Bridging spatial and temporal surface pressure dynamics for gust aerodynamic modeling

Abstract

The low-altitude economy promises new options for urban transport and logistics but is constrained by gust-induced, highly unsteady aerodynamics that threaten flight safety. We present a Graph Transformer framework that fuses a surface-pressure graph with temporal attention to predict gust-induced unsteady aerodynamic loads. Sparse pressure taps are encoded as a full-link graph aligned with the airframe topology. Comparative studies show the necessity of a full-link graph, compared to streamwise or crosswise links, for precise gust modeling and highlight the complex gust flow patterns that include crosswise flows. The attention mechanism learns gust-onset-based attention patterns, enabling the model to identify and prioritize critical temporal phases of gust events. By correlating the complex spatiotemporal scales, the unified framework demonstrates robust performance on various challenging gust scenarios, delivering consistent and accurate multi-output predictions. Our framework provides a practical path for robust gust modeling, contributing to flight safety, thereby advancing the low-altitude economy.

Dashuai Chen and colleagues propose a graph-transformer framework that predicts gust-induced aerodynamic loads from sparse pressure measurements. The approach improves understanding of unsteady gust effects relevant to low-altitude flight.

Introduction

The rise of the low-altitude economy (LAE), especially in urban aero mobility, is opening up new horizons for air mobility and commerce. The development of LAE has been gaining substantial support in countries such as the United States and China, where the integration of advanced aircraft into urban aerospace is treated as a critical next step in environmental protection, economic growth, and city development.

Technological advances in battery life, artificial intelligence, and automation are making low-altitude operations more efficient and convenient. This progress opens up new avenues for economic growth, potentially creating jobs and spurring innovation in many sectors, as shown in Fig. 1. For instance, the non-contact nucleic acid testing utilizing drones could play a crucial role in detecting and preventing mass outbreaks^1,2. The electric vertical take-off and landing (eVTOL) vehicles that are scheduled to be piloted in six cities in China promise to substantially reduce traffic congestion, noise pollution, and carbon emissions³.

Fig. 1. Conceptual illustration of gust scenarios and UAV applications in urban low-altitude environments.

The left column depicts representative urban gust sources: vortex shedding from tall buildings, wingtip vortices from large aircraft, and buoyancy-driven updrafts from urban heat island effects. The right column highlights various UAV missions exposed to such gusts, including contactless delivery, urban air taxis, and agricultural operations. In the center, a delta wing model interacts with unsteady flow: the left half side shows mild vortex structures under calm conditions, while the right half side shows intensified, irregular flow under gust conditions. The surrounding vortex field is colored by velocity magnitude, with the colorbar indicating typical gust and flight speeds (0–26 m s⁻¹) encountered in urban environments. Sensor locations on the wing surface feed into an AI “brain” schematic, symbolizing intelligent gust perception and control.

Although the promotion of LAE is being accelerated, small aircraft designed for LAE currently only operate cautiously in calm weather conditions. A major challenge remains for LAE aircraft to fly safely and efficiently under complex aerodynamic conditions, which are commonly encountered in urban environments^4–6. As illustrated in Fig. 1, skyscrapers generate unpredictable gusts, high-intensity tail vortices, and wind shear⁷, while the urban heat island effect creates strong updrafts in some parts of cities⁸. Additionally, With the increasing number of civil airliners in operation, wingtip vortices have become a persistent concern in aviation safety and air traffic management^9,10, particularly during take-off and landing below 1000 meters, which is common operation altitude for LAE aircraft.

From the perspective of safety and economy, promoting the LAE on a large scale requires that the LAE aircraft is capable of carrying out missions in various weather conditions, not just under calm ones. However, unstable weather patterns present serious challenges to the robust control of LAE aircraft. Vortices, wind shear, and other kinds of gusts can rapidly and adversely affect the flow field around the aircraft, which leads to large-scale flow separation on the surface of aircraft, or a sharp change in surface pressure. Figure 1 illustrates the differences in the vortex field around a delta wing under calm conditions and during gust encounters. The pronounced vortex structure on the right side highlights the strong impact of the gust. These abrupt flow field changes result in dramatic fluctuations in aerodynamic loads, which may fluctuate 200% or more in a very short time^11,12. The sharp gust load changing may cause the fuselage to shake and, in severe scenarios, potentially lead to stalling and a fall. Therefore, precise aerodynamic load predictions are essential for flight control systems to maintain stability even in gusty environments^13–16.

