive evaluation and constructive summary, which greatly encourages us.\n\n**Note to reviewers:**  We conducted additional experiments and analyses in response to the valuable suggestions from other reviewers. Detailed experimental configurations are provided in [**Table 1(Click to View)**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%201.jpg), and supplementary results can be found anonymously here:  [**Anonymous Supplementary Tables**](https://anonymous.4open.science/r/ICMLRebuttal-F329)\n\nBased on insightful feedback from other reviewers, we have further strengthened our manuscript with the following detailed analyses and clarifications:\n\n- **Computational Efficiency:** We explicitly compared DragSolver against state-of-the-art deep learning surrogate models (PTV3, Mamba3D, PointGPT) under limited training data and varying noise conditions. These comparisons clearly demonstrate DragSolver’s superior predictive accuracy and competitive inference speed ([**Table 2**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%202.jpg) and [**Table 3**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%203.jpg)).\n\n- **Robustness under Limited Data and Noise:** We evaluated DragSolver extensively under significantly reduced training samples (as low as 5%–30%) and challenging noise scenarios (up to 40% dropout, ±4° rotation, 9% random noise), confirming its consistent robustness and strong generalization capabilities ([**Table 2**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%202.jpg) and [**Table 3**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%203.jpg)).\n\n- **Uncertainty Quantification:** We clarified and differentiated epistemic (MC Dropout) from aleatoric (random sampling) uncertainties through additional targeted experiments, emphasizing their complementary roles in robust uncertainty estimation ([**Table 6**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%206.jpg)).\n\n- **Choice of Point Cloud Representation:** We provided explicit rationale and empirical evidence supporting our choice of point clouds over meshes, highlighting advantages in computational efficiency, flexibility, and generalizability for large-scale automotive aerodynamic analysis ([**Table 4**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%204.jpg)).\n\n- **Wheelbase Normalization:** We further clarified the practical necessity and benefits of our wheelbase normalization strategy through comparative experiments, confirming it significantly reduces training instability due to scale variations while preserving original geometric proportions ([**Table 5**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%205.jpg)).\n\nWe believe these comprehensive revisions effectively address key concerns, further improving the manuscript’s clarity, rigor, and broader impact.\n\nAgain, we sincerely thank the reviewer for their encouraging and supportive comments.", "claims_and_evidence": "The authors make several claims about each major component of their method (mentioned 4 parts) and either adequately discuss the motivation behind these claims and/or support them experimentally later in the experimental section.", "methods_and_evaluation_criteria": "The choice of datasets and benchmarks is adequate and accurately reflects the current state of research in this direction. The evaluations are rigorous and demonstrate the effectiveness of the approach from multiple perspectives.", "theoretical_claims": "No theoretical claims, theorems, or proofs are presented in the paper.", "experimental_designs_or_analyses": "The design of the experiments sufficiently covers different aspects of the proposed approach, including both the accuracy of in-distribution predictions and the generalization ability of the model. Rigorous ablations are also much appreciated and support the claims made about the method. Additionally, the method is compared against a number of popular 3D architectures, further strengthening the case for its effectiveness.", "relation_to_broader_scientific_literature": "The paper clearly situates itself within the existing literature by thoroughly discussing prior work on estimating physical properties of shapes, particularly in the context of CFD-free predictions. It provides sufficient detail on previous approaches, highlighting their limitations and demonstrating how the proposed **DragSolver** framework addresses these challenges, namely through the four discussed components of the method.", "essential_references_not_discussed": "No essential references are missing.", "other_strengths_and_weaknesses": "In short, the major Strengths of the paper are: \n\n* The proposed DragSolver framework is clearly structured, with each component thoroughly justified and experimentally supported. The paper includes rigorous ablation studies that strengthen its claims.  \n\n* The experiments cover multiple aspects, including in-distribution accuracy and generalization ability, and the method is compared against modern 3D architectures, demonstrating its effectiveness.  \n\n* The paper thoroughly discusses prior work, highlights the limitations of existing approaches, and evaluates the method on relevant, up-to-date datasets, ensuring its alignment with the current state of research.