Airfoil2Vec: Spectral Geometry-Conditioned Neural Surrogate Models for Airfoil Aerodynamics and a Downforce-Generating CFD Dataset

Haitz Sáez de Ocáriz Borde; Flavio Savarino; Andrei Cristian Popescu; Pietro Innocenzi; Pantelis Papageorgiou; Xerxes Xian Chong

physics.flu-dynSep 25, 2026

Airfoil2Vec: Spectral Geometry-Conditioned Neural Surrogate Models for Airfoil Aerodynamics and a Downforce-Generating CFD Dataset

Authors: Haitz Sáez de Ocáriz Borde, Flavio Savarino, Andrei Cristian Popescu, Pietro Innocenzi, Pantelis Papageorgiou, Xerxes Xian Chong

Organizations: Ratio Labs

Abstract

We introduce a dataset of approximately 10,000 Reynolds-Averaged Navier-Stokes (RANS) simulations of steady, incompressible, two-dimensional subsonic flow around downforce-generating NACA 4-digit airfoils, targeting aerodynamic regimes relevant to automotive and motorsport applications (openly available on https://huggingface.co/datasets/ratiolabs/downforce-airfoils). Using this resource, we study geometry-conditioned neural surrogates for fast flow prediction, comparing neural fields with neural ODEs, MLPs with graph-based models, and several spectral geometry-conditioning methods. We further propose Airfoil2Vec, an airfoil-specific spectral geometry encoder that combines the joint contour spectrum with separate spectral representations of camber and thickness, for predicting continuous pressure and velocity fields. We evaluate generalization through angle-of-attack interpolation, interpolation and extrapolation to unseen NACA 4-digit geometries, and generalization to unseen non-NACA airfoils. The resulting surrogate accurately captures aerodynamic quantities and qualitative flow features while providing orders-of-magnitude speedups over conventional computational fluid dynamics.

Figures & tables

Figure 1: Example of a few inverted (negative camber) NACA 4-digit airfoils parameterized by camber

m

, position of maximum camber

p

, and thickness

t

Figure 2: Complementary spectral views used by Airfoil2Vec for representative airfoils.

Figure 3: Controlled camber and thickness variations across the Airfoil2Vec spectral views. In the camber sweep, variations in the airfoil predominantly modify $C_{\mathcal{A}}$ while the thickness representation remains comparatively stable; in the thickness sweep, the converse behavior is observed for $H_{\mathcal{A}}$ . The joint $Z_{\mathcal{A}}$ responds to both types of variation, motivating the additional separated Airfoil2Vec views.

Encoding	Val. MSE	MAE $p$	MAE $v_{x}$	MAE $v_{y}$
No Fourier	$1.08{\times}10^{-3}$ $\pm$ $3.92{\times}10^{-4}$	0.0222 $\pm$ 0.0049	0.0221 $\pm$ 0.0034	0.0147 $\pm$ 0.0017
Fourier ( $\sigma{=}45,l=256$ )	$\mathbf{1.77{\times}10^{-4}}$ $\pm$ $\mathbf{1.29{\times}10^{-5}}$	0.0138 $\pm$ 0.0008	0.0119 $\pm$ 0.0005	0.0106 $\pm$ 0.0010

Table 1: Validation MSE (mean

\pm

std over 3 seeds) for baseline vs. Fourier encoding.

$k$	2	4	6	8	10
GCN	2.17	2.37	2.59	2.54	2.75
GAT	2.10	2.26	2.41	2.55	2.70
GATv2	2.41	2.63	2.78	2.91	3.08
SGC	2.02	2.18	2.31	2.47	2.71
ChebNet	1.98	2.07	2.16	2.30	2.55
GIN	2.13	2.24	2.38	2.56	2.82

