Signal-Aware Conditional Diffusion Surrogates for Transonic Wing Pressure Prediction

Víctor Francés-Belda; Carlos Sanmiguel Vila; Rodrigo Castellanos

physics.flu-dynApr 13, 2026

Signal-Aware Conditional Diffusion Surrogates for Transonic Wing Pressure Prediction

Authors: Víctor Francés-Belda, Carlos Sanmiguel Vila, Rodrigo Castellanos

Organizations: Theoretical and Computational Aerodynamics Group Flight Physics Department. Subdirectorate General for Aeronautical Systems Spanish National Institute for Aerospace Technology (INTA) · Department of Aerospace Engineering Universidad Carlos III de Madrid Madrid, Spain · Aerial Platforms Department Subdirectorate General for Aeronautical Systems Spanish National Institute for Aerospace Technology (INTA)

Abstract

Accurate and efficient surrogate models for aerodynamic surface pressure fields are essential for accelerating aircraft design and analysis, yet deterministic regressors trained with pointwise losses often smooth sharp nonlinear features. This work presents a conditional denoising diffusion probabilistic model for predicting surface pressure distributions on the NASA Common Research Model wing under varying conditions of Mach number, angle of attack, and four control surface deflections. The framework operates on unstructured surface data through a principal component representation used as a non-truncated, reversible linear reparameterization of the pressure field, enabling a fully connected architecture. A signal-aware training objective is derived by propagating a reconstruction loss through the diffusion process, yielding a timestep-dependent weighting that improves fidelity in regions with strong pressure gradients. The stochastic sampling process is analyzed through repeated conditional generations, and two diagnostic metrics are introduced, the Local Reliability Index and Global Reliability Index, to relate sampling-induced spread to reconstruction error. Relative to the considered deterministic baselines, the proposed formulation reduces mean absolute error and improves the reconstruction of suction peaks, shock structures, and control surface discontinuities. The sampling-induced spread exhibits strong correspondence with surrogate error, supporting its interpretation as a qualitative reliability indicator rather than calibrated uncertainty quantification.

Figures & tables

$b$ =	design space dimension
$\mathbf{C}$ =	design matrix
$C_{p}$ =	pressure coefficient vector
$\mathbf{c}$ =	conditioning vector (flight condition sextuplet)
$c$ =	chord, m
$\mathbf{D}$ =	database’s matrix
$d$ =	number of dataset’s samples

Table 1

Flight condition	M	α	$\bm{\delta_{1}}$	$\bm{\delta_{2}}$	$\bm{\delta_{3}}$	$\bm{\delta_{4}}$
FC 1	0.689	$\mathrm{4}\,\mathrm{1}\mathrm{0}\mathrm{0}$	4.000	-0.510	$-2.220$	$-1.100$
FC 2	0.802	$\mathrm{4}\,\mathrm{0}\mathrm{1}\mathrm{0}$	12.100	-12.750	$8.200$	$-1.460$
FC 3	0.852	$\mathrm{-}\mathrm{2}\,\mathrm{2}\mathrm{8}\mathrm{0}$	-4.300	-17.550	$-4.370$	$-0.220$

Table 2: Sextuplet vector of the selected visualization test cases. All angles are expressed in degrees.

Figure 1: (a) Flight envelope for the NASA- CRM configuration in the $\mathrm{M}$ - $\alpha$ plane. (b) Representative $C_{p}$ distribution over the upper surface of the wing, corresponding to the flight condition vector ( $\mathrm{M}$ , $\alpha$ , $\delta_{1}$ , $\delta_{2}$ , $\delta_{3}$ , $\delta_{4}$ ) $=(0.6,2.5^{\circ},-8.6^{\circ},-8.0^{\circ},-6.36^{\circ},-1.38^{\circ})$ .

Architecture
Activation	ELU
Projection size ( $r$ )	1024
Network depth	5
Feature width sequence	512-256-128-64-32
Diffusion parameters
Diffusion steps ( $T$ )	1000

Table 3: Hyperparameters and architectural specifications of the proposed DDPM framework.

Figure 3: MAE distribution between actual and predicted

C_{p}

fields for the different models and database sets. Visualization test cases are denoted by symbols. Dashed lines represent the global mean MAE for each model. Note that the distribution tails are truncated at the sample extrema (minimum and maximum observed errors).

Figure 4: Chordwise pressure distributions for actual and surrogate

C_{p}

fields across span percentages

\eta=0.3,0.6,0.9

. The position over the airfoil is normalized with the chord

c

Figure 5: Error distribution over the upper surface of the wing for DDPM-S (top row), and MLP (bottom row) models.

Figure 6: Visualization test cases together with the three maximum- and minimum-error flight conditions for the DDPM-S within the training and testing CH spaces.

Figure 7: Convergence of the model’s statistics as a function of ensemble size

N

. Bar symbol denotes mean values across the

l

flight conditions.

Figure 8: Spatial deviation of the model. The rows display the actual fields ( $\mathbf{x}$ ) along with the ensemble means ( $\bm{\mu}_{\theta}$ ) and standard deviations ( $\bm{\sigma}_{\theta}$ ) computed from $N=1000$ independent samples for each flight condition.

Figure 9: (a) LRI curves as a function of

\omega

and the Mach number associated with the flight condition. (b) GRI- MAE -

\mathcal{I}

relationship.

Conditional Diffusion Models

Conditional diffusion models generate or infer data by steering iterative denoising with observations, labels, text, physical constraints, or other inputs.

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Aerodynamic surrogate models trained on high-fidelity CFD data reproduce numerical predictions of both scalar outputs and entire fields accurately, yet their predictive fidelity is limited by systematic discrepancies between CFD and experimental observations. We present an experimentally grounded correction framework that adapts a CFD-trained deep learning surrogate using wind-tunnel PSP measurements. A Geotransolver surrogate trained on 2,300 high-fidelity CFD simulations of the NASA CRM wing-body configuration, spanning geometric variation, Mach 0.70-0.85, and angles of attack 0 to 4 degrees, reproduces the CFD integrated aerodynamic forces and pitching moment to R2 > 0.99 but does not match the experimental data. To incorporate experimental information without retraining the surrogate, a correction network is trained on spatially registered PSP measurements at two freestream Mach numbers (0.70 and 0.85) across the same angle-of-attack range, learning the discrepancy between the surrogate-predicted and experimentally measured surface-pressure distributions. At Mach 0.85 the correction substantially improves agreement with PSP, particularly at the wing suction peak, shock location, and subsequent pressure recovery, reducing both the magnitude of the prediction error and the fraction of wetted surface on which it exceeds 0.05 in Cp, and it does so from a limited experimental dataset without modifying the pretrained surrogate parameters. On held-out angles of attack the grounded surrogate agrees with measurement to within 2.3-2.7% of the measured Cp range, and outperforms direct interpolation between the measured conditions at every state tested. Experimental measurements can therefore ground a large-scale simulation-trained surrogate by learning systematic CFD-to-experiment discrepancies while preserving its generalization capability and computational efficiency.

Abstract

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