Data-informed lifting line theory

Arjun Sharma; Jonas A. Actor; Peter A. Bosler

physics.flu-dynMar 30, 2026

Data-informed lifting line theory

Authors: Arjun Sharma, Jonas A. Actor, Peter A. Bosler

Organizations: Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, 87185, USA

Abstract

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, $b^{2}/S$
$b$ =	Wingspan (physical)
$S$ =	Planform area
$y$ =	Normalized spanwise coordinate ( $-1\leq y\leq 1$ )
$c(y)$ =	Chord length at $y$
$\Gamma(y)$ =	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+ $\lambda$ )/2	4, 5, 7.5, 10, 15, 20, 30, 40, 50, 75, 100
Angle of Attack (AoA)	$-6^{\circ}$ to $6^{\circ}$ (step $0.05^{\circ}$ )
Taper Ratio ( $\lambda$ )	0.25, 0.5, 0.75, 1.0
Sweep Angle	$-30^{\circ}$ , $-15^{\circ}$ , $0^{\circ}$ , $15^{\circ}$ , $30^{\circ}$
Twist	$-5^{\circ}$ , $-2.5^{\circ}$ , $0^{\circ}$ , $2.5^{\circ}$ , $5^{\circ}$
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	0.54
2	Absent	Shallow	Black-box	7.281e-03	3.163e-01	0.57
2	Absent	Shallow	Grey-box	2.019e-02	1.794e-01	0.57
3	Present	Deep	Black-box	7.262e-03	1.446e-01	0.42
3	Present	Deep	Grey-box	6.632e-03	6.076e-02	0.42

Table 3: Test loss and relative error across network architectures and modeling settings.

Panel of Figure 2 — 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.

Panel of Figure 3 — Figure 3: Spanwise lift coefficient $c_{l}(y)$ distributions for different wing and flow configurations labeled on each panel.

Panel of Figure 4 — Figure 4: Spanwise induced-drag coefficient $c_{d}(y)$ distributions for different wing and flow configurations labeled on each panel.

Panel of Figure 5 — Figure 5: Lift-curve slope, normalized by AR $(1+\lambda)/2=b/c_{\text{root}}$ =100, versus 1/AR with sweep, twist, and taper-ratio ( $\lambda$ ) values within the training-data range. Gray shading marks OOD region (AR(1+ $\lambda$ )/2<4 or $b/c_{\text{root}}<4$ ).

Panel of Figure 6 — Figure 6: Lift-curve slope, normalized by AR $(1+\lambda)/2=b/c_{\text{root}}$ =100, versus 1/(AR) for wing shapes with sweep/twist/taper ratio outside the training-data range. Entire domain is OOD and shaded grey.

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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.

Abstract

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