Table 2: Validation MSE for different graph neighbourhood sizes

k

\sigma=45

(

\times 10^{-4}

Model	Val. MSE ( $\times 10^{-4}$ )	MAE $p$ ( $\times 10^{-2}$ )	MAE $v_{x}$ ( $\times 10^{-2}$ )	MAE $v_{y}$ ( $\times 10^{-2}$ )
GCN (No Fourier)	$8.17\pm 0.31$	$1.57\pm 0.16$	$2.33\pm 0.34$	$1.48\pm 0.23$
GCN (Fourier)	$\mathbf{2.16\pm 0.06}$	$\mathbf{1.39\pm 0.04}$	$\mathbf{1.24\pm 0.12}$	$\mathbf{1.35\pm 0.12}$
GAT (No Fourier)	$7.99\pm 0.28$	$1.78\pm 0.12$	$2.01\pm 0.21$	$1.55\pm 0.18$
GAT (Fourier)	$\mathbf{2.10\pm 0.07}$	$\mathbf{1.33\pm 0.05}$	$\mathbf{1.73\pm 0.14}$	$\mathbf{0.90\pm 0.06}$
GATv2 (No Fourier)	$8.87\pm 0.55$	$1.82\pm 0.50$	$2.12\pm 0.66$	$1.69\pm 0.64$
GATv2 (Fourier)	$\mathbf{2.46\pm 0.21}$	$\mathbf{1.56\pm 0.18}$	$\mathbf{1.55\pm 0.26}$	$\mathbf{1.00\pm 0.15}$

Table 3: Val loss (mean

\pm

std over 3 seeds) for GNN architectures (

k=2

) with and without Fourier encoding.

# Integration Steps	Validation MSE $\times 10^{-4}$
2	$1.74\pm 0.24$
4	$1.53\pm 0.09$
8	$\mathbf{1.52\pm 0.22}$
16	$1.80\pm 0.59$
32	$1.84\pm 0.41$

Table 4: MLP-NODE val MSE for different numbers of integration steps (mean

\pm

std over 3 seeds).

	Connectivity sweep ( $s=4$ fixed)				Integration-step sweep ( $k=k^{\star}$ fixed)
Model	$k=1$	$k=2$	$k=4$	$k=8$	$k^{\star}$	$s=1$	$s=2$	$s=4$	$s=6$
GCN	1.60	1.58	1.51	1.26	8	1.35	1.44	1.23	1.21
GAT	1.09	1.17	1.23	1.34	1	1.05	1.34	1.16	1.45
GATv2	1.20	1.28	1.30	1.58	1	1.25	1.26	1.28	1.32
SGC	1.25	1.37	1.28	1.34	1	1.22	1.49	1.41	1.52
ChebNet	1.30	1.25	1.45	1.47	2	1.29	1.42	1.49	1.55

Table 5: GNN-NODE hyperparameter ablation. For the graph-connectivity sweep (left), the number of integration steps is fixed at

s=4

. For the integration-step sweep (right), graph connectivity is fixed to the best value

k^{\star}

found for each model in the connectivity sweep. Validation MSE is reported in units of

\times 10^{-4}

GNN	GAT	GATv2	GCN	GIN	SGC	ChebNet
	$(k{=}1,s{=}1)$	$(k{=}1,s{=}1)$	$(k{=}8,s{=}6)$	$(k{=}4,s{=}2)$	$(k{=}1,s{=}1)$	$(k{=}2,s{=}1)$
Validation MSE	$1.19\pm 0.19$	$1.19\pm 0.26$	$1.35\pm 0.06$	$1.09\pm 0.26$	$1.42\pm 0.18$	$\mathbf{1.07\pm 0.11}$

Table 6: Final validation MSE for the selected GNN-NODE configuration of each GNN operator. Each column reports the selected graph connectivity

k

and number of integration steps in parentheses. Results are mean

\pm

standard deviation over 3 seeds (

\times 10^{-4}

Configuration	MLP-NF	GNN-NODE	MLP-NODE
Baseline	$1.77\pm 0.13$	$1.07\pm 0.11$	$1.52\pm 0.22$
Poly2Vec	$1.48\pm 0.49$	$1.03\pm 0.07$	$\mathbf{1.42\pm 0.17}$
Airfoil2Vec	$\mathbf{1.41\pm 0.14}$	$\mathbf{0.98\pm 0.05}$	$1.94\pm 0.14$

