TandemFoilSet: Datasets for Flow Field Prediction of Tandem-Airfoil Through the Reuse of Single Airfoils
Authors: Wei Xian Lim
Loh Sher En Jessica
Zenong Li
Thant Zin Oo
Wai Lee Chan
Adams Wai-Kin Kong
Venue: ICLR 2026 Poster (2026)
Keywords: Physics-informed Graph Neural Network; Tandem-Airfoil; Flow Field Prediction; CFD; Aerodynamics;
Primary area: datasets and benchmarks
TL;DR: We introduce TandemFoilSet, a paired set of 5 tandem-airfoil + 4 single-airfoil CFD datasets (8,104 simulations total) and baseline benchmarks to enable scalable ML flow-field prediction for tandem-airfoil interactions.

Abstract: Accurate simulation of flow fields around tandem geometries is critical for engineering design but remains computationally intensive. Existing machine learning approaches typically focus on simpler cases and lack evaluation on multi-body configurations. To support research in this area, we present **TandemFoilSet**: five tandem-airfoil datasets (4152 tandem-airfoil simulations) paired with four single-airfoil counterparts, for a total of 8104 CFD simulations. We provide benchmark results of a curriculum learning framework using a directional integrated distance representation, residual pre-training, training schemes based on freestream conditions and smooth-combined estimated fields, and a domain decomposition strategy. Evaluations demonstrate notable gains in prediction accuracy. We believe these datasets will enable future work on scalable, data-driven flow prediction for tandem-airfoil scenarios.

Published as a conference paper at ICLR 2026
TANDEMFOILSET: DATASETS FOR FLOW FIELD PREDICTION OF TANDEM-AIRFOIL THROUGH THE REUSE OF SINGLE AIRFOILS
Wei Xian Lim"
Loh Sher En Jessica"
Emerging nanoscience Research Institute Nanyang Technological University 50 Nanyang Ave, Singapore 639798 weixian001@0 @e ntu. edu.
College of Computing and Data Science
Nanyang Technological University 50 Nanyang Ave, Singapore 639798
1chs00376e ntu. edu.sg
Zenong Li
Thant Zin Oo
College Computing and Data Science Nanyang Technological University 50 Nanyang Ave, Singapore 639798 $ize0018@e ntu. edu.
College of Computing and Data Science Nanyang Technological University 50 Nanyang Ave, Singapore 639798
2oo ac.
Wai Lee Chan
Adams Wai-Kin Kong
School Mechanical and Aerospace Engineering Nanyang Technological University 50 Nanyang Ave, Singapore 639798 wailee aelle__hha___eee sutd. edu. sg
College of Computing and Data Science Nanyang Technological University 50 Nanyang Ave, Singapore 639798 AdamsKeng@ntu. edu. sg
ABSTRACT
Accurate simulation of flow fields around tandem geometries is critical for engi- neering design but remains computationally intensive. Existing machine learning approaches typically focus simpler cases and lack evaluation multi-body configurations. To support research in this area, we present FandemFoilSSe five tandem-airfoil datasets (4152 tandem-airfoil simulations) paired with four single-airfoil counterparts, for total of 8104 CFD simulations. We provide bench- mark results of curriculum learning framework using a directional integrated distance representation, residual pre-training, training schemes based on freestream conditions and combbne- estimated fields, and a domain decomposition strategy. Evaluations demonstrate notable gains in prediction accuracy. We believe these datasets will enable future work on scalable, data-driven flow prediction for tandem-airfoil scenarios.
INTRODUCTION
In engineering design, simulating flow around geometries using computational fluid dynamics (CFD) is critical step for component modelling. Studying partial differential equations (PDEs) with intricate, multi-body geometries is particularly important for applications like high-lift aircraft wings (Rumsey & Ying. 2002) and wind farm wake interactions (Deskos et al., 2020). Due to the complexity these problems, analytical solutions are generally unavailable. In practice, such complex geometries can be built by assembling simpler shapes, with the downside of increased mesh resolution and domain size requirements, leading to high computational cost (Spalart & Venkatakrishnan, 2016).
'Equal contribution
1 "ooresppnndigg author for fluid dynamics 1 CCurently at Engineering Product Development, Singapore University of Technology and Design, Singapore Cooresponding author for machine learning

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Published as a conference paper at ICLR 2026
Previous work on accelerating flow field prediction using neural networks (NNs) has largely focused on single-object cases with abundant data (Pfaff et al., 2021; Lino et al., 2021; Fortunato et al., 2022; Cao et al., 2023). Industries such as acrospace and marine have previously generated simulation data for single-body geometries like airfoils and hydrofoils, presenting an opportunity to exploit existing datasets for more efficient prediction of flow fields around multiple geometries. However, the extension to interacting multi-body systems, such as tandem configurations, a sequentially arranged airfoil setting, remains underexplored. Such configuration representative of many engineering applications, including compressor blades in turbomachinery (McGlumphy et al., 2007), unmanned aerial vehicles (Yin al.. 2021; Okulski & Lawryńezuk, 2022), hydrofoil systems for maritime vessels (Maram et al., 2021), and race car engineering (Azmi et al., 2017).
To address this gap, we make the following contributions:
TandemFoilSet !; A collection of five high-fidelity tandem-airfoil flow field datasets paired with four single-airfoil datasets under various flow settings. It the first to comprehensively capture interactions between front and rear airfoils, providing foundation for benchmarking data-driven surrogate models in tandem-airfoil aerodynamic environments.
Novel benchmark curriculum learning scheme in which models first trained on single- airfoil flow fields and then fine-tuned on tandem- configurations in a memory-efieiiet multi-network. approach which, on average, outperforms the baselines by 65%.
First use of freestream condition (Ferm, 1990) as a physics prior for residual pre-training and smooth combination of single-airfoil flow predictions, enabling reduced training com- plexity and improved accuracy.
Extension of directional integrated distance (DID) (Jessica et al., 2024) to represent two-body flow fields, effectively encoding directional pressure and velocity distributions in tandem-airfoil configurations. This is the first application of DID in setups with more than one object.
We believe that these datasets and benchmark results will enable further research into data-driven prediction of flow around tandem objects.
2 PRELIMINARIES AND RELATED WORK
2.1 PRELIMINARIES
Graph Construction The CFD simulation mesh M is represented as a graph G EV, where V and E are the sets of nodes and edges, respectively. The mesh nodes are directly represented as graph nodes E V, and the faces between them as bi-directional edges (1. j). (j. i) E E.
Geometry Representations This work employs the shortest vector (SV) and DID methods intro- duced by Jessica et al. (2024) to encode geometries as input node features. The SV represents the shortest vector from node to the geometry, while each DID) value captures the average distance between the node and the geometry within given angular segment, up to a maximum distance dmaz- These were originally demonstrated only on single-object geometries. The DID was numerically calculated via the procedure in Alg. in Appx. E. Although extending the DID to multiple geometries is conceptually straightforward, the numerical calculations grow significantly more complex with each additional object. These challenges are discussed Appx. E.
Solid Bodies within a Flow Flow around solid bodies is fundamental problem in fluid mechanics and, in applications such as aircraft design, is typically modelled within an infinite flow region (Wu, 1976). In such settings, the influence of solid bodies can be viewed as localised force field that accelerates or diverts the flow, with the velocity gradually recovering towards a freestream condition large distances (Fan & Li, 2019). Mathematically, the flow velocity U can be decomposed
U U= - U",
where U is the freestream condition and U' the flow perturbation induced by the body. This decomposition analogous to transformations from inertial to body frames (Speziale, 1998), and
The datasets and the generation code can he accessed here.
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is applicable across turbulent flows (Speziale, 1998) and potential flows (Collicott et al., 2017). This underlying principle informs the design of the smooth-com and residual pre-training techniques in Secs, 4.1 and 4.3, which aim to enhance learning efficiency when leveraging freestream conditions. The datasets constructed reflect these flow characteristics to support future studies exploring rrestream-innormed models.
2.2 RELATED WORK
Previous NN-based strategies for accelerating CFD simulations have explored various architectural adaptations. Graph neural networks (GNNs) have become a leading approach, outperforming MI.Ps and CNNs (Pfaff et al., 2021; Ogoke et al., 2020; Bonnet et al., 2022). Chen et al. (2021) improved performance by incorporating both node and edge features. Encoolerprroessoo-dccode with such graph convolutions were subsequently adopted by Pfaff et al. (2021) and Sanchez-Gonzale et al. (2020), and further improvements in scalability and information propagation speed were achieved through multi-scale designs (Lino et al., 2021; Fortunato et al., 2022; Cao et al., 2023; Gladstone et al., 2024). These developments largely shaped network and convolutional design for CFD prediction, though relatively little attention has been given by these to leveraging existing datasets or embedding physical priors.
architectures
Physical priors have been incorporated through different means by other works. Raissi et al. (2019) integrated physical equations directly into the training loss of NNs, technique later extended to graph convolutional networks (GCNs) (Würth et al., 2024). While effective in reducing reliance on training data, these approaches were evaluated primarily on scenarios without internal objects. Physics-based solvers were also incorporated into the model itself (Obiols-Sales al., 2020; De Avila Belbute-Peres et al., 2020; Lim et al., 2024). Physics-inspired features and loss terms were combined (Libao et al., 2023), building on Pfaff et al. (2021). However, none of these approaches address the interaction of flows between two objects in tandem configurations. The concept of using coarse simulations as a basis for improved predictions has also been explored using CNNs (Kochkov et al., 2021). More recently, Jessica et al. (2024) and Lim et al. (2025) applied residual learning techniques, previously used in image super-resolution (Zhang et al., 2018; Yang et al., 2019), to GCNs for fluid prediction, though this too relied on coarse simulations. Domain decomposition for grid-based simulations has been addressed using fixed-size subdomains and CNN-based architectures (Mao et al., 2024), but did not directly address mesh- based scenarios with tandem shapes.
Finally, public datasets only feature single-body flows (Bonnet et al., 2022; Schillaci et al., 2021; Liu al., 2024; Agarwal al., 2024; Yang et al., 2024) and have limited focus on interactions between multiple bodies like tandem airfoils. On the other hand, multi-geometr data like that of Bartoldson al. (2023) were not paired with their single-object counterparts. In all, the existing datasets did not facilitate research of reusing onjeet data to improve tandem-object flow field prediction.
3 SAND
TandemFoilse FoilSe is comprehensive collection of nine 2D datasets that span wide range of configu- rations across both aerial and ground-effect environments. Built primarily from four-digit NACA airfoil (MPXX) the datasets vary systematically to cover diverse aerodynamic characteristics. They are categorized into two domains: seven air- and two land-based datasets.
The acrial datasets simulate cruise and takeoff conditions. Cruise datasets assume ideal freestream (far from ground), while Takcoff datasets include ground effect. For both, NACA parameters are uniformly sampled M, P E [0, 6] and XX E [5, 25], covering both symmetric and asymmetric airfoils. The land-based Race Car datasets model inverted airfoils as simplified 2D spoilers operating near the ground. Their NACA parameters are also uniformly sampled with M E [2, 9), P E [2, 8), and X e [5, 20). Additionally, five high-lift airfoils (CH10, R423, FX74-C1.5- S1210) were included expand aerodynamic diversity. With 8104 cases, including 4152 tandem- airfoil configurations, HandemFoilSed is the first public tandem- airfoil dataset of its kind, supporting advanced studies in aerodynamics modelling and machine learning-basee based flow prediction.
LA5055,
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Table I: Total number of cases, average grid cells and the range of Re and AOA well stagger, gap, and height (sec Fig. 1) normalised by the chord length, for TandemFoilSet. *Asterisk marks indicate the datascts are in tandem-airfoil configuration. More are available in Appx. A.
DATASET
CASES
AVE. CELLS
Re
AOA [1]
STAGGER
HRIGHT
SINGLE = 0
1014
122788
500
0
*CRUISE AOA= 0
781
351313
500
0
[0.5,2] 2]
[-0.4,0.]