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2y8ZiTGjVz", "reviewer": "Reviewer_mQmd", "summary": "The paper presents a Transformer-based framework designed for predicting the aerodynamic drag coefficient of automotive designs directly from 3D vehicle models. This work is motivated by the high computational costs and inefficiencies of traditional Computational Fluid Dynamics (CFD) simulations and wind tunnel experiments, which, despite their accuracy, are often too slow for rapid design iterations. The authors propose DragSolver as a deep learning-based surrogate model that integrates multi-scale feature extraction, heterogeneous scale normalization, surface-guided gating, and epistemic uncertainty estimation to improve the reliability and generalizability of aerodynamic predictions.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s insightful and constructive comments. Here, we respond to each concern in detail.\n\n**Note to reviewers:**  We conducted additional experiments and analyses in response to your valuable suggestions. These results will be explicitly included in the revised manuscript. Detailed experimental configurations are provided in [**Table 1(Click to View)**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%201.jpg), and supplementary results can be found anonymously here:  [**Anonymous Supplementary Tables**](https://anonymous.4open.science/r/ICMLRebuttal-F329)\n\n---\n\n**Q1: Computational Efficiency and Comparison with Deep Learning-based Surrogates**  \n\n**A1:** We appreciate the reviewer’s comment on computational efficiency. The reported inference time (0.9–5 seconds per shape) includes 10 inference passes for uncertainty estimation and data loading overhead. To address this concern, we conducted additional efficiency comparisons with state-of-the-art models (PTV3, Mamba3D, PointGPT; [**Tables 2**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%202.jpg) and [**3**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%203.jpg)). DragSolver consistently achieves superior accuracy, particularly under limited training data (5%–30%) and high-noise conditions where other models often fail to converge. A single-pass inference (without uncertainty estimation) for the entire test set (1154 samples) takes only 11.98–13.25 seconds, demonstrating DragSolver’s practical computational efficiency. Additionally, automotive aerodynamic optimization typically requires feedback at intervals of seconds or minutes during design iterations rather than millisecond-level speed, thus DragSolver fully meets real-world requirements. We will clarify this context explicitly in the revised manuscript.\n\n**Q2: Data Dependency (Fully Supervised Learning)**  \n\n**A2:** We appreciate the reviewer’s insightful comment regarding data dependency. Although our current method indeed uses fully supervised learning and thus relies on labeled data, we specifically evaluated DragSolver’s effectiveness under significantly reduced training data conditions (as low as 5%–30%, see [**Table 2**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%202.jpg)). The results demonstrate DragSolver’s remarkable ability to maintain high accuracy (*R²*>0.92 at 10% training data) and superior stability compared to state-of-the-art surrogate models (PTV3, Mamba3D, and PointGPT), which largely fail to converge at these limited training ratios. These findings highlight DragSolver’s efficiency in leveraging limited labeled data. Nevertheless, integrating semi-supervised or physics-informed learning approaches to further reduce data dependency remains an important future direction.\n\n**Q3: Robustness to Noise and Incomplete Data**  \n\n**A3:** We appreciate the reviewer’s valuable suggestion regarding robustness to real-world noisy or incomplete 3D data. To address this important concern, we conducted extensive experiments with varying levels of realistic noise and data augmentation ([**Table 3**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%203.jpg)), including substantial random dropout (up to 40%), rotations (±4°), translations (0.03), and noise addition (9%) to simulate common real-world imperfections such as missing data, occlusions, or sensor artifacts. The results clearly show that DragSolver consistently achieves superior accuracy and stability under these challenging conditions, significantly outperforming state-of-the-art surrogate models (PTV3, Mamba3D, and PointGPT), which often fail to converge effectively under higher noise intensities. This strongly supports DragSolver’s robustness and suitability for real-world deployment. Nevertheless, explicitly testing DragSolver on industrial CAD data and actual 3D scans remains an essential direction for future research, which we will highlight in the revised manuscript.