Table 7: Geometry-encoding ablation. Validation MSE (

\times 10^{-4}

), mean

\pm

std over 3 seeds. GNN-NODE uses ChebNet (

k=2

, steps

=1

Test	Airfoil	$\alpha$ $[^{\circ}]$	MLP-NF			MLP-NODE			GNN-NODE
			$v_{x}$	$v_{y}$	$p$	$v_{x}$	$v_{y}$	$p$	$v_{x}$	$v_{y}$	$p$
AoA interpolation	NACA4207	-5	0.73	7.36	0.03	0.54	3.44	0.02	0.59	3.41	0.02
Geometry interpolation	NACA4412	0	0.66	11.68	0.03	0.53	8.34	0.01	0.58	7.93	0.02
Geometry extrapolation	NACA8604	-5	0.99	7.31	0.04	0.91	5.36	0.03	1.09	5.81	0.03

Table 8:

\mathcal{L}_{2}

-norm based error (%) of

v_{x}/U_{\infty}

v_{y}/U_{\infty}

p/p_{\infty}

states over the full computational domain for models MLP-NF, MLP-NODE, GNN-NODE. Three tests are performed, one per subset of the validation set: AoA interpolation, geometry interpolation and geometry extrapolation. For each test and state variable, the lowest error across the three models is colored in green.

Figure 4: Comparison of (first row) $v_{x}/U_{\infty}$ and (second row) $v_{y}/U_{\infty}$ between OpenFOAM and the MLP-NODE prediction. This case illustrates the model’s ability to extrapolate airfoil geometry, as the NACA8604 was not included in the training set at any AoA. Overall $\mathcal{L}_{2}$ -norm based error $[\mathcal{L}_{2,v_{x}},\mathcal{L}_{2,v_{y}},\mathcal{L}_{2,p}]=[0.91,5.36,0.03]\%$ .

Figure 5: Pressure coefficient at $\alpha=-5^{\circ}$ for a few airfoil geometries demonstrating AoA interpolation capability of the models. The models were trained on each of these geometries at every AoA except the one shown, which belongs to the AoA interpolation subset of the validation set.

Figure 6: $v_{x}/U_{\infty}$ profiles at 5 streamwise stations in the $\alpha=-10^{\circ}$ wake of NACA airfoils with 8% camber and 4%-18% thickness representative of thin and thick cambered airfoils. Truth: solid colored lines. MLP-NODE prediction: solid black lines.

Figure 7: Spectral noise analysis of $v_{y}/U_{\infty}$ for NACA8604 at $\alpha=-5^{\circ}$ (Appendix G ): prediction-to-truth energy ratio $E_{p}/E_{t}$ , spectral coherence $\gamma^{2}(\kappa)$ and cumulative tail RMS $\sigma_{\kappa}$ versus wavenumber $\kappa$ (cycles/ $c$ ) in the (top) farfield patch and (bottom) near-wake patch. The gray band marks $\kappa<\kappa_{c}$ , excluded from the high-frequency residual $\sigma_{\text{hf}}$ .

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 8 — Figure 8: Grid-independence study and selected mesh for the NACA6412 airfoil at $M_{\infty}=0.2$ and $Re=4.66\times 10^{6}$ .

Ratio of specific heats, $\gamma$	1.4
Specific gas constant, $R$ ( $\mathrm{J\,kg^{-1}\,K^{-1}}$ )	287.1
Free-stream velocity magnitude, $U_{\infty}$ ( $\mathrm{m\,s^{-1}}$ )	68.06
Free-stream density, $\rho_{\infty}$ ( $\mathrm{kg\,m^{-3}}$ )	1.225
Free-stream temperature, $T_{\infty}$ ( $\mathrm{K}$ )	288.15
Free-stream pressure, $p_{\infty}$ ( $\mathrm{Pa}$ )	101341.63
Free-stream dynamic viscosity, $\mu_{\infty}$ ( $\mathrm{kg\,m^{-1}\,s^{-1}}$ )	$1.789\times 10^{-5}$

Appendix

Table 9: Flow conditions.

Figure 9: Performance of simpleFoam on the 276,000 cells mesh. (Left) Clock time per job in seconds vs. number of CPUs for parallel execution. (Right) Completed jobs per hour on a 128 CPU machine. The optimal set-up to complete the largest number of experiments in one hour is highlighted with the boxes.

