We present a data-driven framework that extends the predictive capability of classical lifting-line theory (LLT) to a wider aerodynamic regime by incorporating higher-fidelity aerodynamic data from panel method simulations. A neural network architecture with a convolutional layer followed by fully connected layers is developed, comprising two parallel subnetworks to separately process spanwise collocation points and global geometric/aerodynamic inputs such as angle of attack, chord, twist, airfoil distribution, and sweep. Among several configurations tested, this architecture is most effective in learning corrections to LLT outputs. The trained model captures higher-order three-dimensional effects in spanwise lift and drag distributions in regimes where LLT is inaccurate, such as low aspect ratios and high sweep, and generalizes well to wing configurations outside both the LLT regime and the training data range. The method retains LLT's computational efficiency, enabling integration into aerodynamic optimization loops and early-stage aircraft design studies. This approach offers a practical path for embedding high-fidelity corrections into low-order methods and may be extended to other aerodynamic prediction tasks, such as propeller performance.
Figures & tables
AR =
Aspect ratio, b2/S
b =
Wingspan (physical)
S =
Planform area
y =
Normalized spanwise coordinate ( −1≤y≤1 )
c(y) =
Chord length at y
Γ(y) =
Bound circulation at y
l(y) =
Lift per unit span at y
Table 1
Figure 1: Neural network architecture: two parallel subnetworks (collocation; wing and flow) combine after optional convolution/dense layers to output spanwise lift, spanwise drag, and totals.
Parameter
Values
AR(1+ λ )/2
4, 5, 7.5, 10, 15, 20, 30, 40, 50, 75, 100
Angle of Attack (AoA)
−6∘ to 6∘ (step 0.05∘ )
Taper Ratio ( λ )
0.25, 0.5, 0.75, 1.0
Sweep Angle
−30∘ , −15∘ , 0∘ , 15∘ , 30∘
Twist
−5∘ , −2.5∘ , 0∘ , 2.5∘ , 5∘
Airfoils
NACA 2440, 0040, 0020, 1320
Table 2: Parameters used to generate the in-distribution dataset. A random subset of 400,000 samples is used for training and validation; the remainder are included in the test set.
Architecture
Conv Layer
Depth
Model Type
test loss
Rel. Err.
Rel. Err. GB/BB
1
Absent
Deep
Black-box
8.261e-03
2.306e-01
0.54
1
Absent
Deep
Grey-box
1.106e-02
1.239e-01
2
Absent
Shallow
Black-box
7.281e-03
3.163e-01
0.57
2
Absent
Shallow
Grey-box
2.019e-02
1.794e-01
3
Present
Deep
Black-box
7.262e-03
1.446e-01
0.42
3
Present
Deep
Grey-box
6.632e-03
6.076e-02
Table 3: Test loss and relative error across network architectures and modeling settings.
Figure 2: Training (a,c) and validation (b,d) loss histories for four architectures under black-box and grey-box formulations with an initial learning rate of 0.001.
Figure 3: Spanwise lift coefficient cl(y) distributions for different wing and flow configurations labeled on each panel.
Figure 4: Spanwise induced-drag coefficient cd(y) distributions for different wing and flow configurations labeled on each panel.
Figure 5: Lift-curve slope, normalized by AR (1+λ)/2=b/croot =100, versus 1/AR with sweep, twist, and taper-ratio ( λ ) values within the training-data range. Gray shading marks OOD region (AR(1+ λ )/2<4 or b/croot<4 ).
Figure 6: Lift-curve slope, normalized by AR (1+λ)/2=b/croot =100, versus 1/(AR) for wing shapes with sweep/twist/taper ratio outside the training-data range. Entire domain is OOD and shaded grey.
This paper investigates the feasibility of using residual learning to improve unsteady aerodynamic load prediction for aeroelastic applications. The machine learning technique selected for the study is the long short-term memory (LSTM) neural network, which is used for its suitability for sequential data with aerodynamic memory effects. The approach is investigated for the NLR 7301 airfoil benchmark using high-fidelity CFD lift data for prescribed pitch and plunge motions in the transonic flow regime in the presence of shock motion. An analytical unsteady aerodynamic model based on the Wagner function is used as a physics-based baseline, and the neural network is trained to learn the difference between the CFD lift coefficient and the Wagner prediction. The residual model is compared with a direct neural-network model trained to predict the CFD lift coefficient. The comparison includes feature and normalization studies, external benchmark cases, and leave-one-out and leave-family-out generalization tests across a range of sinusoidal and non-sinusoidal motions. The residual model performs best when its inputs align with the Wagner formulation variables, generally giving lower error and more consistent performance across training runs, though the direct model remains more accurate for some high-frequency cases. The residual model also generalizes better in the leave-one-out and leave-family-out tests, with a smaller increase in error than the direct model when entire motion families are withheld from training. Overall, the results indicate that residual learning shows promise as a modular approach for augmenting classical low-order aerodynamic theories, especially when the physics baseline removes a structured part of the aerodynamic response and leaves a lower-variance correction for the neural network to learn.
Accurate machine-learning models for aerodynamic prediction are essential for accelerating shape optimization, yet remain challenging to develop for complex three-dimensional configurations due to the high cost of generating training data. This work introduces a methodology for efficiently constructing accurate surrogate models for design purposes by first pre-training a large-scale model on diverse geometries and then fine-tuning it with a few more detailed task-specific samples. A Transformer-based architecture, AeroTransformer, is developed and tailored for large-scale training to learn aerodynamics. The methodology is evaluated on transonic wings, where the model is pre-trained on SuperWing, a dataset of nearly 30000 samples with broad geometric diversity, and subsequently fine-tuned to handle specific wing shapes perturbed from the Common Research Model. Results show that, with 450 task-specific samples, the proposed methodology achieves 0.36% error on surface-flow prediction, reducing 84.2% compared to training from scratch. The influence of model configurations and training strategies is also systematically studied to provide guidance on effectively training and deploying such models under limited data and computational budgets. To facilitate reuse, we release the datasets and the pre-trained models at https://github.com/tum-pbs/AeroTransformer. An interactive design tool is also built on the pre-trained model and is available online at https://webwing.pbs.cit.tum.de.
Yunjia Yang, Babak Gholami, Caglar Gurbuz +2
Technical University of Munich, Garching, D-85748, Germany · BMW Group, Bremer Str. 6, Munich, D-80788, Germany
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.
Nitin Nagesh Kulkarni, Dheeraj Vemula, Yin Yu +2
Luminary AI 101 S Ellsworth Ave, San Mateo, CA, USA