SINGLE A0A=
1014
122788
500
5
*CRUISE AOA= 5
784
351315
500
5
lo.5. 2]
-0.4, 0.4
"TAKROFF
784
271316
500
5
00.5, 2]
[ -.22, 0.6
[0.4. I]
SINGLE RANDOM
1025
111370
10 10
[-5,
"CRUISE RANDOM
900
210181
109,5 ,5 10
[-5.6]
[0.5, 2]
[-0.8, 0.8]
SINGLE INVERTED
87108
100 5 109]
10.0]
1.1]
*RACE
900
130276
[10%, 5 109]
[100.0]
[-0.5,0.00
[0.05,0.] 0.1]
1.11
3.1 DATASET CATEGORISATION AND CONFIGURATION
The datasets are categorised based on flow conditions, providing mix of low and high Reynolds number, Re, scenarios. The Cruise and Takeoff datasets are designed to capture stable aerodynamic characteristics under fixed and low Re 500 conditions, with angles of attack (AOA) set at 0 or 5 The Takeoff dataset incorporates ground effect, with height variations between the airfoil and the ground, simulating conditions typical of UAV takeoff as illustrated in Fig. (a)). These configurations enable the analysis of steady aerodynamic behaviours under controlled flow conditions. In contrast, the Cruise Random and Race Car datasets operate under high Re, randomly selected from the range [10* 5 x 10%] and AOA varying between [- 10 7.). The Race Car dataset captures aerodynamic interactions near the ground, rep- resenting the downwash effect caused by inverted air- foils, a critical feature in regh-performance automotive applications (Fig. 1(b)). It includes a mix of standard NACA airfoils and five high-lift airfoils, enhancing the diversity of aerodynamic shapes.
(a)
Stagger (S)
Gap (G)
Height (H)
(b)
Stagger (S)-
Gap (G)
Height (H)
Figure 1: The schematic drawings tandem- airfoil configuration with ground effect, such as (a) Takeoff and (b) Race Car.
The datasets are summarised in Tab. 1. The near-field x commponent velocity, n, and kinematic pressure, p, contours of selected cases are shown in Fig. 5. In experiments, the datasets are uniformly sampled in a 8:1:1 ratio for training, validation, and test sets, respectively, unless otherwise stated.
3.2 MESHING AND SIMULATION SETUP
Two meshing methods were employed to simulate the aerodynamic properties of the airfoils. For single airfoils, the blockMesh utility from OpenFOAM-v2II a C-grid hexahedral mesh. For tandem-ain airfolls,,, the overset meshing method (Benek et al., 1986) with a combination of background meshes and overset grids generated by Pointwise (Karman & Wyman, 2019) was used. This allowed flexible manipulation of airfoil orientation while maintaining quality.
(Jasak et al., 2007) was used to create
For low Re cases, the freestream velocity was 0.146 m/s with a kinematic viscosity of 2.92 x 10-4 4 m²/s. resulting in Re- 500. Mesh resolution was tuned to achieve boundary layer resolution of yy s 1, ensuring accurate modelling of near- near-wall flow characteristics. High Re simulations setups have higher grid density accommodate the high flow velocities. Further details are in Appx. A.
Simulations were done using the simpleFoam solver for single airfoil setups and overSimple Foam for candem--irfoll setups. Both used the to SST turbulence model (Menter et al., 2003), employing Aavier-Stokes (NS)
the SIMPLE algorithm (Caretto et al., 2007) for steady-state incompressible equations. Detailed configurations of solver settings and turbulence model parameters in Appx. A.
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3.3 DATASET VALIDATION AND VERIFICATION
For all datasets, we performed mesh independence studies to guarantee numerical convergence. For high Re datasets, validated simulation results against experimental data from literature, ensuring that the chosen mesh configuration maintained high accuracy across a range of AOAs. For tandem- airfoil configurations, we replicated previous studies on tandem-airfoil ground effects to benchmark the simulation setup, confirming consistency with established results. These comprchensive checks ensure that the dataset accurately represents aerodynamic phenomena relevant to both low and high Re conditions. The full details of the procedures and results are in Appx. A.
4 BENCHMARKING SETUP
This section introduces the schemes developed to optimise the use of existing single-airfoil datasets for predicting flow around tandem-airfol configurations-2 traditionally resource-intensive task. The overall framework is illustrated in Fig. and proceeds as follows:
1. A NN is pre-trained on single-airfoil cases to predict flow fields from geometry representa- tions and boundary conditions, with freestream conditions serving as prior estimate fields for residual training.
2. The pre-trained network is used to predict flow fields around single airfoils. The z-velocity, y-velocity, and pressure fields are then smoothly combined to generate preliminary tandem- airfoil flow fields.
3. The NNs for the multiple-goometry geometry task are initialised with the weights of the pre-trained single-geometry model.
4. These networks are then residually trained predict tandem airfoil flow fields, using the smoothly combined fields as estimates.
Gicerrry
91.B.
Model
Pealdall Prodicin
Piud Prettcton
(c)
Candta and Pabi
Figure 2: Overview the benchmarking method, using (a) freestream based residual pre-training, (b) smooth- combining, and (c) combined field based residual training. Here, the multiple-goometr model is portrayed as single network for simplicity. Note that multi-NN as shown in Fig. 4 may be used instead.
The multpple-goometry model may consist of multiple neural networks. Full technical details cach stage are provided to establish benchmark for future research single-to-tandem flow prediction.
4.1 SMOOTH--CMMBINIH
This section introduces the procedure for combining multiple fields to obtain a cost-effective amalga- mate field. Let ..,,...y,, denote the L fields to be combined. The combined field ÿ at node is then computed as: 7(1) 71 (i) (1) + (1) (1)
(1)

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.......77....................... YL represents the weight of the respective original fields in the combined field. These will be assigned based on their absolute deviation from a reference field yo-
Here, 7i.....
yo (i) (i)
(
(2)
Do (i) y1 (7) + vo
yr (3)
At nodes where all fields do not deviate from the reference field, y (i) =.... yy yo (i), the weights can be set to (t)...... (i) = L/L. This results in the final combined field exactly matching the reference field these nodes, i.e., ( (z).
Deviation from Freestream When combining flow fields (see Fig. 2(b)), we can assign weights to each field based their deviation from the freestream. Let y/ 1/2 U 1. U2 2 be flow fields such as the x-velocity fields. Let Yo = U. be the freestream flow field with no internal geometry. Then:
(i) (1) (1( + (2)) -2)))
U (1) U.(1
UOI [0((0] (20
n U (0) - (1) (1) 22)
The approach is guided by the physical principles outlined in Sec. 2.1, which suggest that the influence of solid body in flow field can be conceptualised as deviation from the freestream, U' Hence, em- ploying weights based on these deviations creates a combined field that preserves the influences both airfoils. This novel concept offers both accuracy and efficiency due to its simplicity and the extra-low com- putational cost of the freestream conditions.
8.6
6
0.0
Figure 3: for the front airfoil (NACA 0024), deseribed in Eqn. (2). Note that 7/2 for the aft airfoil (NACA 4424) given by 1 7/
Figure 3 provides an example of the resulting weights. Note that, where the flow fields are very similar but not equal the freestream, the weights may still significantly favour the field that is most different from the freestream, such as the blue area close to the front (left) airfoil.
4.2 DID CALCULATION FOR MULTIPLE OBJECTS
As mentioned, the SV and DID are incorporated into the neural network inputs as geometry represen- tations. While both were utilised in previous applications, this marks the inaugural use of DID in multiple-object scenario. Numerically calculating DID values using the original algorithm becomes progressively more complex and time-consuming with the addition of each object.
Deviation from Maximum We propose an alternative procedure that capitalises on the smooth- combining scheme, using the deviation from the maximum value dmaz weights. Here, Yo dmazs and V,..... y2 are the individual DID fields to be combined. This alternative method is detailed in Alg. in Appx. E, providing estimated DID representation of multiple geometries in significantly reduced time-frame, ensuring computational efficiency of DID in a multi-object setting.
4.3 RESIDUAL TRAINING
In this section, we will introduce two residual training schemes, one supporting the utilisation of pre-training, and the other capitalising on the smooth- techniques outlined in the previous section. Residual training, extensively used in image super-resolution. involves utilising an estimate to ease the learning. Let the network output be Ú. Instead of directly predicting the flow field U gt. the model is trained to predict the residual field gt Üest and minimise the loss function,
Ż ž= c (Ugt. UÜests
boundary +{c(Ug
(U I (J+Ues) Üeet) intemal
where Ô est represents estimated flow field and is weight parameter. In CFD cases, this estimate often takes the form cost-effective lower-resolution simulation result (Jessica et al., 2024).

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Freestream Conditions Rather than lower-resolution result, this paper suggests innovative approach: using freestream conditions as estimate for single geometries, or setting U est U oc- This concept, like the smooth-combining procedure, aligns with the physical principles discussed in Sec. 2.1 that assert that freestream conditions should serve of the field. In contrast to lower-resolution fields, freestream conditions do not need to be derived from any physics- based simulator, so its computational cost is minimal. The residually pre-trained network is used not only predict the flow fields of single airfoils for smooth-combining, but also to initialise the weights of the network for predicting tandem-airfoil. This initialisation can improve the final prediction performance. Figure ((a) illustrates the freestream based residual pre-training.
reliable estimate for the majority
Combined Flow Fields To estimate the flow field of tandem-airfoil, we propose employing com- bined flow fields, obtained through the smooth-combining procedure discussed the previous section and illustrated in Fig. 2(b), or setting U est = U. The combined flow field may still differ from the target tandem-airfoil flow field, but it provides cost-effective estimate for improving the learning through residual training. The combined field-based residual training process is visually presented in Fig. 2(c), where it can be seen as part of the consolidated pre-training, smooth mooooth-ommiiingg. and residual training method. Both residual training procedures in the proposed method use different estimates, one based on the freestream conditions and the other on the combined flow fields, which are unlike previous methods based on low-resolution simulation results.
4.4 MULTI-NN INFERENCE PROCEDURE
SV. DID, Iniet values
Frost HN
Multi-NN inference involves training multiple NNs to predict CFD domains with tandem geome- tries. While the multiple-geometty model shown in Fig. 2(c) can be conceivably handled by a single NN, predicting a hundem-geomen case after be- ing pre-trained on solely single-geomes geometry instances would be particularly demanding. The multi-NN technique hence ensures that each NN exclusively predicts a field with at most one airfoil, mitigating these challenges.
From Proticttoo
SV, DID, Overlap
Back
Back Prediciio
sv DED, Overlap valune
HN
Upper Prodiction
To implement this procedure, the domain parti- tioned into distinct sub-graphs, In our implementa- tion, these partitions segregate the graph into front, back, upper, and lower flow fields, with overlapping regions to ensure continuity and synchronisation between the sub-fields. An individual NN is then trained to predict each sub-field. The detailed steps involved in the multi-NN inference procedure are depicted in Fig. 4, and can be summarised as fol- lows:
letet and Overtep
Lower HN
Lower Prediction
Figure 4: Inference of tandem-airfoil fields using multi-NN inference procedure. Overlap regions (highlighted) are updated by the most recent NN.
The CFD domain is subdivided into front, back, upper, and lower flow fields. Inputs like the SV, DID or estimated fields are segmented accordingly.
2. The front flow field is predicted, with inlet values serving input feature for nodes along the inlet. The rest of the field will receive a zero-value here.
3. The predicted values within the overlap regions between the front and back sub-graphs are utilised as the corresponding input features predict the back flow field.
4. The previous step repeated, employing both the inlet values and appropriate overlap data from the front and back fields, to compute the upper and lower fields.
5. All sub-fields are combined to generate the final prediction for the complete flow field, with the latest and most updated prediction used for the overlap regions.
The dependency structure of our multi-NN approach is tied to geometric settings where the divi- sion of flow fields aligns with predefined boundaries and inlets. This structure ensures coherent
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information propagation across sub-domains, similar to domain decomposition in CFD (Chan & Mathew, 1994). Upper and lower fields can be predicted using diverse strategies (c.g., freestream conditions, interpolation), reduce the required NNs. Sub-domains may even be omitted if deemed unnecessary, providing flexibility for diverse CFD applications, similar to domain decomposition customisation (Lim al., 2023). This procedure is then applied to all experiments conducted in this work, except the single-NN variants in Experiment 3, which evaluate the effectiveness of Multi-NN.