$\sigma$ sweep w/ $l=256$							$l$ sweep w/ $\sigma=45$
	5	15	30	45	60	75	64	128	256	512
$(\times 10^{-4})$	2.16	2.00	2.16	1.77	2.08	2.08	1.96	1.89	1.77	2.31
	$\pm$ 0.21	$\pm$ 0.06	$\pm$ 1.08	$\pm$ 0.13	$\pm$ 0.43	$\pm$ 0.31	$\pm$ 0.27	$\pm$ 0.30	$\pm$ 0.13	$\pm$ 0.60

Appendix

Table 10: Fourier ablation validation MSE (mean

\pm

std) vs.

\sigma

and encoding dimension.

Set	Subset	Geometries	AoAs per geometry	Flow fields
Training	–	860 NACA geometries	10 (all but $-5^{\circ}$ )	8,600
Validation	AoA interpolation	the 860 training geometries	1 ( $-5^{\circ}$ )	860
	Geometry interpolation	10: NACA1406, 2406, 4409, 4609, 6409, 6609, 1412, 2412, 4412, 6412	11	110
	Geometry extrapolation	37: NACA8404, 8504, 8604, 8704, 8804; $MP17$ , $MP18$	11	407
Total		907		9,977

Appendix

Table 11: Composition of the training and validation sets. Every geometry is simulated at 11 AoAs,

\alpha\in\{0^{\circ},-1^{\circ},\dots,-10^{\circ}\}

. In the geometry extrapolation subset,

MP17

and

MP18

denote NACA

MP17

and NACA

MP18

for

M\in\{7,8\}

and

P\in\{1,\dots,8\}

Figure 10: Comparison of (first column) $v_{x}/U_{\infty}$ , (second column) $v_{y}/U_{\infty}$ and (third column) $p/p_{\infty}$ between OpenFOAM and the MLP-NODE prediction. Streamlines are superimposed. This case is representative of the capability of the model to interpolate AoA since the model was trained on the NACA4207 using all but $\alpha=-5^{\circ}$ . Overall $\mathcal{L}_{2}$ -norm based error $[\mathcal{L}_{2,v_{x}},\mathcal{L}_{2,v_{y}},\mathcal{L}_{2,p}]=[0.54,3.44,0.02]\%$ .

Figure 11: Comparison of (first column) $v_{x}/U_{\infty}$ , (second column) $v_{y}/U_{\infty}$ and (third column) $p/p_{\infty}$ between OpenFOAM and the MLP-NODE prediction. This case is representative of the model’s ability to interpolate airfoil geometry, since the model was trained on geometries similar to the NACA4412 at $\alpha=0^{\circ}$ , as shown. Overall $\mathcal{L}_{2}$ -norm based error $[\mathcal{L}_{2,v_{x}},\mathcal{L}_{2,v_{y}},\mathcal{L}_{2,p}]=[0.53,8.34,0.01]\%$ .

Figure 12: Pressure coefficient at several $\alpha$ for two airfoil geometries featuring medium values of camber and thickness. This demonstrates geometry interpolation capability of the models: both airfoils belong to the geometry interpolation subset of the validation set and were not seen during training at any AoA.

Panel of Figure 13 — Figure 13: Pressure coefficient at several $\alpha$ for a few airfoils with high camber (8%), variable position of maximum camber and very thin (4%) and thick (18%) geometric features. This demonstrates geometry extrapolation capability of the models: all airfoils shown belong to the geometry extrapolation subset of the validation set and were not seen during training at any AoA.