\n\n**Q4: The Paper is Narrow in Scope (Automotive Drag Prediction)**  \n\n**A4:** We appreciate the reviewer’s valuable point. Automotive drag prediction itself is an important and challenging problem with significant industrial impact, directly influencing vehicle energy efficiency and emissions. Moreover, automotive aerodynamics is widely recognized as a classical and representative scenario in aerodynamic shape optimization. Although our current study specifically targets automotive applications, the methodologies proposed in DragSolver—such as multi-scale feature extraction, surface-guided gating, and uncertainty quantification—can naturally generalize to broader aerodynamic and hydrodynamic design problems (e.g., aerospace, naval, and structural engineering). We will clarify both the intrinsic importance of automotive drag prediction and the potential broader applicability of our methods in the revised manuscript.", "claims_and_evidence": "One of the strengths of the paper is the multi-scale feature extraction mechanism, which effectively captures both global shape characteristics and fine-grained local geometric details that influence aerodynamics. Another notable contribution is the surface-guided gating mechanism, which suppresses irrelevant internal structures such as seats and dashboards that are often present in 3D scans but do not impact external aerodynamic behavior.", "methods_and_evaluation_criteria": "The paper just used fully supervised training. It does not provide a detailed computational efficiency analysis in comparison to alternative CFD-based surrogate models", "theoretical_claims": "There were no theoretical claims in the paper.", "experimental_designs_or_analyses": "The paper compares perfomrance on 3-different datasets.", "relation_to_broader_scientific_literature": "The paper is focused on automotive drag prediction and as such narrow in scope.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "Although DragSolver is significantly faster than traditional CFD simulations, the inference time of 0.9 to 5 seconds per shape could still be a limitation in real-time design applications where instant feedback is necessary. A comparison with other deep learning-based surrogates in terms of computational cost would further contextualize the trade-offs involved in adopting this approach.\n\nAnother limitation is the reliance on fully supervised learning, which requires a large number of high-fidelity training samples with ground-truth Cd values obtained from CFD or wind tunnel experiments.\n\nWhile the study evaluates DragSolver across multiple datasets, it does not explicitly address real-world deployment challenges, such as robustness to noisy or incomplete 3D scans. Industrial CAD models and real-world scans often contain missing data, occlusions, or sensor artifacts that could impact the model’s predictions.\n\nHence, while this is a great concept, further research is needed to optimize computational efficiency, reduce data dependency, and improve robustness to real-world noise.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "None"}}, {"review_id": "aAOjfCGHZZ", "reviewer": "Reviewer_pidM", "summary": "The authors present DragSolver, a Multi-scale Transformer for processing car point clouds to estimate drag coefficient for automotive designs in real-world applications. To adapt traditional transformer architectures to this new task, the authors propose multiple designs to achieve more trustful results, like 1) multi-scale feature extraction, 2) heterogeneous scale normalization to process vehicles in diverse sizes, 3) surface guided gating to alleviate distractions caused by irrelevant interior designs, and 4) Monte Carlo output drop layer to  estimate the range of drag coefficient for real-world applications. The results show the superiority of the architecture design.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s insightful and constructive comments. We have conducted additional experiments and analyses based on your suggestions, and these results will be explicitly included in the revised manuscript. \n\n**Important Note:**  To clearly demonstrate the thoroughness of these additional experiments, detailed experimental configurations are provided in [**Table 1 (Click to View)**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%201.jpg). Additional results mentioned in our responses are provided anonymously here:  [**Anonymous Supplementary Tables**](https://anonymous.4open.science/r/ICMLRebuttal-F329)\n\n---\n\n**Q1: Point Clouds vs. Meshes**\n\n**A1:** We appreciate the reviewer’s insightful comment regarding the choice of point clouds over meshes. Indeed, mesh-based methods can effectively encode surface connectivity and local geometric details, but their performance heavily depends on the quality and consistency of mesh discretization. Achieving high-quality mesh generation often requires substantial domain expertise and careful tuning specific to each geometric structure, significantly increasing complexity and cost. To ensure general applicability, flexibility, and ease of use—especially for large-scale automotive datasets—we adopted point clouds. Point clouds naturally bypass issues related to mesh discretization and topology, simplify data processing pipelines, and enable straightforward implementation of various data augmentation techniques (e.g., random sampling, rotation, noise addition), enhancing robustness and generalization. Moreover, to address your concern about capturing local geometric details, we explicitly evaluated DragSolver's accuracy across different point cloud sampling densities ([**Table 4**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%204.jpg)). Our experiments confirm that DragSolver maintains strong predictive accuracy even with significantly reduced point counts (as few as 10,000 points), effectively capturing both local and global aerodynamic features necessary for accurate drag estimation. We will explicitly clarify this rationale in the revised manuscript.