Gusts have commonly been modeled using simplified profiles such as sinusoidal, trapezoidal, or top-hat shapes^17,18, and quantified by basic metrics like the gust ratio or flow angle^12,19–21. A more general description was later provided through a dimensional framework based on three nondimensional parameters—the gust ratio, encounter width, and development rate²². These parameters together provide a complete and universal description of transverse gusts, enabling systematic comparisons across studies. The introduction and definitions of these nondimensional parameters are provided in Supplementary Note 1. In our study, we adopt this dimensional framework to describe the gusts used in both simulations and experiments.

Classical unsteady aerodynamic theories, including indicial-response formulations and linear state-space models, were derived under the assumptions of small perturbations, attached flow, and quasi-steady circulation^23–25. These frameworks underpin conventional stability-and-control analyses but lose accuracy once strong separation or vortex shedding arises during gust encounters. Nonlinear extensions such as the Goman-Khrabrov framework incorporate empirical time-scale dynamics to reproduce stall hysteresis, and closed-loop implementations based on Theodorsen’s linear theory have demonstrated partial lift regulation during transverse gusts at gust ratios up to GR = 1.0^26–28. However, subsequent studies²⁷ have shown that the aerodynamic response in these conditions, which are dominated by leading edge vortex formation and complex circulation dynamics, deviates substantially from linear predictions. These findings further emphasized that effective control requires real time estimation of gust induced lift, which can be inferred from surface pressure measurements.

Accurate prediction of gust-induced aerodynamic loads from sparse surface pressure is therefore essential not only for understanding fundamental unsteady mechanisms but also for enabling robust closed-loop gust control. The present study builds upon these insights by developing a data-driven Graph Transformer framework capable of capturing the coupled spatial-temporal evolution of pressure fields directly from sparse measurements.

Data-driven methodologies^29–33 have shown great potential in modeling gust aerodynamic phenomena. These approaches are particularly well-suited to address the nonlinearities that are inherent in gust aerodynamics^11,12,34,35. An effective and practical approach is to apply the neural networks to map the sparse surface pressure measurements into unsteady aerodynamic loads. Surface pressure serves as a reliable indicator of the fluctuations of unsteady flows and is the primary contributor to aerodynamic loads. Multi-layer perceptron (MLP)³⁶ has been employed for the estimation of unsteady aerodynamic loads^11,37,38 by leveraging neural networks to identify and learn the intricate patterns between sparse surface pressure and aerodynamic loads. However, MLP models cannot account for the influence of historical information, which is crucial for capturing the inherent delays in unsteady aerodynamic load responses caused by the inertia³⁹. Such delays make it challenging for sparse pressure measurements at current time slots to fully capture the hysteresis and memory effects of unsteady flow fields. Long short-term memory (LSTM)⁴⁰, a type of recurrent neural network with a gating mechanism, is highly effective at capturing long-term dependencies, making it well-suited for predictions involving time series events, such as gust flows^41,42.

Despite their progress, existing data-driven models overlook key physical characteristics required to capture the transient and nonlinear nature of gust flows. Such flows exhibit strong spatial-temporal nonlinearities⁴³ that remain inadequately represented in most surrogate frameworks. Conventional neural networks often act as black boxes, providing numerical accuracy but limited physical interpretability. Moreover, many studies treat aerodynamic coefficients such as lift, drag, and moment as independent outputs, training separate networks for each quantity. This fragmented strategy increases model complexity and hinders flexible deployment in practical control systems. The generalization capability of these proxy models also remains uncertain, as validation has largely been restricted to narrow operating envelopes. Recent approaches—such as the cluster-based Bayesian model⁴⁴ and transition-network framework⁴⁵—introduce interpretability and probabilistic reasoning, yet still exhibit limitations in both predictive accuracy and generalization across flow regimes. Developing a unified and physically informed learning architecture that accurately predicts gust-induced aerodynamic loads while revealing interpretable spatiotemporal dynamics is therefore essential.