	NACA8404			NACA8504			NACA8604			NACA8704			NACA8804
$\alpha$ $[^{\circ}]$	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN
0	1.53	1.24	1.35	1.60	1.16	1.13	1.82	1.42	1.43	2.23	1.84	2.14	2.77	2.65	2.31
-1	1.38	1.00	1.00	1.50	1.10	1.04	1.73	1.40	1.40	2.14	1.71	1.94	2.58	2.71	2.44
-2	1.32	1.00	1.02	1.39	1.11	1.05	1.59	1.39	1.34	2.06	1.67	1.83	2.60	2.97	2.90
-3	1.24	1.01	1.03	1.30	1.33	1.20	1.49	1.56	1.50	2.25	2.12	2.34	3.10	3.63	3.79
-4	1.22	1.20	1.18	1.31	1.67	1.53	1.66	2.01	2.01	2.80	2.68	3.03	3.60	3.95	4.26

Appendix

Table 12: Wake

v_{x}/U_{\infty}

\mathcal{L}_{2}

-norm error (%) in the field of view

x/c\in[1,2]

y/c\in[-0.2,0.2]

for models MLP-NF, MLP-NODE, GNN-NODE on NACA airfoils with 8% camber and 4% thickness, representative of thin cambered airfoils.

	NACA8418			NACA8518			NACA8618			NACA8718			NACA8818
$\alpha$ $[^{\circ}]$	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN
0	1.69	0.88	1.68	1.63	0.93	1.62	1.81	1.32	1.87	3.01	2.41	2.30	3.71	3.25	3.39
-1	1.77	1.03	1.46	1.71	1.11	1.38	1.96	1.56	1.80	3.17	2.68	2.52	3.68	3.27	3.24
-2	1.87	1.22	1.38	1.82	1.32	1.34	2.20	1.88	2.00	3.43	3.01	2.91	3.87	3.59	3.64
-3	1.98	1.39	1.45	1.97	1.53	1.48	2.50	2.22	2.37	3.64	3.27	3.28	4.02	3.79	3.99
-4	2.12	1.54	1.59	2.15	1.74	1.71	2.72	2.41	2.67	3.75	3.41	3.50	4.16	4.01	4.18

Appendix

Table 13: Wake

v_{x}/U_{\infty}

\mathcal{L}_{2}

-norm error (%) in the field of view

x/c\in[1,2]

y/c\in[-0.2,0.2]

for models MLP-NF, MLP-NODE, GNN-NODE on NACA airfoils with 8% camber and 18% thickness, representative of thick cambered airfoils.

Case	Prediction above $\kappa_{c}$	$E_{p}/E_{t}$	$\gamma^{2}$	$\sigma_{\text{hf}}$	$\sigma_{\text{inc}}$
Exact	$F_{p}=F_{t}$	$1$	$1$	$0$	$0$
Attenuation / omission	$F_{p}=aF_{t}$ , $0\leq a\leq 1$	$a^{2}\leq 1$	$1$	$(1-a)\sqrt{T}\leq B$	$0$
Coherent amplification	$F_{p}=aF_{t}$ , $a>1$	$a^{2}>1$	$1$	$(a-1)\sqrt{T}$ , $>B$ if $a>1+B/\sqrt{T}$	$0$
Spatial displacement	$F_{p}=F_{t}\,e^{-2\pi\mathrm{i}\,\mathbf{k}\cdot\mathbf{s}/L}$	$1$	$\leq 1$	$\leq 2\sqrt{T}$	$\leq\sqrt{T}$
Additive noise	$F_{p}=F_{t}+F_{n}$	$1+E_{n}/E_{t}$	$\approx E_{t}/(E_{t}+E_{n})$	$\approx(\sum E_{n})^{1/2}$	$\approx\sigma_{\text{hf}}$

Appendix

Table 14: Response of the spectral diagnostics above

\kappa_{c}

to controlled departures of the prediction from the truth, derived from their definitions.

T=\sum_{\kappa\geq\kappa_{c}}E_{t}\leq B^{2}

is the truth-tail energy and

E_{n}

the energy of additive noise uncorrelated with the truth (expected values). For omission (

a=0

E_{p}=0

and

\gamma^{2}

is undefined.

Settings
	Farfield	Near-wake
Patch center $(x_{c},y_{c})$	$(0.50,\,2.00)$	$(1.55,\,0.00)$
Patch extent	$x\in[0,1]$ , $y\in[1.5,2.5]$	$x\in[1.05,2.05]$ , $y\in[-0.5,0.5]$
Side $L$	$1$	$1$
Grid $n_{g}\times n_{g}$	$512\times 512$	$512\times 512$
Grid spacing $\Delta=L/(n_{g}-1)$	$1.96\times 10^{-3}$	$1.96\times 10^{-3}$

Appendix

Table 15: Probe-patch settings and derived scales of the spectral noise analysis for NACA8604 at

\alpha=-5^{\circ}

, state

v_{y}/U_{\infty}

(lengths in chords

c

, wavenumbers in cycles/

c

). Cell sizes

h

are Delaunay edge lengths of the cell centers inside the patch (minimum / median / 95th percentile).