5
BENCHMARKING RESULTS
In this section, we evaluate the effectiveness of our schemes through four experiments, assessing: (i) the multi-object DID representation method, (ii) the smooth-combining, pre-training, and residual training schemes, (iii) the multi-NN inference, and (iv) the overall framework in varying flow condi- tions. We employ the five datasets in TandemFoilSet, and two highly influential GNN architectures: MeshGraphNet (MGN) (Pfaff et al., 2021) and invariant edge-GCNN GCNN (IVE) (Chen et al., 2021), to test performance across various flow conditions and convolutional types.
5.1 EXPERIMENT 1: MULTI-OBJECT DID EFFECTIVENESS
In the first experiment, to assess the effectiveness of the multiple-object DID representation, compare MGN performance with and without DID on the Cruise A and Takeoff datasets. No additional methods applied. As shown in Tab. the DID significantly improves performance for both datasets, with over 91% % reduction in MSE test loss for the Cruise AOA=0" case. This suggests that the DID representation remains effective when estimated using the smooth-combining method.
Table 2: MSE (x10-3 performance evaluation of DID.
MODEL DATASET
CRIISE AOA-0"
TAKROFF
MGN SV MGN + SV DID
11.51 5.48
8.17 3,37
1.03 0.60
91.1%
3.74 199
54.22%
5.2 EXPERIMENT ABLATION STUDY
The second set of experiments evaluates the effectiveness of pre-training, smoott--cobiningg, and residual training in various geometric shapes and configurations on both MGN and IVE. Hence, three datasets with fixed and low Re conditions were considered: Cruise AOA=0%, Cruise AOA=50 and Takeoff. We compare the baseline against the following four combinations of the suggested schemes:
PRE: A single-airfoil model is pre-trained, and its weights are used to initialise the networks of the main tandem-airfoil model before training.
PRE-FREE + COMB: Additionally, the freestream condition is used as an input feature to the pre-trained single-airfe airfoil model, and the combined flow fields from the single-airfoil network are used as input features to the main tandem -irrooll model.
RES-FREE RES-COMB: The single-airfoil model is residually trained using the freestream conditions as the estimate fields. Likewise, the main tandem-airfoil model residually trained using the combined flow fields as estimate fields. However, its weights are not initialised with those of the single-airfoil model.
PRE-RES-FREE RES-COMB: The pre-trained single-airfoil model is freestream-based resid- ually trained. Its weights are used to initialise the main combined field-based residually trained tandem-airfoil model.
The single-airfoil dataset with the same AOA value (09 5%) as the tandem-airfoil dataset is used. All models, including the baselines, incorporate both SV and DID. Likewise, multi-NN inference was used in all models. The outcome of this ablation study is shown in Tab. 3.
The importance of the smooth- combined field, as either an input or for residual training, is demon- strated by the last three models always outperforming the first two significantly, with up 70% improvement of accuracy over the MGN and IVE baseline models. While the best performance was occasionally achieved through pre-training residual training alone, the comprehensive method with
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Table 3: MSE (x10 performance evaluation of ablation study. Average improvement is measured relative to baseline over three datasets. The best performance is bold while the second best is underlined.
MODEL/ / DATASET
CRUISE AOA=0"
CRUISE AOA=5"
TAKEOFF
IMPROV.
MGN (BASELINE)
1.03 0.60
1.34 + 0.67
3.74 + 1:95
MGN PRE
1.04 : 0.62
-0.7%
1.21
1 0.64
10.1%
3.69
1 2.01
1.27%
3.55%
MGN PRE-PREC
0,42 0.50
59.3%
0.74
0.58
44.7%
1.31
0.67
64. 9%
56.3%
MGN- RES-PREE + RES-COMB
0.49 + 0.67
52.0%
0.68
+ 0.37
48.9%
1.24
+ 0.67
66.9%
55.9%
MGN PRE-RES-FREE RES-COMB
0.45 1: 0.69
56.1%
0.67
1: 0.50
49.4%
1.12
1 0.57
70.0%
58.5
IVE (BASELINE)
0.85 4: 0.55
1.05 4: 0.69
2.53
4: 1.24
PRE
0.98 0.55
-16.0%
+ 0.59
2.73
1.37
-7.74%
-7.94%
IVE PRE-FREE COMB
0.60 + 0.40
29.2%
0.73 + 0.40
30.0%
0.89
+ 0.39
6..00%
41 11.4%
RES-FREE RES-COMB
0.54 :: 0.34
36.2%
0.62
1: 0.36
40.4%
0.77
1 0.36
%
8%
IVE- PRE-RES-PREE RES-COMB
0,52 1: 0.29
38.9%
0.63 1: 0.30
40.3%
0.83
1: 0.37
67.3%
48.8 %
Table 4: MSE (x10 10 (x)0-)) aerodynamic performance analysis of the airfoil boundary cells, lift, ct and drag, Cd- coefficients for Cruise A and Takeoff datasets, using the comprehensive model relative to the baseline.
CRUISE AOA=5" MSF
TAKEOFF MSF
MODEL. / DATASRT
MSE,
MSF
MSF
MSF
MGN (BASELINE)
2.52 + 2.19
0.05 + 0.07
0.32 +: 0.29
7.61 + 7.27
0.15 0,38
0.23 0.16
PRE-RES FREE RES-COMB
0.84
0.05
0:08
0.21 : 0.40
1.55 1.32
0.04 0.07
0.13 0.26
IMPROVEMENT
58.2%
0%
34.4%
79.6%
73.3%
43.5%
both schemes together shows the most consistent performance, often having errors within an order below the standard deviation of the best model. Overall, these results suggest that the freestream condition and combined-fiee based residual training both individually enhance the performance of models across various scenarios, but perform most consistently when used in conjunction with one another. As it is observed that the MGN models demonstrate a higher average improvement from the schemes at over 55%, subsequent experiments focused on the MGN architecture.
Also, to analyse the model's prediction on the aerodynamic performance of the uundem-airfols Tab, 4 presents the MSE of the airfoil boundary cells, as well as the lift and drag coefficients for Cruise A and Takeoff datasets. The comprehensive method reduced error from the baseline by up to almost 80%, and showed superior capability more complex flow fields as the Takeoff dataset.
5.3 EXPERIMENT 3: MULTI-NN EFFECTIVENESS
In the third experiment, assess the effectiveness of the multi-NN inference by comparing MGN performance in a single-NN versus a multi-NN setup, where the field is split into front and back sub-fields. The upper and lower fields were excluded due to memory limitations. The experiment uses just the Cruise A dataset to showcase its performance. As in Tab. 5, multi-NN outperforms the single-NN in both models, suggesting that separate and specialised NN predictions enhance accuracy,
Table 5: MSE (x10-3) performance evaluation of Multi-NN on Cruise A dataset.
MODEL/ DATA SCHEME
SINGLE-NN
MULTI-NN
IMPROVEMENT
COMB
1.66
0.49 0.67
70%
MGN PRE-RES REE-FREE + RES COMB
1,51 + 1.61
0.45 # 0.69
70%
5.4 EXPERIMENT 4: EFFECTIVENESS IN VARYING FLOW CONDITIONS
In the final experiment, we assess the model under varying conditions using the Cruise Random dataset through two sampling styles: Uniform and Extrapolation. In Extrapolation, the data reflecting the highest and lowest 5% of the AOA, Re, Stagger, or Gap value range is used as the test set, while the training and validation sets are uniformly sampled from the middle 90% range. In addition, the Race Car dataset is also used with uniform sampling to examine the model performance in sophisticated flow conditions. Table shows that the MSE test losses are higher than the datasets with fixed and low Re flow conditions (compared to Tab. 3), which expected due to the varied conditions without increase in dataset model size. The proposed model still achieves up to 94% and 65% reductions in MSE test losses Cruise Random and Race Car datasets, respectively, demonstrating its effectiveness.

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Table 6: MSE performance evaluation on Cruise Random and Race Car datasets with varying flow conditions.
MODEL/ / DATA SCHEME
UNIPORM
ADA
Rs
STAGGER
GAP
RACE CAR
MGN (BASELINE)
1.79 1.38
2.03 1.96
4.85
1.82
1.74 1.66
1.95 1.68
0.61
MGN PRE RES FREE RES-CEOS COMB
0.10 + 0.13
0.18 0.24
0.36
0.53
0.13 0.17
0.14 0.20
0.21 0.29
IMPROVEMENT
94.4%
91.1%
92.6%
92.5%
92.8%
65.6%
5.5 EXPERIMENT MULTI-TASK TRAINING WITH MIXED SINGLE- AND TANDEM-AIRFOI. DATASETS.
Beyond the proposed curriculum scheme, we also evaluate a simple multi-task setup in which a model trained jointly on both single- and tandem-airfoil cases. Table 7 reports results for the Cruise A and Cruise Random datasets under this setting. Compared to the baseline multi-task model, our proposed curriculum (PRE-RES-FREE) RES-COMB) model reduces the MSE values by at least one order of magnitude, corresponding to a 66.3% and 94.4% relative improvements for Cruise AOA=5 and Cruise Random datasets, respectively, showcasing its superior performance.
Table 7: MSE performance comparison between multi-task setup and the proposed curriculum scheme for the Cruise A0A=5 and Cruise Random datasets.
MODEL/ DATASET
CRUISE AOA=5' (x10-3)
CRUISE RANDOM
TASK +RE-COME MGN FRE-RES-FREE RES-COMB
1.04
0.67 + 0.50
0.10 0.13
IMPROVEMENT
66.3%
94.4%
5.6 ADDITIONAL. RESULTS
Additional experiments with Transformer-bias based architecture, Transolver (Luo et al., 2025), which further confirm the robustness of our transfer-learinng pipeline, are reported in Appx. (see Tab. 26). Neural network prediction times showed 76% reduction in wall time compared to simulations (Appx. C). A parametric study of DID parameters demonstrated that remains robust and consistently outperforms the baseline (Appx. F). The smooth-combining method and linear interpolation approaches, and showed to be the best performing (Appx. H). Model accuracy was validated across varying tandem-airte positions and flow parameters, with the method consistently outperforming the baseline (Appx. 1). Navier-Stokes residual analysis confirmed high prediction accuracy, with residuals significantly below maximum thresholds (Tab. 25).
evaluated against freestream
5.7 EXTENSION TO THREE-BAR -ODD CONFIGURATIONS
While our primary evaluation focuses on two-body tandem configurations, we additionally examine more complex three-airfoil setting to probe the scalability of our geometry encoding and multi-NN training pipeline. Specifically, we construct new three-airfoil tandem dataset under the same flow conditions as the Cruise A subset and benchmark both the baseline MGN and our best variant (MGN PRE-RES-FRED error reductions both overall and rear-boundary metrics on this three-body dataset, providing direct evidence that the proposed framework extends beyond the original two-body configuration without architectural changes.
RES-COMB). As detailed in Appx. K, our method achieves substantial
CONCLUSION
This paper introduces TandemFoilSet: five kandem-airfor datasets with four paired single-airfoil datasets to support future research on flow prediction in multi nirti-airfoil scenarios. It also utilises a smooth-combining technique that extends the DID representation multi-object flow fields, a pre- training and residual training procedure leveraging freestream and combined fields, and multi-NN inference scheme for tandem geometries. Our benchmark method was evaluated across two baseline settings and the datasets, demonstrating consistent improvements in prediction performance. Due to computational limitations, we were not able to test the method on diree-dimensional objects or other geometries. For future work, the generalising capabilities of the method should be further tested in, for instance, multiple-airfoil cases like in a turbomachinery stage, and bluff body configurations such cylinder sphere.
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A DATABASE GENERATION
A.1 DATASET CHARACTERISTICS AND CONFIGURATION
In this study, TandemFoilSet is created to form a comprehensive collection of five tandem- and four single-airfoil datasets that explore a wide range of aerodynamic configurations in both aerial and ground-effect environments. They mainly consist of four-digit (MPXX) NACA airfoil, with systematically varied parameters to capture diverse aerodynamic characteristics. The datasets can be broadly categorized into two application domains: air- and land-based configurations.
For aerial configurations, five datasets simulate cruise and takeoff scenarios. The Cruise datasets assume an idealized freestream setting (i.e., height, H - oo), while the Takeoff datasets incorporate ground effect through finite height values. In both cases, the NACA airfoil parameters are uniformly sampled with camber magnitude M [0, 6]. camber location P E [0, 6]. and thickness XX E [5, 25]. These settings ensure a diverse distribution of symmetric and asymmetric airfoils across typical flight conditions.