\n\n**Q2: Wheelbase Normalization**\n\n**A2:** We acknowledge the reviewer’s concern regarding normalization of wheelbase length. Although directly using actual vehicle dimensions is intuitive, significant scale differences (e.g., Ford F-450 vs. Peel P50) can cause substantial training instability and poor generalization, especially with limited or noisy training data. Our comparative experiments ([**Table 5**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%205.jpg)) clearly demonstrate that training without wheelbase normalization results in notably higher errors and instability (e.g., Relative *L²*: 0.0057 without normalization vs. 0.0014 with normalization, under limited 30% training data conditions). Importantly, our normalization strategy uniformly scales vehicle geometries to a consistent reference length without altering their inherent shapes, thus significantly reducing scale-related instability and improving predictive accuracy. In practical scenarios, predictions can be effortlessly rescaled back to the actual dimensions of the vehicles. We will explicitly clarify this point in the revised manuscript.\n\n**Q3: MC Dropout vs. Point Cloud Sampling**\n\n**A3:** We appreciate the reviewer’s insightful comment. Indeed, Dropout is commonly employed during training to prevent overfitting. However, in our method, MC Dropout is specifically utilized during inference as an approximate Bayesian approach to estimate epistemic (model) uncertainty [1], which differs fundamentally from the aleatoric (data) uncertainty captured by random point cloud sampling. To clearly illustrate this distinction, we conducted additional comparative experiments ([**Table 6**](https://anonymous.4open.science/r/ICMLRebuttal-F329/Table%206.jpg)). The results show that random sampling alone primarily addresses aleatoric uncertainty and thus cannot adequately capture epistemic uncertainty, which MC Dropout specifically targets. Moreover, combining both methods provides a balanced estimation of both uncertainties, leading to a more robust prediction. We will explicitly clarify this distinction in the revised manuscript.\n\n[1] Gal, Yarin, and Zoubin Ghahramani. \"Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning.\" *ICML*. PMLR, 2016.", "claims_and_evidence": "The presentation is clear.", "methods_and_evaluation_criteria": "1. In line 60, the author emphasize that both local details and global shape influence drag. Why do the authors choose point cloud over mesh as the model input? The point cloud is not always an efficient 3D representation as it requires a vast number of points to capture high-frequency patterns like sharp edges. Also, reconstructing meshes from point clouds leads to artifacts, indicating that point cloud might not be an ideal way that preserves sufficient local feature for estimating drag.\n\n2. In line 210, the authors normalize the sizes of different vehicles by normalizing the wheelbase length to a fixed number. However, for real-world applications, the wheelbase length is not always standardized, *e.g.* applying a fixed wheelbase length for `Ford F-450` and `Peel P50` surely leads to an unexpected result. Therefore, why not directly use their actual size in the real world?\n\n3. In line 264 (Sec. 3.5), the authors propose MC Dropout for the output layer (Module 4 in Fig. 2) for diverse estimations to provide an error interval estimation. Dropout is mostly a technique adopted during training to prevent overfitting. Can this be accomplished with a point cloud sampling method to produce diverse inputs?", "theoretical_claims": "The theoretical claims are well-addressed.", "experimental_designs_or_analyses": "The experiments are well-designed. The ablations are sufficient.", "relation_to_broader_scientific_literature": "This paper is closely related to point cloud processing methods like PointNet, PointNet++, ... as this paper focuses on designing 3D point cloud operators to estimate drag coefficients for vehicles. The paper is also closely related to DrivaerNet, DrivaerNet++, DrivaerML as they all focus on estimating drag coefficients.", "essential_references_not_discussed": "All related works are included.", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces a transformer-based method for estimating the automotive drag coefficient from 3D vehicle models as an alternative to traditional CFD simulations. 
Reviewers initially had concerns about the use of point clouds over meshes and the lack of comparison to more traditional CFD-based approaches, rather than the provided comparisons to other learning-based approaches.
However, the paper provides substantial benchmark experiments where DragSolver demonstrates good performance (in and out-of-distribution) compared to other recently proposed learning-based approaches (point cloud / mesh-based, transformer or SSM backbone). 
The authors provide rigorous ablations and added additional ones during the rebuttal phase, alleviating many of the raised concerns by the reviewers. 
Even though this paper is an edge case, I think it is a valuable contribution and I urge the authors to release the code upon acceptance as promised.
Decision: Accept (poster)