CoPiT: A conditional Position-induced Transformer for aerodynamic field prediction on unstructured meshes

Junfeng Chen; Yongchen Fan; Zhonghua Qiao; Kailiang Wu

physics.comp-phOct 23, 2026

CoPiT: A conditional Position-induced Transformer for aerodynamic field prediction on unstructured meshes

Authors: Junfeng Chen, Yongchen Fan, Zhonghua Qiao, Kailiang Wu

Organizations: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China · Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China · Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

Abstract

High-fidelity Reynolds-averaged Navier--Stokes (RANS) simulations remain a standard tool for aerodynamic analysis, yet each solve on an unstructured mesh is computationally expensive, making many-query tasks such as design exploration, optimization, and multi-condition evaluation costly. Neural operators provide mesh-flexible surrogates for mapping geometry and operating conditions to flow fields; however, in many existing architectures, global parameters such as the freestream state and design variables are concatenated with pointwise inputs, leaving their influence on intermediate features implicit. This paper proposes CoPiT, a Conditional Position-induced Transformer that builds on the Position-induced Transformer (PiT) as its geometric backbone and injects a global condition vector through multi-stage feature-wise linear modulation (FiLM), thereby making parametric modulation explicit while preserving geometry-based attention weights. The model is evaluated on two-dimensional AirfRANS airfoil fields, including scarce-data and extrapolation regimes, and on three-dimensional BlendedNet++ blended-wing-body surfaces. On the AirfRANS full-data test set, CoPiT reduces surface and volume losses by approximately 54% and 24% relative to Transolver, respectively, and achieves the lowest overall normalized test MSE among the compared baselines on BlendedNet++. Matched ablations show that FiLM conditioning generally improves upon input concatenation, while the preferred placement is task-dependent. CoPiT thus provides a practical operator-learning surrogate for parametric aerodynamics on unstructured meshes.

Figures & tables

Figure 1: Schematic of a surrogate-assisted aerodynamic design workflow. Offline CFD generates training data; a learned operator then provides approximate evaluations during design exploration.

Figure 2: Overview of CoPiT architecture. Surface features are encoded onto a latent mesh, then refined by

L

position-attention layers, and decoded at query points. Global conditions modulate processor and decoder features through FiLM.

Figure 3: Feature-wise Linear Modulation (FiLM) [ 25 ] in CoPiT. A global condition vector

\mathbf{c}

is mapped to channel-wise scale

\gamma(\mathbf{c})

and shift

\beta(\mathbf{c})

, which modulate instance-normalized features. The width

D

printed on the feature matrices is the number of channels being modulated, denoted

q

in the text.

Model	Surface loss $\downarrow$	Volume loss $\downarrow$	$\rho_{d}\uparrow$	$\rho_{l}\uparrow$
MLP	0.112950	0.045625	$-$ 0.1169	0.9129
GraphSAGE	0.066220	0.040903	$-$ 0.3033	0.9651
PointNet	0.092511	0.112178	$-$ 0.0220	0.9376
Graph U-Net	0.038649	0.056590	$-$ 0.1384	0.9667
Transolver	0.009120	0.037414	0.1444	0.9981
CoPiT	0.004228	0.028599	0.8388	0.9994

Table 1: AirfRANS full-data test results. The best point estimate in each column is shown in bold . The CoPiT row reports random seed 0; statistics over three seeds are given in the text.

Figure 4: AirfRANS full-field maps on one full-data test case ( $u_{x}$ , $u_{y}$ , $p$ , $\nu_{t}$ ): CFD reference, CoPiT prediction and absolute error. Truth and prediction share a color scale per variable; the error uses a separate scale.

Figure 5: Near-wall view of the same AirfRANS test case as Fig. 4 .

Figure 6: Surface pressure and wall shear on one AirfRANS test case ( airFoil2D_SST_31.812_… ). Top: CoPiT; bottom: Transolver. Blue: CFD; orange: prediction. Wall shear is evaluated from the predicted velocity with a no-slip wall condition.

Panel of Figure 7 — Figure 7: True versus predicted $C_{d}$ (left panels) and $C_{l}$ (right panels) on the AirfRANS full-data test set.

Model	Surface loss $\downarrow$	Volume loss $\downarrow$	$\rho_{d}\uparrow$	$\rho_{l}\uparrow$
MLP	0.219169	0.120089	$-$ 0.2423	0.9235
GraphSAGE	0.194539	0.137171	$-$ 0.1390	0.9813
PointNet	0.182676	0.147593	$-$ 0.0501	0.9493
Graph U-Net	0.147319	0.112956	$-$ 0.0950	0.9760
Transolver	0.086225	0.180010	0.0673	0.9938
CoPiT	0.049273	0.237275	0.4231	0.9963

Table 2: AirfRANS scarce-data test results (200 training cases).

Model	Surface loss $\downarrow$	Volume loss $\downarrow$	$\rho_{d}\uparrow$	$\rho_{l}\uparrow$
MLP	2.089834	0.318465	$-$ 0.1465	0.6423
GraphSAGE	0.179743	0.318270	0.0130	0.9271
PointNet	0.201310	0.332888	0.0062	0.8985
Graph U-Net	0.205886	0.309486	0.0280	0.9045
Transolver	0.044703	0.332677	0.0898	0.9477
CoPiT	0.045225	0.278162	0.1645	0.9741

Table 3: AirfRANS Reynolds-number extrapolation test results.