For land-based configurations, two Race Car datasets model the behavior of inverted airfoils as simplified 21) representations of spoiler operating near the ground. Here, the NACA parameters are also uniformly sampled, with M E [2, 9). P E [2, 8). and XX E [5, reflecting common high-performance automotive aerodynamic profiles. To further enrich the dataset with complex, high-lift geometries, five additional non-NACA airfoils: CH10, E423, FX74-CL5-140, LA5055, and S1210, were included in the land-based configurations.
In general, the datasets can also be categorised based on their flow conditions, providing balanced exploration of both low and high Reynolds number, Re, scenarios as tabulated in Tab. The Cruise and Takeoff' datasets are generated under fixed low Reynolds number conditions (Re = 500), with angles of attack (AOA) set 0% or 5 The Takeoff dataset further distinguished by the inclusion of ground effect, where height variations between the airfoil and the ground are introduced to capture near-ground aerodynamic interactions. In contrast, the Cruise Random and Race Car datasets are designed to capture high Re conditions, where Re is randomly sampled from the range [10*, 5 x 10%), and AOA is varied between [-10-, 77). The Race Car dataset is also subjected to ground effect, specifically configured to simulate the acrodynamic characteristics of high tigh-performance vehicles, where inverted airfoils generate substantial downwash forces.
Figure illustrates representative flow field visualisations for the Cruise AOA=5", Takeoff, Cruise Random, and Race Car datasets, showcasing the diversity of aerodynamic interactions captured.
The x-component velocity, u, and kinematic pressure, p, contours highlight the differences in flow behavior across configurations, including the noticeable ground effect in the Takeoff and Race Car datasets.
In short, Tandem encompasses 8104 cases, including 4152 tandem-airfoil configurations, making it one of the largest publicly available tandem-airfoil datasets for machine learning applica- tions. The datasets are divided into training, validation, and test sets following an 8:1:1 ratio, ensuring a balanced representation of aerodynamic characteristics across all partitions. The distribution of key geometric and flow parameters across the training, validation, and test sets is illustrated in Figs. 6 to For extrapolation experiments, data representing the top and bottom 5% of the AOA or Re ranges are isolated for testing, while the remaining 90% are uniformly sampled to a 8 ratio for training and validation.
A.2 MESH GENERATION
For single airfoil datasets, two-dimensional C-grid type of hexahedral mesh is adopted where the airfoil situated at the center of the domain with a domain size of chord lengths as shown in Fig. 10 (a), whereas the near-field views of the airfoil are illustrated on the left of Fig. 10, showing (b) symmetric and (c) chambered NACA airfoils. For low Re datasets, a total number of 278 airfoil cells is used upon grid convergence study in Appx. A.4 with both the leading edge, ATLE, and trailing edge, ArI grid sizes equal to 5 mm. the first cell thickness adjacent to the airfoil wall, AEEL = mm such that y x 1 to fully resolve the boundary layer around the airfoil geometry without adopting any wall function. Near the domain boundary, the far-field grid resolution, A fara = 0.2 m with a total expansion ratio of 40 along the outwards direction, resulting in total of 188 cells.
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Figure 6: Distribution of stagger and gap parameters in train, valid, and test sets for (top row) Cruise A and (bottom row) Cruise =OA=5 datasets.
For tandem-airfoil datasets, the the computational domain is divided into a background mesh gener- ated by blockMesh and two independent overset meshes created using Pointwise (Karman & Wyman, 2019), which defines the grids surrounding the airfoil as presented in Fig. 11 for configuration with ground effect. set up an overset computational domain, each subdomain is required to assign 'zonelD' via setFields and/or topoSet utility to identify the types of mesh and further classify the cell types as shown in Fig. II. For background mesh, its 'zoneID' always set to 0, whereas other overset meshes can take any number but must contain overset boundary patch where the information is shared. Figure (a shows the red zones having the airfoil shapes identified as hole cells (i.e. omitted during simulation), surrounded by interpolated cells (white contour lines along the airfoils) of the background mesh to couple with the overset meshes. For the overset meshes, as illustrated Fig. 11 (b), the hole (red) cells are identified where they are outside of the background mesh. The internal
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Figure 8: Distribution of stagger, gap, Re, and AOA parameters in train, valid, and test sets for Cruise Random dataset.
cells of overset meshes are labeled as calculated type (blue), whereas the overset boundary cells are labeled interpolated type couple with the background mesh.
For the Takeoff dataset, the background mesh with ground has a domain size of 24 x 11 m which consists of total 240000 structural grids with the smallest grid resolutions (Ar 0.01 m) between -0.5 < & < 4.5 and -1 < = < 0.5 where the overset meshes are located at. For the Cruise datasets, the background domain size larger, having 320000 cells with domain size of 24 x 20 m, and the finest resolution grids range between -0.5 < < 4.5 and -0.5 < 2 < 0.5. The grid size is geometrically increased 0.02m within the extension of 0.5m from the finest grid region; thereafter, the grid size is then further increased to 0.2m geometrically to domain boundaries. 28 four-digit NACA airfoils are meshed as overset meshes using Pointwise (Karman Wyman, 2019). This helps to reduce the total number of cells while retaining the quality of the mesh. Similar to the single airfoil dataset, the thickness of the first cell adjacent to the airfoil wall is ØBL = mm to keep y se 1. The wall boundary cells are then extruded normally until achieving maximum grid size of 0.01m, resulting in approximately 15000 cells for each overset mesh
The same meshing method used in the Cruise and Takeoff datasets is used in high Re datasets (both Cruise Random and Race Car datasets). Each simulation contains approximately a total number of 798 airfoil cells with ArLE ArTE 0.2mm. ArgI. 2 x 10-5 is used such that reduce
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0.1
2
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0.0G
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100
80
60
40
60 80 100
15
10
5
10 15
20
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0
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20
Train Set Counts
Valid Set Counts
Test Set Counts
Tmin Counts
90
Valid Set Counts
25
Test Set Counts
25
20
20
15
15
70
10
60
0.4
0.6
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0.4
G/c
G/c
G/c
Re [10")
2.5
2.5
Re 110
2.5
3
3
5
5
7.5
7.5
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80
60
40
60
80
15
10 5
10
15
15
10
5
0
10
15
Train Set Counts
Valid Set Counts
Test Set Counts
Figure 9: Distribution of stagger, gap, height, Re, and AOA (a) parameters train, valid, and test sets for Race Car dataset.
0.4
Abahii
0.3
0.2
0.1
5
0.2
[m)
00.3
-0.2
0.2
0.4
0.6
0.8
1.2
x m
0.4
0.2
0.1
A
00.1
-10
0.2
-8
-6
-2 0
8 10
x [m]
-022
0.2
0.8
1.2
Figure 10: (a) A type C-grid mesh is adopted where the airfoil situated at the center of the domain. The near-field views of the airfoil illustrated the left side, showing (b) symmetric and (c) chambered NACA airfoils
the non-orthogonality of the mesh and to ensure the 30 < y < 100 for the wall function to work efficiently. For the datasets that utilize the overset technique, the background mesh consists of domain size of roughly -21 x 10. or 21 x 20 for simulation with or without ground effect. The flow and geometry interaction regions, where the overset meshes placed, are refined to have grid size Az Ay = 0.01 m, then geometrically coarsened to domain boundaries at a cell-to-cell expansion ratio of 1.1.
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(a) Background mesh
10
0.2
0.4
interpolated cells
0.8
-05
0.5
1.5
2.5
3
10
x [m]
(b) Overset meshes
0.2
interpolated cells
0.4
0.0
0.8
0.5
0.5
1.5
2.5
3
-0.5
0
0.5
1.5
x [m]
x [m]
Figure 11: Calculated cells (blue cells), interpolated cells (white cells), and hole (red cells) defined for (a) background and (b) overset meshes. The left column represents the Takeoff meshes, while the right column represents the Race Car meshes.
A.3 BOUNDARY CONDITIONS AND SOLVER SETTINGS
Tables and tabulate the applied boundary conditions the steady-state flow variables, namely the ensemble-average velocity, U and kinematic pressure, which is divided by density, p. and the additional turbulent-related variables, such as turbulent kinetic energy, k. turbulent specific dissipation rate. w. and turbulent kinematic viscosity, Mt, are introduced due adopting k SST turbulence model. No slip condition is used on the airfoil wall, however, slip condition (normal velocity is zero) is applied on the ground to avoid the viscous boundary layer formation. Freestream boundaries refer to the outer domain boundary for which the positive flux (outflow) uses zeroGradient whereas the negative flux (inflow) uses fixedValue.
Table 8: The boundary conditions used for the datasets considering low Re flow condition.
Fields
Airfoil Walls
Freestream boundaries
Ground
U
m/
noSlip zeroGradient
freestream Venestr U.c
slip
p
m2
freestreamPrestie ressure
k
m
/
fixedValue kwall
freestream lioo freestream Woo freestream Vtoc
zeroGradient
w
s
fixedValue Wwall
m
fixedValue
Table 9: The boundary conditions used for the datasets considering high Re flow condition.
Fields
Airfoil Walls
Freestream boundaries
Ground
U
m/s
noSlip
freestream Velocity Uno
slip
/
zeroGradient
FreestreamPressuree Poo freestream koo
k
m
/s2
kLowRe FoowReWallFunn
WallFunction
zeroGradient
w s
freestream Who freestream Vi
14
m²/s
nutLowRe WallFunction
For low Re datasets, the freestream velocity, Uo. 0.146 m/s and the kinematic viscosity, 2.92 x 10 m²/s such that the Reynolds number, Re = 500. Gauge pressure is considered, hence the freestream pressure, Poc 0 m² /s2. The turbulent freestream values are chosen based
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recommendations in Menter (1994),
Vioo = 10 V
(4a)
Weo C.
(4b)
lio0 VrooWoo
(4c)
where L is the approximate length of the computational domain, Cy = 2 for thin airfoil, XX > 20, and Cw 2 20 for thick airfoil, XX > 20. For the wall boundary condition, the following is recommended,
kwalt
(5a)
Wwall 10
(5b)
BLUBL
where 31 = 0.075. For high Re datasets, the Uce is calculated based on the Re, which varies between 10 and 5 x 106 with 1.461 10-5 and the chord length of the airfoil as the characteristic length. Note that, the characteristic length in Race Car dataset is the effective chord length, Ceff. between the leading edge of the main airfoil and the trailing edge of the secondary airfoil. omegaWallFunctio is used as the wall functions of the w field, estimating y by blending between the viscous and inertial sublayer estimations with binomial function (Menter, 2001). The corresponding wall functions for k and v4 are kLowRe FallFunntion and nurLowRe Wall respectively.
simpleFoam, an incompressible steady-state turbulence flow solver, is used generate the single airfoil datasets, whereas all the tandem airfoil datasets, which involve overset meshes, are simulated by using overSimple Foam, the simpleFoam counterpart. Both solvers discretize the steady-state incompressible NS equations coupled with & SST turbulence model (Menter et al., 2003), using SIMPLE algorithm (Caretto et al., 2007). The temporal and spatial discretization schemes used in each solver are summarized in Tab. 10 together with the convergence criteria. The resulting linear equations are solved using preconditioned (bi-)conjugate gradient (PCG) and stabilized preconditioned (bi- Jconjugate gradient (PBiCGStab) for symmetric and asymmetric matrices, respectively. The selected preconditioners are diagonal incomplete- Sccomplee-Choleesky.. (DIC), and diagonal incomplete LU, (DILU) for symmetric and asymmetric, respectively. The convergence criteria of each field vary in between the order of 10-6 to 10-5
Table 10: The discretization schemes chosen for dataset generation
Terms
Discretization Scheme
Time (0/Ot)
Steady state
Gradient (V)
2nd order Gauss linear
Divergence (V.)