		$\sigma_{\text{hf}}$			$\sigma_{\text{inc}}$
Patch	Model	$v_{x}$	$v_{y}$	$p$	$v_{x}$	$v_{y}$	$p$
Farfield	MLP-NF	$3.5\times 10^{-3}$	$4.7\times 10^{-3}$	$2.4\times 10^{-4}$	$3.5\times 10^{-3}$	$4.7\times 10^{-3}$	$2.3\times 10^{-4}$
	MLP-NODE	$2.0\times 10^{-3}$	$1.8\times 10^{-3}$	$1.6\times 10^{-4}$	$1.9\times 10^{-3}$	$1.7\times 10^{-3}$	$1.5\times 10^{-4}$
	GNN-NODE	$1.7\times 10^{-3}$	$2.5\times 10^{-3}$	$1.3\times 10^{-4}$	$1.6\times 10^{-3}$	$2.4\times 10^{-3}$	$1.3\times 10^{-4}$
	omission bound $B$	$1.1\times 10^{-4}$	$1.0\times 10^{-4}$	$5.6\times 10^{-6}$	–	–	–
Near-wake	MLP-NF	$4.0\times 10^{-3}$	$3.9\times 10^{-3}$	$5.6\times 10^{-4}$	$2.9\times 10^{-3}$	$3.9\times 10^{-3}$	$5.5\times 10^{-4}$

Appendix

Table 16: High-frequency residual

\sigma_{\text{hf}}

and its incoherent part

\sigma_{\text{inc}}

for NACA8604 at

\alpha=-5^{\circ}

per model, state and patch (velocities in units of

U_{\infty}

, pressure in units of

p_{\infty}

). Cutoffs from the truth-tail criterion: farfield

\kappa_{c}=(5,\,4,\,4)

and near-wake

\kappa_{c}=(29,\,18,\,5)

cycles/

c

for

(v_{x},\,v_{y},\,p)

. The last row of each block is the omission bound

B=(\varepsilon\operatorname{Var}_{w}q_{t})^{1/2}

, the level reached by a prediction that omits the truth tail (Appendix G ); it is not an upper bound for every noise-free prediction.

	Selig S1223			Clark Y			Eppler E423			Wortmann FX 63-137			NASA LS(1)-0413
$\alpha$ $[^{\circ}]$	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN	NF	MLP	GNN
0	2.09	1.81	2.08	0.76	0.62	0.69	1.79	1.53	2.00	1.41	1.42	1.67	1.22	1.21	1.27
-1	2.17	1.86	2.11	0.75	0.60	0.65	1.82	1.53	1.99	1.46	1.47	1.72	1.26	1.28	1.32
-2	2.26	1.92	2.16	0.76	0.61	0.65	1.87	1.56	2.00	1.52	1.53	1.78	1.30	1.34	1.37
-3	2.36	2.01	2.23	0.78	0.61	0.67	1.94	1.61	2.04	1.60	1.61	1.84	1.32	1.37	1.39
-4	2.51	2.15	2.37	0.80	0.62	0.69	2.04	1.70	2.11	1.69	1.70	1.93	1.35	1.41	1.41

Appendix

Table 17:

\mathcal{L}_{2}

-norm based error (%) of

v_{x}/U_{\infty}

over the full computational domain for models MLP-NF, MLP-NODE, GNN-NODE. Five non-NACA airfoils listed: Selig S1223, Clark Y, Eppler E423, Wortmann FX 63-137 and NASA LS(1)-0413.