Model	Surface loss $\downarrow$	Volume loss $\downarrow$	$\rho_{d}\uparrow$	$\rho_{l}\uparrow$
MLP	0.876219	0.836325	0.0378	0.8608
GraphSAGE	0.763761	0.765209	0.0546	0.9078
PointNet	0.584604	0.870350	0.1223	0.9357
Graph U-Net	0.696672	0.843954	$-$ 0.1955	0.9344
Transolver	0.237925	0.423247	0.0692	0.9957
CoPiT	0.261624	0.727504	0.5921	0.9579

Table 4: AirfRANS angle-of-attack extrapolation test results.

	MLP	GraphSAGE	PointNet	Graph U-Net	Transolver	CoPiT
Params	19,988	29,204	75,244	65,820	5,996,868	1,494,026

Table 5: Trainable parameter counts (AirfRANS full-data models). The six-channel CoPiT checkpoints used for the other splits have 1,494,538 parameters.

Variant	Surface $\downarrow$	Volume $\downarrow$	$\rho_{d}\uparrow$	$\rho_{l}\uparrow$
Geometry-only PiT	0.859223	1.013652	0.3449	0.7797
Concatenation only	0.009882	0.035483	0.6478	0.9995
Widened concatenation	0.008969	0.037486	0.7104	0.9993
Processor-only FiLM	0.007099	0.037301	0.6774	0.9991
Decoder-only FiLM	0.002370	0.026978	0.8273	0.9992
Complete CoPiT	0.004228	0.028599	0.8388	0.9994

Table 6: AirfRANS full-data test ablation (same evaluator as Table 1 ). Single-run point estimates; the lowest field loss and highest rank correlation in each column are in bold .

Model	Overall $\downarrow$	$C_{p}$	$C_{f,x}$	$C_{f,z}$
CoPiT	0.0518	0.0175	0.0978	0.0400
Transolver	0.0568	0.0188	0.1107	0.0411
FiLMNet	0.0691	0.0202	0.1356	0.0516
PointNet	0.0715	0.0236	0.1336	0.0572
GraphSAGE	0.0859	0.0346	0.1416	0.0817
Graph U-Net	0.1327	0.1320	0.1614	0.1046

Table 7: BlendedNet++ forward surface regression under a common reimplementation protocol (test set,

N=2{,}498

). Single-run normalized MSE; the lowest point estimate in each column is shown in bold .

Figure 8: BlendedNet++ surface $C_{p}$ on test_case_0674 : CFD reference, CoPiT and Transolver predictions (shared color scale), and absolute errors (shared color scale).

Figure 9: BlendedNet++ streamwise skin friction $C_{f,x}$ on the same test case as Fig. 8 : CFD, CoPiT and Transolver (shared color scale), and absolute errors (shared color scale).

Figure 10: BlendedNet++ spanwise skin friction $C_{f,z}$ on the same test case as Fig. 8 : CFD, CoPiT and Transolver (shared color scale), and absolute errors (shared color scale).

Variant	Overall $\downarrow$	$C_{p}$	$C_{f,x}$	$C_{f,z}$	Params
Concatenation only	0.0741	0.0263	0.1382	0.0577	1.19M
Hybrid (broadcast+FiLM)	0.0570	0.0203	0.1066	0.0442	1.50M
FiLM-only (CoPiT)	0.0530	0.0181	0.0991	0.0417	1.50M

Table 8: BlendedNet++ conditioning ablation on the official forward test set (

N=2{,}498

). Single-run normalized MSE; lowest point estimate in each column in bold . All variants use the same PiT backbone and Euclidean positional metric. Metric on one random 20,000-point subset per case, as recorded at the end of training.

Dataset	Model	Wall-clock
AirfRANS Full	CoPiT	11.6 h
AirfRANS Full	Transolver	19.3 h
BlendedNet++	CoPiT (FiLM-only)	25.8 h
BlendedNet++	Transolver	21.3 h
BlendedNet++	Concatenation only	24.1 h
BlendedNet++	Hybrid	25.8 h

Table 9: Single-run training wall-clock time on one A100. AirfRANS Full uses 398 epochs; BlendedNet++ uses 100 epochs on the official

8{,}992/1{,}000/2{,}498

split.

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Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models. Evaluating design changes through numerical simulation requires meshing the continuous geometry, a computationally expensive and often brittle process that can require manual intervention and replaces the continuous representation with a discrete approximation. Most neural surrogates accelerate the simulation, but inherit this representation gap by relying on meshes, point clouds, voxels, or other sampled approximations of geometry. We introduce CANTO, a transformer neural operator that maps directly from continuous CAD geometry to physical fields, without meshing the input geometry. We develop a theoretical framework for learning operators from geometric manifolds to function spaces of physical fields, representing geometry through sequences of parametric patches. CANTO instantiates this framework by directly tokenizing non-uniform rational B-spline (NURBS) patches from their control points, knot vectors, and weights, and predicts continuous surface and volume fields at arbitrary query locations. We evaluate CANTO on four automotive and aircraft aerodynamics industry benchmarks: AhmedML, WindsorML, DrivAerML, and HiLiftAeroML. CANTO achieves state-of-the-art accuracy on most evaluated surface and volume prediction tasks, including a 19.8% reduction in surface-pressure relative

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