2nd order Upwind biased
Laplacian (V2)
2nd order Gauss linear
Interpolation
2nd order linear interpolation
A.4 DATASET VALIDATION AND VERIFICATION
A.4.1 DATASETS WITH Re = 500 FLOW CONDITION
To validate the low Re datasets, the grid sensitivity study was conducted on three different levels of grid resolutions as tabulated in Tab. 11 with their corresponding lift, c₁, and drag, cd, coefficients. Note that the thickness of the first cell adjacent to the airfoil wall is maintained at y < for all three grid resolutions. The q and ca differences between the fine and dense cases are less than 0.5%, suggesting that the solutions are converged. Thus, the fine grid resolution is used for the datasets with low Re flow conditions.
For the tandem-airfoil configurations, the parametric study of tandem-airfoil simulations with ground effect in Yin et al. (2021) is replicated for validation purposes. The optimum computational domain size described in Appx. A.2 is obtained upon verifying the boundaries are far enough that the near- field flow interaction between two amdem-airfoiin is not affected by the numerical boundaries. In this validation, the airfoil geometry is fixed using NACA 0012 airfoil with 5 angle of attack
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Table 11: The mesh independent analysis for NACA 0012
Re
500 and 5
Resolutions
Total number of cells
C₁
Ca
Coarse Fine
30832 122952
0.25881
0.18191
0.26326
0.18179
Dense
512864
0.26254
0.18175
under a low Reynolds number of 500. Figure 12 compares and shows that the simulation results (red lines) of c₁, ca, and c//c are in excellent agreement with the reference solutions (Yin et al., 2021) (black lines) for parameter study stagger, S, height, H, and gap, G, respectively.
0.6
0,6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
G
0.3
3
0.3
3 0.3
0.2
0.2
0.2
G/c - 0.2,
Glc 0.2
0.1
11/c 0,4
0.1
0.1
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0.5
1.5
-0.4
0.2
0.2
0.4
0.40.50.6..70.800
S/c
G/c
H/c
0.2
0.2
0.2
0.15
0.15
0.15
Ref. Front
0.1
Ref. Back
3
0.1
0.1
Front
0.05
Back
0.05
S/c = 0,5
0.05
Hic 0.6
0
0.5
1.5
2
00,4 0,2
0.2 0.4
0.4
0.6
0.8
S 3/6
G/c
H
3.5
3,5
3.5
3
3
3
2.5
C/C
2.5
C/
2.5
2
2
1.5
1.5
1.5
0.5
1.5
2
-0.4-0,
0
0.2 0.4
1.4.5.0.600
70.80.9
S/c
G/c
H/c
Figure 12: (top row) Lift coefficient, (middle row) drag coefficient, ca. and (bottom row) lift-to- drag ratio, q/ ca as a function of (left column) stagger, (middle column) gap, and (right column) height distances.
A.4.2 DATASETS WITH IIGH Re FLOW CONDITION
The mesh independence study for datasets that consider high Re flow conditions was conducted with NACA 0012 airfoil at Re x 10th and = 0 flow condition. The domain size 20 chord lengths found sufficient shown in Fig. 13(a). Thereafter, three different grid resolutions were conducted for grid sensitivity analysis. and the са results are tabulated in Tab. 12, indicating the fine grid resolution is converged. With the fine grid resolution and same Re, comparison with experimental data (Abbott & Von Doenhoff, 2012; Ladson, 1988; McCroskey, 1987) was then conducted with a range of angles of attack between - 16 to 18 shown in Fig. 13(b). The simulation results agree
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with the references, especially between -00 to 10", thus, the fine grid resolution is adopted for the generation of the datasets, which consider high Re. For the Race Car dataset, the analyses of computational domain size and accuracy of e and ca presented in Fig. 14, indicating that the adopted domain size and grid resolution are well aligned with the reference Grabis & Agarwal (2019).
Table 12: The mesh independent analysis for NACA 0012 Re 6 10c and 0
Resolutions
Total number cells
Ca
Coarse
60876
0.007575
Fine
111036
0.007594
Dense
345280
0,007565
(a) 0.008
(b) 2
0.0078
0.0076
0
0.0074
0.0072
0.007
-2
0
100
200
300
400
-20
10
10
20
Domain size [c]
1'
Figure 13: Analyses of (a) domain independence and (b) accuracy for high Re datasets (NACA 0012, Re x 10S, a = 09). In (b), the simulated c is given by a solid line, while its experimental coun- terparts are denoted by dotted-line (MoCroskey, 1987), circles (Ladson, 1988), and squares (Abbott Von Doenhoff, 2012).
(b)
(c) 0.2
XX
1.5
1.5
0.15
C
3.1
0.5
Ref.
0.5
Sim.
0.05
0.5
20
40
60
80
100
0
0.5
1.5
2
2.5
3
0
0.5
1
1.5 2 2.5 3
domain size [c]
H/c
H/c
Figure 14:
Analyses (a) domain independence, benchmarking of (b) and (c) Ca of Race Car datasets through Grabis & Agarwal (2019).
B INPUT AND NEURAL NETWORK PARAMETERS
were used to calculate the DID estimates. These were chosen to give arcs each spanning j degrees, centred at intervals. Also, the maximum distance value used was dmaz = The weighing parameter a = 1 was used for the training loss.
In all experiments, the angle segments (0), 07) E [(- (
15x
Table 13 shows the neural network model parameters and training parameters used in the experiments.
The input features (and their sizes) to all models consist of the node positions (2), SV (2), DID (8), inlet/overlap values (3) and freestream/combined field (3) when residual training was done, making the input layer size 15 without residual training and 18 with. Likewise, the number of output features
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Table 13: Neural network parameters
MGN
IVE
NUMBER OF HIDDEN LAYERS
15 128, 128, [128, 128, 128
8
HIDDEN LAYER SIZE (NODE)
128
[128, 256, .... 256 128 (64, 128, 256, 256
HIDDEN LAYER SIZE (EDGE)
LOSS FUNCTION
MSE ADAM LAMBDA DECAY 4 k Ao) 5.0E-05
MSE
OPTIMIZER
ADAM LAMBDA DECAY (1+ & Au) 2.0E-04
LEARNING RATE SCHEDULER
LAMBDA FUNCTION
INITIAL LEARNING RATE (Ap)
Table 14: Number of GPUs used in parallel
DATASET MMODEL.
MGN
IVE
CRUISE AOA=0"
4
CRUISE AOA=5
4
TAKEOFF AOA=5"
4
4
CRUISE RANDOM
4
RACE CAR
4
of all models was 3, for the x-velocity, y-velocity and pressure fields. We use the standard mean squared error (MSE) loss function and Adam optimizer to train the neural networks. A custom decay function is used for the learning rate, defined in Tab. 13. A standard z-score style normalisation is used for all trainings with statistical mean and standard deviation the training dataset, yielding non- zero values of normalised freestream/combined field. All models trained using balf-precision, NVIDIA GeForce RTX 4080 SUPER and RTX 4080 graphics cards and distributed data parallel with the number of GPUs as specified in Tab. 14.
C SIMULATION AND PREDICTION TIMINGS
This section compares the average wall time required to simulate a flow scenario compared to predicting the flow fields using neural network. 'aable 15 shows the average wall time per simulation for each dataset used in training and testing. Note that, each simulation runs in parallel with 64 CPUs. Likewise, Tab. 16 shows the average time per double-airfoil case for each step involved in the neural network prediction using the MGN + PRE-RES-FREE RES-COMB model.
Table 15; Average simulation timings in term of wall time, which uses 64 CPU cores and total CPU times.
NUMBER OF AIRFOILS
DATASET
WALL TIME (S)
ToTAL CPU TIMES (S)
SINGLE A
304.42 + 32.59
19483 + 2086
SINGLE AIRFOIL
SINGLE A
152.64 14.99
9769 + 959
SINGLE RANDOM
249.71 149.04
15981 = 9539
SINGLE INVERTED
364.73 +: 206.25
23343 = 13200
AVERAGE
267.88
17144
CRUISE AOA=0"
252.02 +: 56.63
16129 = 3624
DOUBLE AIRFOIL
CRUISE A
284.93 127.49
18236 8159
TAKEOFF AOA=5
292.96 246.70
18749 = 15789
CRUISE RANDOM
435.47 + 132.39
27870 + 8473
RACE CAR
560.56 + 545.15
35876 = 34890
AVERAGE
365.19
23372
Note that the airlol cases meshed using the standard C-grid mesh, while an overset or "chimera" mesh is used in the double-airfoil cases. The background mesh uses a rectangular mesh and the overset mesh uses handerafted mesh. The background mesh cells are then connected to
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Table 16: Average neural network prediction timings on single GPU.
STAGE
OPERATION
SIMULATION TIME (8)
READ CFD FILE CALCULATE GEOMETRIC FEATURES (x2) INFERENCE (x2)
12.00 0.72
SINGLE AIRFOIL PREDICTIONS
3.34 + 0.27
0.30 0.00
READ CFD FILE PROCESS OVERSET MESH
18.97 + 0.72
DOUBLE AIRFOIL PREDICTION
CALCULATE GEOMETRIC FEATURES
17.03 + 1.02
COMBINE FIELDS
8.60 0.68
INFERENCE
1.10 0.00
EXPORT CFD FILE TOTAL
4.22 0.39
65.57 + 3.80
their nearest neighbour cells in the overset mesh to avoid importing two disjointed graphs. Hence, importing double-airfoil mesh takes longer than single-airfoil mesh.
From Tabs. 15 and 16, we can see that the average total prediction time comes up to only 65.57 seconds. This a 99.7% reduction from the average CPU times 23372 seconds takes to simulate a double-airfoi case using OpenFOAM.
D TRAINING SCHEMES AND TIMINGS
This section presents the average time required by the various NNs in the multi-NN model to train. Note that while the NNs had a maximum epoch of 300, an early-stopping mechanism was utilised, such that training would cease if the validation loss did not improve after 20 epochs, indicating convergence, Additionally, the MGN front models would have minimum epoch of 200 to ensure sufficient training, due having more turbulent validation losses.
0.2
Front
(b) Back
0.15
Training Validation
0.15
Loes
0.1
Lees
0.1
0.05
0.05
50
100)
1.50
200
50
100
150
200
Epach
Epoch
0.05
preppeepeeppe
0.05
Upper
(d) Lower
0.04
0.04
1,085
0.03
0.02
0.02
0.01
0.01
5
10
15
20
25
10
20
30
40
50
Epoch
Epoch
Figure 15: Training and validation loss curves of the various sub-domains.
The training and validation loss curves as shown in Fig. 15 for each sub-domain: (a) front, (b) back, (c) upper, and (d) lower, exhibit consistent convergence. For the front and back sub-domains, the validation losses closely follow training losses across 200 epochs, indicating good generalization to unseen data. Similarly, for the upper and lower sub-domaing both losses converge rapidly and remain stable, suggesting signs of overfitting. It is worth noting that the size of the computational domain (graph) is large. Prior to domain decomposition, the model may have been underfitting, and this structural decomposition allows the networks to effectively capture sub-domain-specine flow
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features without increasing the risk of overfitting. These results collectively demonstrate that approach maintains a balance between model complexity and generalization.
The average training times of the MGN models using the Takeoff AOA=5" dataset shown in Tab. 17. The single-airfoil models had larger training set sizes, leading to longer training times. Likewise, due the simplicity of the upper and lower fields, the upper and lower NNs take the least training time to converge. Hence, these categories are separated.
Table 17: Average neural network training timings
NN
MODEL
AVERAGE TRAINING TIME (8)
MGN (BASELINE)
27,067
SINGLE-AIRFOL
MGN + FREE
28.020
21,883
MGN RES- FREE
10,562
MGN (BASELINE
10,452
MGN PRE-FREE
15,739
FRONT BACK
MGN PRE-FRER COMB
14,056
14,517
MGN + RES-FREE RES-COME
11,878
MGN + PRE-RES-FREE + RES-COMB
20,459
MGN (BASELINE)
2,853
MGN + PRE-FREE
2,644
UPPER AND LOWER
MGN + PRE-FREE + COMB
1,905
2,191
MGN RES-FREE RES-COME
1,770
MGN PRE-RES-FREE RES-COMB
1,782
To facilitate fair comparison of cost accuracy trade-offs, we report the average training wall-time and peak GPU memory usage for the different training variants used in this work. All measurements are obtained on the same GPU and Takeoff dataset for the MGN model. The four regimes Tab. 18 correspond to: (a) the baseline direct training model, (b) the PRE-FREE (pretrain-only) modle, (c) the tresidual-only model, and (d) the full PRE-RES-FREE FREE + RES-COMB
RES-FREE + curriculum model.