Figure 14: Comparison of $v_{x}/U_{\infty}$ at $\alpha=-5^{\circ}$ between OpenFOAM and the MLP-NODE prediction. This case illustrates the model’s ability to extrapolate airfoil geometries beyond NACA 4-digit airfoils using only NACA training data. Five non-NACA airfoils are represented: Eppler E423, Clark Y, Selig S1223, Wortmann FX 63-137 and NASA LS(1)-0413.

Figure 15: Full flow field comparison of (first column) $v_{x}/U_{\infty}$ , (second column) $v_{y}/U_{\infty}$ and (third column) $p/p_{\infty}$ between OpenFOAM and the MLP-NODE prediction for Selig S1223 airfoil at $\alpha=-5^{\circ}$ .

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Neural Field Ensembles for Aerodynamic Surface Prediction: Winning Solution to the ONERA CRM Wall Distribution 2025 Challenge

Machine-learning surrogate models offer a promising alternative to high-fidelity Computational Fluid Dynamics (CFD) simulations for aerodynamic analysis and design. However, constructing accurate surrogates for realistic aircraft configurations remain challenging due to complex geometries, multiple flow regimes, and limited training data. This work presents the methodology that achieved first place in the ONERA CRM Wall Distribution Regression Challenge, which focuses on predicting pressure and skin-friction coefficient distributions over the NASA Common Research Model wing-body-pylon-nacelle configuration under different operating conditions. The proposed approach formulates the problem as a conditional neural field mapping spatial coordinates, surface normals, and operating conditions to aerodynamic wall quantities. Fourier feature encoding, a relative squared error objective aligned with the challenge metric, ensemble learning, and

k

-fold cross-validation are progressively introduced to improve prediction accuracy and exploit the limited training data. Beyond presenting the final methodology, the paper documents the successive model design choices that led to the winning solution through a comprehensive ablation study and discusses several alternative approaches that were investigated but ultimately discarded. On the hidden competition test set, the proposed methodology achieves an overall score of 8.81, outperforming the strongest organizer-provided baseline, which achieved a score of 8.64, while requiring approximately three orders of magnitude fewer trainable parameters. These results illustrate that carefully designed coordinate-based neural fields constitute an efficient and robust framework for aerodynamic surrogate modeling on complex geometries under limited-data conditions.

Lionel Salesses, Caroline Sainvitu, Tariq Benamara

Cenaero, Gosselies, Belgium

Neural Surrogate Modeling Implicit Neural Representations Computational Fluid Dynamics

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Apr 20, 2026

Faster by Design: Interactive Aerodynamics via Neural Surrogates Trained on Expert-Validated CFD

Computational Fluid Dynamics (CFD) is central to race-car aerodynamic development, yet its cost -- tens of thousands of core-hours per high-fidelity evaluation -- severely limits the design space exploration feasible within realistic budgets. AI-based surrogate models promise to alleviate this bottleneck, but progress has been constrained by the limited complexity of public datasets, which are dominated by smoothed passenger-car shapes that fail to exercise surrogates on the thin, complex, highly loaded components governing motorsport performance. This work presents three primary contributions. First, we introduce a high-fidelity RANS dataset built on a parametric LMP2-class CAD model and spanning six operating conditions (map points) covering straight-line and cornering regimes, generated and validated by aerodynamics experts at Dallara to preserve features relevant to industrial motorsport. Second, we present the Gauge-Invariant Spectral Transformer (GIST), a graph-based neural operator whose spectral embeddings encode mesh connectivity to enhance predictions on tightly packed, complex geometries. GIST guarantees discretization invariance and scales linearly with mesh size, achieving state-of-the-art accuracy on both public benchmarks and the proposed race-car dataset. Third, we demonstrate that GIST achieves a level of predictive accuracy suitable for early-stage aerodynamic design, providing a first validation of the concept of interactive design-space exploration -- where engineers query a surrogate in place of the CFD solver -- within industrial motorsport workflows.

Nicholas Thumiger, Andrea Bartezzaghi, Mattia Rigotti +5

IBM Research · Dallara Automobili

PDE Surrogate Modeling Graph Neural Operators Neural Operators Computational Fluid Dynamics Automotive Aerodynamics

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Abstract

Neural Surrogate Modeling

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Faster by Design: Interactive Aerodynamics via Neural Surrogates Trained on Expert-Validated CFD