Table 18: Average neural network training wall-time and peak GPU memory usage for different training variants.
Regime
Total Training Time (s)
Max GPU Memory (GB)
(a)
10,452
72.88
(b)
14,517
72.88
11,878
76.73
(d)
20,459
76.75
As seen in Tab. 18, the residual-only configuration (c) adds only a modest overhead over the baseline (a), while the full curriculum (d) roughly doubles the training time but operates within similar peak memory envelope.
E A DISCUSSION ON THE DID
In this section, we will discuss the challenges of calculating the DID as done in the original work and present the justification for using the smooth-combine method to estimate it instead.
As mentions previously, the DID in Algorithm I. Although extending the theoretical definition of DID to multiple geometries is conceptually straightforward, the numerical caleulations grow significantly more complex with each additional object. These challenges are indicated in red within Alg. 1, and are illustrated in Figs. 16 and 17.
estimated numerically following the procedure outlined
The first challenge is determining whether the point on the object boundary & is obstructed from the point of reference As shown in Fig. 16(a), in single object scenario, it suffices ascertain
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Algorithm DID calculation. Steps that gain complexity with additional objects are shown in red.
Input: nodes V; positions [[24, Mc) E V} boundary indices bd E V & is the boundary of geometry]: angle segments maximum dmaax
13.00
0 0
DID
for E
J 1 do DID, +-1
0
for E V do
[distance between and k. vk [k bdl: (0, < 8..2 < 65) and unobstructed from d -- minimumtd, dmmox) DID, average
propartion of (8. 8') where (k is unobstructed from 6), vk E [A bd (0) < 0s.k < 03)
wa=DID, end for
we) dmar
append DID; DID, end for
append DID; DID Return: DID)
that either boundary face adjacent to k is on the side of the object that faces However, as seen in Fig. 16(b), there is the possibility that k is obstructed from by the boundary faces of another object. Determining obstruction is process that increases in complexity with the addition of every object.
(a)
Figure 16: Determining obstruction of boundary point from the reference point (a) single-object case and (b) doublc-object case. Note how a boundary point that is unobstructed in the first case may obstructed by another object in the second case.
(a)
(b)
(c)
Figure 17: Determining the angular range that faces object boundary (shown in green) in (a) single-object case. (b) overlapping double-object case, and (c) non overlappren gouble-object case. Note that the angle segment (0,, 8%) used was (0, n 22 in (a) and (b) and (T/4, 3 in (c).
Likewise, the second challenge i is obstructed by object. As shown in Fig. 17(a), in singular object scenario, this proportion can be represented as one continuous segment using the minimum and maximum value of 0,,k, the angle which & is with respect to i. However, in double object scenario, this proportion may be represented as one continuous segment as seen in Fig. 17(b), or two separate segments as seen in Fig. 17(c). Multiple objects involve pair-wise comparisons of each object in determining whether they overlap (as in the former case) or not (as in the latter case), greatly increasing complexity.
determining the proportion of the angular range (0y. where
To circumvent these challenges, smooth-combining method using the deviation from the maximum value dmaz was utilised to estimate the DID fields for multiple objects in this paper, detailed in Alg. 2.
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635
675
0.75
523
023
0.2
-050
-2.5-
-0.0
-3.00
-1.00
-3.00
1.0
100
1.00
4025
-225
-025
4.25
-.00
-1.0
-1.00
DID Direct (A)
DID Smooth-Combine (B)
A-8
Figure 18: Comparison the DID field calculated directly with that from smooth--cmbiied estimate. Above: Angular range is [0, T|. Below: Angular range is [T/2, 3 3r/2].
Algorithm 2 DID estimation for number of objects.
Input:
positions [[=1; 9.) : E boundary indices bd,
boundary geometry
V geometries
I E {1...., L:: angles segments [0,, 03) : J maximum value dmaa
Vo dmaz
for = 1 to do
y1 DID; calculated using Alg. boundary
end for
DIDest + combined field y calculated using Equ. and Equ. 2, with (Mo- Mr.))
Return: DID...
While there is difference between the resulting comobh-commined fields from the direct DID calculation, it is important to highlight that these differences are minimal. To illustrate this, the DID fields for two angle segments from a direct calculation and smooth-combining, as well as their difference, are shown in Fig. 18.
As can seen from the figures, the most significant difference occurs in points when both objects within its angular range. In these areas, the smooth-comee coootthcommiintiion will overestimate the DID when the objects do not overlap (blue regions), and underestimate the DID when the objects overlap (red regions). On the other hand, when only one object is in the angular range of i, both the direct numerical calculation and the unobstructed boundary point the objects using relatively uniform weight. Hence, there is little difference between the two. Importantly, the smooth-combined calculation produces a close estimate without harsh lines.
combined calculation calculates its average distance to every
The timings and maximum memory usage of the DID calculation done directly was estimated using a small sample of the datasets. is compared against the estimation using the smooth-combine scheme in Tab. 19. Note that the direct calculation was not optimised, and some steps that could be done in parallel were instead done in succession. Doing them in parallel would decrease the timing but increase the memory overhead.
Table 19: Comparison of DID calculation times and memory overhead
CALCULATION TYPE
AvG. TIME PER FIELD (S)
MAX. MEMORY USAGE (GB)
DIRECT WITH SMOOTII-COMBINE
5221.641
25.5
3,492
23.3
As can be seen, using the smooth-combined estimate saves a significant amount calculation time for similar amount of memory required, making it the ideal choice. For dataset of size 784, the direct DID calculations would take an estimated 47 days. The accuracy performance of the direct DID is hence irrlevvnt when the goal is produce faster results than numerical simulations.
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F SENSITIVITY STUDIES ON THE DID
To ensure fair comparison with the baseline methods, hyperparameters such learning rate, network depth, and layer sizes were kept consistent with the baseline settings. This minimizes variability and ensures that the observed improvements are due to our proposed approach rather than hyperparameter tuning. However, sensitivity to certain domain-specinf parameters, such as the maximum DID distance, dmaz and the number angle segments used DID computation. could impact performance. These parameters influence the granularity of the directional distance representation and its ability to capture relevant physical interactions. Sensitivity studies were conducted using the model MGN PRE-RES-FREE RES-COMB on the Cruise AOA= 5 dataset.
The results, as shown in Tab. 20, reveal the impact of varying dmaz and the number of angle segments on MSE.
Table 20: MSE performance evaluation for sensitivity studies on DID parameters.
dmax
No. OF ANGLE SEGMENTS
MSE(x10 (x 10
8
0.6810..50
2.5
8
0.8510.38
5
4
0.8910.44
5
16
0.51
Reducing dmaz to 2.5 increases MSE, likely because a smaller dmee limits the model's ability to capture longer-range interactions. Similarly, decreasing the number of angle segments to 4 also leads to higher emrоr,, suggesting that fewer angle segments reduce the directional resolution of the DID contrast, increasing the number of angle segments to 16 improves the performance at the expense of higher computation time for the DID features. Compared with the baseline MGN, whose MSE is 1.34 x 10-2 on the same Cruise AOA= 5 dataset, the variations of these results are relatively minor, indicating the proposed method's 's robustness and insensitivity to these parameters.
representation.
G FEASIBILITY OF INDIVIDUAL DID OF EACH OBJECT
Computing single DID for both objects simultaneously is primarily a practical decision aimed at improving efficiency and scalability. While Alg. 1 can technically compute a single DID for multiple objects simultaneously, its numerical complexity increases significantly with each additional object, resulting in slower training and inference speeds. For instance, Alg. required 5222 seconds to compute the DID for tandem airfoils, whereas Alg, 2, which computes separate DIDs for individual airfoils and then combines them, completed the task in only 3.5 seconds.
Additionally, calculating separate DIDs for each object would increase the input size proportionally to the number of objects. If the dimension of a single object's 's DID is N and there are M objects, the total input size would scale as N x M. leading higher memory requirements and computational load on GPUs. By combining the DIDs into single representation, our approach maintains scalability and significantly reduces computational overhead. Algorithm 2 strikes an effective balance, allowing for efficient handling of multi-object scenarios without sacrificing performance, discussed in Appx. E.
To further evaluate the feasibility of using individual DIDs for each object, experiment was conducted to compare the performance and resource usage of individual DIDs versus a single combined DID using the model MGN PRE- RES-FREE RES-COME on the Cruise = 5 dataset. The results, as tabulated in Tab. 21, reveal that the single combined DID achieves better computational efficiency and prediction accuracy.
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Table 21: Performance evaluation of experiment using single combined and individual DID on Cruise =OA= 5
datasct.
METHOD
AVERAGE GPU MEMORY USAGE (GB)
MSE (x 10 1)
SINGLE COMBINED DID INDIVIDUAL. DID
16.64
0.68
23.37
0.80++0442
H
SHMOTH-COMBINGE METHOD VALIDATION
To validate the effectiveness of the smooth- method, compared its performance against freestream and a simple linear interpolation weighted by the distance to each airfoil as defined,
( (1)- -U1()) + (1 (1-)(1) U2(1),
d2
(6)
(1) d1(1) d2
where dj and d2 are the shortest distances to front (leading) and back (trailing) airfoils, respectively, Figure 19 illustrates the weighting field, 71. generated from distance-based linear interpolation, showing a smooth gradient between the two airfoils. The comparison between (a) freestream, (b) distance-based linear interpolation, and (c) smooth-combining methods is presented in Fig. 20, which shows the absolute error contours of the combined velocity components and pressure fields relative their corresponding ground truths. The smooth-combining method demonstrates the lowest errors, particularly the downstream and flow interaction regions, where the freestream and linear interpolation methods show pronounced inaccuracies.
1.0
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Figure 19: Distance-based linear interpolation weight values for the front airfoil, 71-
This qualitative observation is supported by the quantitative results in Tab. 22, where the smooth- combining method achieves the lowest mean absolute error (MAE) with respect to ground truths across all evaluated metrics, including velocity components and pressure. Specifically, the smooth- combining method outperforms both freestream and linear interpolation with an overall MAE (x10-3) of 1.46 + 0.31, compared to 1.76 + 0.23 and 6.45 + 0.53 for linear interpolation and freestream, respectively.
Table 22: MAE of combined flow fields via various methods against ground truths.
METHOD VARIABLE
u (x10-2)
" (x10-3)
p (x10-*)
OVERall (x10-9)
FREESTREAM
1.57 + 0.13
3.08 + 0.28
5.45 + 0.58
6.45 + 0.53
LINEAR INTERPOLATION
0.39 + 0.05
1.14
1.95 + 0.25
1.76 + 0.23
SMOOTH-COME COMBINING
0.31 1 0.07
1.07 1 0.16
1.71 0.31
1.46 1 0.31
To further assess the utility of smooth-combining, we conducted additional experiment using the Linear-interpole flow fields as initial estimators for training the model (MGN + PRE-RES-E FES-FREE + CES-COMB) the Cruise A dataset. As shown in Tab. 23, the smooth-combining approach results significantly lower MSE than linear interpolation. These results confirm that smooth- combining not only provides more accurate starting point for further training, but also captures
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Figure 20: Absolute error contours of combined flow field variables u (top row), (middle row), and p (bottom with respect ground
row) via (a) freestream, (b) distance-baan based linear interpolation, and (c) truth u, U, and p.
complex flow interactions more effectively than alternative approaches. Its superior performance in both initial approximation and subsequent training highlights its importance for this framework.
Table 23: Evaluation of model performance using linear interpolated vs smooth-combined flow fields
METHOD
MSE (x10
LINEAR INTERPOLATION
1.15 1 0.58
COMOIIG
0.68 + 0.50
I ANALYSIS OF PREDICTED FLOW FIELDS
To showcase the effects of varying distance between the two airfoils and increasing AOA on the accuracy of our NN model, have crafted Figs. 21 and 22. The initial qualitatively compares the prediction of x-velocity, û. to the ground truth, MGT, for (a) two closely-separated airfoils (S = 0.92, -0.2) with strong influence by the front airfoil the aft airfoil and (b) two distant airfoils (S = 1.8, G = 0.38) that are just mildly interacting with each other. The latter shows contours of x-velocity for approximately (a) positive and (b) negative AOA extremes considered in this work (i.e., [-5 6%]). All the cases illustrate that, visually, there are little differences between the ground truths and corresponding predictions, thus verifying the robustness of NN model for a decent range of separation distance between the two airfoils and AOA.
As the loss function used training the MGN model includes boundary loss component, which directly penalizes errors on the airfoil surface, ensuring that the model learns to capture surface flow characteristics accurately. Figure 23 illustrate the correlation between the ground truth and MGN-predicted lift, Ct and drag, ca. coefficients for three datasets: (a) Cruise A (b) Takeoff, and (c) Cruise Random. For C₁ the data points generally follow the diagonal line, indicating a strong correlation between the MGN predictions and the ground truth values for both front and back airfoils. However, a slight deviation is observed in the Cruise Random dataset, suggesting greater prediction variability under complex flow conditions. For ca. the predicted values also show
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Figure 21: Comparison of (top) ground truth z-velocity, toa, (middle) predicted ú flow fields by model MGN PRE-RES-FREE RES-COMB, and (bottom) normalised z-velocity error flow fields (a) closely-separated and (b) distant airfoils for Cruise AOA=
dataset.
(a) 4.3x10 :AAA 5.94 (b) Re -3.5x10 :AAA -4.79'
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Figure 22: Comparison of (top) ground truth z-velocity, uer and (middle) predicted à flow fields by model MGN + PRE-RES-FREE + RES- CEs-COMB, and (bottom) normalised æ-velocity error flow fields at (a) positive and (b) negative AOA for Cruise Random datasets.
reasonable correlation with the ground truth, though noticeable spread exists, especially for the back airfoil (blue circles). This spread is more sigmificant in the Takeoff and Cruise Random datasets, indicating that the MGN model has difficulties in capturing velocity-relater drag characteristies in these scenarios. These results demonstrate that the MGN model effectively predicts lift coefficients, dominated by pressure, for both front and back airfoils, but its performance on drag prediction more sensitive to variations in flow conditions and the relative position of the airfoil.
To quantify the accuracy of our NN model across different scenarios, we have tabulated the MSE values of our predictions relative to their ground truths under varying Reynolds number, Re, AOA, S, and G Tab. 24. Like the qualitative assessment in Figs. 21 and 22, Tab. 24 confirms once again the consistent robustness of our NN model, with the MSE remaining within remarkable range of 0.10 (an order smaller than the baseline MSE of 1.79 for the uniform training condition in Tab. 6) regardless of Re, AOA, S. and G.
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(8) Cruise AOA=5"
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Figure 23: Correlation of ground truth and MGN-predicted (left) lift and (right) drag coefficients for (a) Cruise A (b) Takeoff, and (c) Cruise Random datasets. White squares and blue circles represent the front and back airfoils, respectively, in the tandem-airfol configuration.
Additionally, evaluated the normalized residual values of the discrete incompressible Navier-Stokes equations of the SIMPLE algorithm with the predicted x- and y-velocity, â and respectively, in Tab. 25. Both variables were predicted with residual that is least two orders smaller than the maximum value of 1, thus reinforcing the accuracy of our NN as the residual for Stokes equations is direct indicator of error relative the exact solution the simulation (Versteeg & Malalasekera, 2007).
Table 24: Normalized MSE of NN predictions under varying Re, AOA, and G.
VARIABLE
RANGE
MSE
10
0.007 + 0.001
Re
10t < Re < 3 x 10t
0.04 + 0.03
Re 3 100
0.21 0.14
AQA < -2
0.07 + 0.08
AOA
-2 < AOA 2
0.08 + 0.09
AOA 2 2"
0.15 0.18
S< 1. S 1.5
0.11 + 0.13
0.10 0.17
S2 1.5
0.08 + 0.08
G < -0.4
0.09 + 0.10
G
-041 S G <.4. 0.4
1.10+0.0 0.15
G 0.4
0.10 0.10
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Table 25: Mean and standard deviation of normalised residuals of discrete incompressible Navier-Stokes
equations SIMPLE algorithm over 10% the random cruise tandem airfoils datasets.
VARIABLE
NORMALISED RESIDUAL (MAXIMUM
0.00203 + 0.00017 0.0168 0.00074
J ADDITIONAL RESULTS WITH BRASDD ARCHITECTURE
To further assess whether the proposed PRE-RES-FRES + RES-COME model is tied to a particular architecture, evaluate Transformer-based surrogate model, denoted Transolver (Luo et al., 2025), on the Cruise A and Cruise Random datasets. We keep the same mesh-based inputs and training scheme as for the GCN-based models in the main text, and simply replace the network architecture with a Transformer encoder.
Table 26 reports the MSE (x10-2) and the relative improvement by comparing Transolver with and without our curriculum learning pipeline, alongside the corresponding GCN baselines (MGN and IVE).
Table 26: MSE (x 10-2) performance comparison between the Transformer-basee model (Transolver (Luo et al., 2025)) and GCN-based models on the Cruise A and Cruise Random datasets.
CRUISE AOA =
CRUISE RANDOM
MODEL DATASET
MSE
IMPR.
MSE
IMPR.
TRANSOLVER
0.103 0.076
18.68 + 17.48
TRANSOLVER PRE FREE RES COMB
0.060 1: 0.082
41.7
6.25 1: 8.88
66.5
MGN (BASELINE)
1.34 0.67
179 +: 138
MGN PRE RES FREE RES COMB
0.67 0.50
50.0
10 1: 13
94.4
IVE (BASHLINE)
1.0% # 0.69
177 + 137
PRE-RES-FREE + RES-COMB
0. 0.63 0.30
40.0
7.91 + 9.30
95.5
Across all three architectures, applying the PRE-RES-FREE RES-ROSEEEEEE model leads to substantial error reductions, particularly on the more challenging Cruise Random dataset where the MSE often drops by more than an order of magnitude. This supports our claim that the proposed curriculum learning scheme is a general, architecture-agnotti component of our contribution rather than a model-specific trick.
Across all three backhones, applying the PRE-RES-FREE RES-COME pipeline reduces the error substantially, especially on the more challenging Cruise Random subset. This supports our claim that the proposed transfer-learning scheme is general, architecture-agnsste component of our contribution, rather than being tailored to a specific GCN model.
K EXTENSION TO THREE-AIRFOIL TANDEM CONFIGURATIONS
To assess whether the proposed framework generalizes beyond the two-body tandem setting, construct an additional dataset consisting of three-airfoil tandem configurations (i.e., three airfoils placed in series along the streamwise direction). All cases share the same inflow conditions as the Cruise A dataset used in the main experiments, so that any performance differences can be attributed to the increased geometric complexity rather than changed flow parameters.
Dataset setup. We vary both the streamwise stagger and vertical alignment between neighboring airfoils. Let the streamwise distance between the leading and middle airfoils be S12, and between the middle and trailing airfoils be S23. Both are sampled in the range
S12, S23 E [0.5, 1.0],
while the vertical gaps between consecutive airfoils are sampled as
G12, G23 E [-0.4, 0.4).
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measured in chord length units relative to the leading airfoil. In total, this yields 125 three-airfoil configurations. For each configuration, we run high-fidelity unsteady CFD simulations using the same numerical setup and temporal sampling in the Cruise A0A=5 tandem dataset described in the main text.
Models and training protocol. We evaluate (i) the baseline MeshGraphNet (MGN) and (ii) our best-perfone pess-performig variant from the main paper. MGN + use the same architecture, DID geometry encoding, and training hyperparameters as the two-body tandem case. We do not introduce any additional tuning specific to the three-body dataset. The only change is that the DID preprocessing now accounts for three airfoils instead of two, which is a direct extension of the original formulation.
+ RES-COME Both models
Quantitative results. Table 27 reports the MSE on the three-airfoil dataset, using the same evalua- tion metrics in the main text. The overall MSE over the full flow field, and the MSE evaluated on mesh nodes lying on the airfoil boundaries, which is more directly related to lift and drag forces prediction.
Table 27: MSE performance on the newly generated three-airfoil tandem dataset (125 configurations). "Overall" reports the MSE over the full flow field, while "Boundary" reports the MSE restricted to nodes on the airfoil surfaces. Model
Overall MSE (x10-2)
Boundary MSE (x10-1)
MGN (Baseline)
6.49 + 2.04
1.80 + 1.10
MGN RES-FRES- + RES-COMB
2.55 = 1.12
0.97 +|2 1.18
Improvement
60. 60.7%
46.1%
Compared to the baseline MGN, proposed variant reduces the overall MSE by approximately 60.7% and the boundary MSE by 46.1% %. These gains are comparable to (and in some cases larger than) those observed in the two-body tandem experiments in the main paper. Importantly, the improvement is achieved without any additional architectural changes or biree-i tuning, suggesting that the combination of DID geometry encoding. frestteam/ccomiind-fieee residual training. and smooth-combingt naturally scales to more complex multi-bod interactions.
Qualitative error analysis. To visualize how well the model captures the three-airfoil flow, Fig. 24 shows flow fields for representative configuration from the new dataset. For each variable, x-velocity u, y-velocity v, and pressure p. we plot (left) the CFD ground-truth field, (middle) the prediction of SIE-RES-FREE + RES- NES-COMB), and (right) the corresponding absolute
our best model (MGN
error GTT pred].
Despite the strong wake-wake interactions and the narrow gaps between airfoils, the predicted fields closely follow the ground truth solution. The absolute error plots remain nearly uniform and very close to zero in the far field, and the noticeable discrepancies airfoil surfaces and in the downstream wake regions. The error magnitudes are small relative to the dynamic range of each variable, and the wake structures behind all three airfoils are well reproduced. These qualitative observations are consistent with the quantitative improvements reported in Tab. 27, indicating that the proposed framework can accurately model complex three-body interactions using the same architecture and training pipeline as in the two-body experiments.
confined to thin shear layers near the
Summary. Overall, the three-airfoil experiments support our claim that the proposed framework is not restricted to two-body tandem configurations. The same architecture and training pipeline transfer to more complex multi-body setups and continue to yield substantial accuracy gains over a strong MGN baseline. Extending the same ideas to 3D geometries (c.g., finite-span wings or cascades) is natural next step, which we leave as future work.
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lemori UCT prod)
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Figure 24: Representative three-airfoil configuration from the new dataset. Rows correspond to x-velocity (top), y-velocity (middle), and pressure p (bottom). For each quantity, the left column shows the CFD ground-truth field, the middle column shows the prediction of MGN + PRE-RES- FREE + RES-COMB, and the right column shows the corresponding absolute error |GT pred]. The errors small and mainly localised near shear layers and wake regions, indicating that the proposed model accurately captures the body wake interactions and near- near-body body flow.
35


Reviews:
[{"review_id": "uzUyTp3vrX", "reviewer": "Reviewer_rNNu", "summary": "This paper introduces TandemFoilSet, a new collection of CFD simulation datasets for tandem-airfoil configurations. It also proposes a novel curriculum learning framework designed to reuse single-airfoil simulation data to predict the more complex tandem-airfoil flow fields. This method is presented as a benchmark, involving techniques like freestream-based residual pre-training and a multi-network domain decomposition approach.\n\nHowever, this paper seems rushed with many internal contradictions. It also blurs its own contribution (a benchmark, or a better training methodology?). In it's current shape, I have to issue reject. But I am open to rebuttal and reconsideration if authors can fix the problems.", "questions": "1.  Which is the correct dataset structure: the \"five tandem / four single\" datasets claimed in the text, or the \"three tandem / six single\" datasets shown in Table 1?\n2.  What is the correct total case count for the dataset: \"over 4000\" (from the abstract) or \"8104\" (from the main text)?\n3.  What specific data normalization (e.g., Z-score, min-max scaling, division by freestream velocity) was applied to the flow fields before training? This is essential for reproducibility and for validating the freestream residual claim.\n4.  Given that the proposed 4-part Multi-NN architecture from Section 4.4 was not tested, can the authors clarify if *all* results in the paper were generated using the simplified 2-part model mentioned in Section 5.3?\n5. Why is your new \"re-use\" method so slow? what can be the limitations?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1.  The primary contribution is a new, large-scale, public dataset for a complex multi-body aerodynamics problem.[1] This is a valuable artifact for the community, as tandem-airfoil configurations are critical in many engineering applications but under-represented in existing benchmarks.\n2.  The dataset appears comprehensive, covering a variety of flow conditions (low and high Reynolds numbers), angles of attack, and geometries, including ground-effect scenarios (Takeoff, Race Car).", "weaknesses": "This paper is not ready for publication and suffers from significant flaws, ranging from internal contradictions to under-evaluated methodological claims.\n\n1.  The paper is rife with basic contradictions, suggesting a rushed submission.\n    *   **Dataset Count:** The text repeatedly claims the collection contains \"five tandem-airfoil datasets\" and \"four single-airfoil datasets\". However, the paper's own **Table 1** summary clearly enumerates a structure of **three** tandem-airfoil datasets and **six** single-airfoil datasets. This is a critical contradiction regarding the paper's primary contribution.\n    *   **Dataset Size:** The abstract claims \"over 4000 fluid simulations\". The main body and appendix state a total of \"8104 cases\".\n    *   **Dataset Naming:** The naming in Table 1 is ambiguous. For instance, the \"TAKEOFF\" dataset appears to contain *both* single and tandem cases, further confusing the \"5 vs. 4\" or \"3 vs. 6\" structure.\n\n2.  The paper's focus is split. It presents itself as a dataset/benchmark paper, but its novelty is heavily invested in a new, bespoke \"curriculum learning\" method. A benchmark paper should ideally use established methods for a fair comparison, not a new method that is itself under-evaluated.\n    *   This is problematic because the proposed method is not thoroughly tested before. The paper details a 4-part \"Multi-NN\" architecture (front, back, upper, lower) in Section 4.4.\n    *   However, in the experiments (Section 5.3), only a 2-part (front, back) model was tested. The paper admits the \"upper and lower fields were excluded due to memory limitations\". This admission undermines the method's own claim of being \"memory-efficient\" and confirms that the paper's core methodological proposal was not actually validated.\n\n3.   The paper's methodological contributions are overstated.\n    *   The \"First use of freestream condition... as a physics prior for residual pre-training\"  is suspect and maybe automatically conducted by previous methods, because previous papers will usually adopt data normalization. If a standard Z-score (mean=0, std=1) normalization was applied, the mean velocity (which is likely the freestream velocity due to the large portion of surrounding meshes) would have been removed automatically. Hence, this \"contribution\" may simply be a implicitly-effect of standard preprocessing.\n    *   Finally, this new method's inference, Table 15, is way to high (about 1/4th of the GT simulation, Table 14); this makes the whole \"re-use\" questionable due to its high inference cost.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "o1aPsRM9ZV", "reviewer": "Reviewer_PCgw", "summary": "This paper introduces TandemFoilSet, a benchmark dataset for tandem airfoils, comprising five distinct configurations and paired single airfoil cases. The authors argue that no public benchmarks currently exist for these situations, which remain relevant in practical engineering contexts. This makes the dataset a meaningful and practical contribution to the CFD and ML-surrogate modeling community.\nThe paper further explores how single airfoil data can be used for data augmentation or as part of an incremental training strategy for a neural surrogate model based on the Mesh Graph Net (MGN) architecture.", "questions": "- Can you confirm that the dataset will be made publicly available upon acceptance? What format will it use (HDF5, VTK, or native OpenFOAM)? Will example loading scripts be provided?\n- Why were only MGN-based variants considered? Could you compare against simpler transfer learning or domain adaptation approaches, or multi-task setups combining single and tandem cases?\n- What do you think is the main contribution of the paper? Is it more a dataset paper or a method paper? The multi-nn inference procedure feels like a methodological description but is stated within a benchmarking setup. I have the impression that the benchmarking setup is somewhat a repurposed method section, and it is unclear why.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The main strength of the paper lies in the rigor and care taken in generating and documenting the dataset. The simulation parameters, convergence studies, and validation details are described thoroughly, which enhances the dataset’s reliability.\n- Tandem configurations are still important for various engineered systems, and having a public dataset here fills a clear gap.\n- The dataset covers both low and high Reynolds numbers (500 up to 5×10⁶), includes ground-effect scenarios, and reproduces results from established experimental studies (Figures 13–14), lending further credibility.", "weaknesses": "- The dataset is restricted to 2D, primarily NACA 4-digit geometries, and only two-body (tandem) configurations. It’s unclear how well the findings or models generalize to 3D or multi-body scenarios.\n- Much of the paper focuses on the benchmark experiments using variants of MGN and a rather elaborate 4-stage transfer learning pipeline. This feels more like a methods paper, even though the main novelty and value lie in the dataset.\n- Only MGN variants are tested. It would strengthen the results to include simpler transfer learning or surrogate baselines for context.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "dHVfVaemM7", "reviewer": "Reviewer_HGdM", "summary": "TandemFoilSet introduces nine 2D CFD datasets (five tandem-airfoil and four single-airfoil counterparts; 8,104 total cases, 4,152 tandem) spanning cruise, takeoff (ground effect), and race-car regimes across wide ranges of NACA parameters, Reynolds number, Angle of Attack, stagger, gap, and height. The paper also proposes a curriculum/transfer pipeline that (i) pretrains on single-airfoil flows, (ii) smooth-combines single-airfoil predictions into a low-cost estimate of tandem flow, and (iii) uses residual training to refine tandem predictions. Benchmarks with MeshGraphNet (MGN) and an invariant edge-GCN (IVE) show significant MSE reductions and improved force-coefficient errors compared to baselines. Datasets, meshing/solver details (OpenFOAM, (k-\\omega) SST), and verification/validation procedures are documented. A notable technical advancement is the extension of Directional Integrated Distance (DID) from single-object to multi-object geometry encoding via a fast “deviation-from-maximum” combination, yielding significant gains over shortest-vector (SV) features alone.", "questions": "I enjoyed reading this paper and how it could encourage the CFD community to adopt more AI tools. Some of these suggestions are (understandably) difficult to execute on a short rebuttable period, but I provide them for the authors to consider for the future. The critical questions I highlight in bold below: \n\n*Distance-field representations (DIR) via SDF/UDF*: You extend DID to multi-object via a clever, fast combination. Have you considered using an (un)signed distance function as the core DIR, leveraging CAD/CSG-style Boolean merges (min/max, smooth-min) to combine primitives and airfoils? SDFs provide smooth gradients, exact surface normals, and robust interpolation; they could be directionally integrated post-hoc (compute DID on top of SDF), or used directly as multi-channel inputs (distance, normal, curvature). This might (i) simplify multi-body composition, (ii) reduce DID compute, and (iii) improve generalization to arbitrary multi-component layouts. Any reason an SDF-centric pipeline wouldn’t fit your smooth-combining and residual-training framework?\n\n**Transformers on meshes / hybrid models:** Your results champion GNNs; however, many recent surrogate winners are transformer-flavored. Anyway to extend your results? Two concrete suggestions would be to try (a) global tokens atop MGN/IVE (attention across far-field neighbors or via landmark nodes) to better capture long-range tandem wake interactions, (b) Fourier/Transformer hybrids (global spectral mixing for long-range interactions, GNN for local inductive bias). This aligns with the multi-NN approach you already have. \n\n*Time-dependent extension*: Everything here is steady. How would you extend TandemFoilSet to unsteady flows (URANS/LES) and time-dependent learning? \n\n**Generalization beyond airfoils / shape systems** You include some non-NACA profiles and race-car ground-effect cases. Could the dataset (maybe in the future) be extended to include bluff-body to stress fundamentally different separation physics? Can you report on how the approach generalizes to configurations (shape, distances, offsets) that are not in the distribution?\n\n**Compute/latency budgeting**: Residual training with freestream and combined-field estimates is appealing because the estimates are cheap. Could you publish wall-time and memory profiles for (a) baseline, (b) pretrain-only, (c) residual-only, and (d) full pipeline so others can match/compare your cost–accuracy tradeoffs?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "First paired single-to-tandem dataset collection enabling principled curriculum transfer for multi-body aerodynamics (including ground effect). The explicit coupling of dataset design with a smooth-combining + residual strategy is clean and practical. Extension of DID to two-body settings with an efficient approximation is a useful geometry-aware feature innovation.\n\n\nDatasets cover diverse operating regimes (fixed and random (Re), AoA; stagger/gap/height sweeps), with mesh studies and literature validation for high-(Re) cases. Clear solver/BC documentation for reproducability! Careful ablations isolate contributions from pretraining, smooth-combining, residual learning, and multi-NN decomposition (front/back/upper/lower subgraphs). Reported gains are substantial and consistent across two GNN families. Includes aerodynamic QoIs (lift/drag, boundary-cell errors) in addition to field MSE which is helpful for practitioner relevance.\n\n\nTimely step toward complex-geometry learning (tandem interactions, ground effect), a frontier problem for CFD surrogates. The “reuse single-airfoil data for multi-airfoil prediction” recipe is likely to influence future multi-body surrogate design.", "weaknesses": "*Scope: steady 2D only*:  All data and benchmarks are steady (RANS) and 2D. This limits conclusions for unsteady or 3D tandem interactions (e.g., vortex shedding, dynamic stall, tip effects), which are central in many applications. The authors acknowledge compute limits and suggest future 3D/multi-airfoil stages, but this remains my main issue\n\n*Architecture diversity*: While the paper focuses on GNNs (MGN, IVE), the current SOTA for accuracy in many learned PDE settings often involves transformer-style operators. Even a small transformer baseline or a clear interoperability plan would bolster the benchmarking story. (See questions for concrete paths.)\n\n*Geometry encoding design space*: DID/SV are well motivated, but the paper could better situate them versus (un)signed distance fields (SDF/UDF), level-set encodings, or constructive solid geometry (CSG) fusion; especially since combining distance fields is a natural operation when moving from single to tandem geometries.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "Iv7svHuxx0", "reviewer": "Reviewer_CkWJ", "summary": "The paper presents a dataset of fluid simulations around tandem airfoils, as opposed to single airfoils.  4000 simulations are included in the dataset.  The authors further demonstrate training models on single-airfoil datasets and then fine-tuning them on tandem-airfoil configurations.  This is done in Section 4.4 using multiple NNs as opposed to a single NN.  The datasets all appear to be two-dimensional.  OpenFOAM is used for the CFD simulations.  If I am reading the tables correctly, the errors of their NN predictions appear to range from 0.5-8% for various tests.  A DID discretization is used to represent the solids/flow field.", "questions": "None", "limitations": "", "rating": 2, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 1, "strengths": "The paper presents some interesting analysis of what is needed to use a discretization like DID when multiple solids are present in a solid-fluid simulation.  This results in some thoughtful usage of multiple NNs and the \"smooth combining\" procedure in Section 4.1.", "weaknesses": "- The paper claims that the proposed dataset is more useful than single-foil datasets due to motivations from real-world engineering, yet, no demonstrations are provided of how this dataset helps solve any real engineering problems better than a single-foil dataset.  The paper is thus unconvincing on this point.\n- 4000 is a relatively small dataset size in the era of deep learning.  It is not sufficiently justified why this is chosen.\n- The data are 2D-only.  Again, the real world is 3D, so it does not seem very useful to have a dataset of mere 2D configurations.\n- I also don't think tandem airfoils occur in isolation in the real world.  For instance, they should be attached to cars or airplanes.  A dataset of real-world objects that happen to have tandem airfoils, would likely be more useful.\n- The paper appears to use a very niche discretization, DID, as opposed to the various FDM, FEM, FVM, or other discretizations that have been popular in CFD for many decades.  This may limit the usefulness of the analysis in the paper, although the tandem-foil geometries could still be used by other researchers (though that is a pretty trivial contribution).\n- Regarding DID and the benchmarks, the high error rates of 0.5-8% would generally not be considered acceptable by CFD researchers.  Classical CFD solvers can get errors down to e.g. machine precision of like 1e-15.  That being said, other methods like PINNs also produce results with high errors.  But there is no comparison of this method to other AI-based methods to justify its performance - perhaps other PINN or similar NN-based methods would do worse, and it's just the size of the dataset that's the constraint.  There are no such comparisons in the paper, though.\n- It is not clear that a method like DID is really practical for CFD when it becomes so much more computationally difficult with each object added to a simulation, when compared to typical one-way solid-fluid coupling algorithms that can handle hundreds of thousands of solids just about as quickly as one.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)