Aerodynamic forces primarily result from pressure differences across an aircraft’s surface. Recent studies have shown that it is possible to reconstruct gust loads using only a limited number of surface pressure measurements¹¹. Despite the advancements, the spatial relationships between pressure points and how these correlations affect the accuracy of gust load predictions have not been thoroughly investigated. Additionally, strong temporal correlations exist in gusts, where the current state is strongly influenced by historical information. Therefore, accurately capturing these temporal dependencies is essential not only for precise gust load predictions but also for a deeper understanding of the gust phenomenon with strong nonlinearities.

In recent years, Graph Neural Networks (GNNs) have emerged as a promising deep learning framework, particularly well-suited for processing graph-structured data^46–51. GNNs have shown strong capabilities in capturing spatial dependencies and patterns in distributed aerodynamic data. Recent advances include the use of graph variational autoencoders for shape optimization⁵² and the prediction of aerodynamic parameters for two-dimensional turbine blades using GNNs trained on small-scale experimental datasets⁵³.

Transformer architectures⁵⁴, known for their efficacy in modeling sequential data across diverse domains^55,56, offer a powerful means to capture long-range temporal dependencies through self-attention mechanisms. Applications in aerodynamics include transformer-based flow field prediction from simulation data⁵⁷ and estimation of aerodynamic parameters for laminar airfoils⁵⁸.

In contrast to the aforementioned studies, this work focuses on a conceptual validation using two-dimensional airfoil data, while the main training and prediction tasks are conducted on three-dimensional experimental data at high Reynolds numbers. Due to the unique characteristics of gust-induced disturbances, traditional approaches based solely on GNNs or Transformers exhibit limited accuracy. This study was initially motivated by the exploration of various graph-based data representations, where features are extracted via GNN and subsequently passed to a transformer for training. To account for differences in the underlying adjacency matrices, we propose three distinct graph structures and systematically compare their performance in the Results section.

In this study, we investigate gust aerodynamic modeling by uncovering spatiotemporal patterns associated with the complex nonlinear characteristics of gust phenomena, through the correlation of sequential sparse surface pressure measurements and corresponding aerodynamic loads. To systematically examine the modeling framework under both controlled and realistic conditions, two separate datasets are employed: a simulation dataset based on low-Reynolds-number 2D airfoil-gust interactions, and an experimental dataset based on high-Reynolds-number 3D gust encounters. These datasets are used independently in this study, with models trained separately on each domain. Our findings demonstrate not only the effectiveness of this spatiotemporal bridging, enabling accurate predictions of multi-aerodynamic forces simultaneously with a single unified model across a wide range of scenarios, but more importantly, reveal the directionality of the connections between surface pressure measurements in spatial scales, and the distribution of contributions to gust loads in temporal series.

Results

To ensure the generalizability of this study, we collected diverse datasets encompassing both numerical simulations and real-world experiments. The first dataset comprises two-dimensional simulations involving an airfoil encountering gusts of varying strengths at low Reynolds numbers. The Reynolds number, defined as Re = Uc/ν, where U is the free stream velocity, c is the chord length of the airfoil, and ν is the kinematic viscosity, is a dimensionless quantity that characterizes the flow status by the ratio of inertial and viscous forces. Low Reynolds numbers correspond to flows with more stable and controlled patterns. This dataset with low Reynolds numbers provides clean, noise-free data, allowing the proposed framework to capture the underlying dynamics in an ideal environment. In contrast, the second dataset originates from experiments involving a three-dimensional aircraft model exposed to gusts of diverse strengths and directions at high Reynolds numbers, which are hundreds of times higher than those in the two-dimensional simulations. The experimental dataset includes measurement noise, presenting a more realistic and challenging validation scenario.

Two-dimensional gusts with low Reynolds numbers

To evaluate the framework under controlled yet nonlinear flow conditions, we first conducted two-dimensional simulations of an airfoil encountering wake-vortex gusts generated by an upstream heaving airfoil, as illustrated in Methods? Both airfoils used a NACA 0012 profile with identical chord length and were separated by x/c = 2. By varying the geometric angle of attack α_f ∈ [−6°, 30°] of the front airfoil, gust strength was modulated, producing a range of vortex structures that impinged on the fixed rear airfoil (α_r ∈ [0°, 20°]). The front airfoil heaved at V = U/4, yielding effective angles ${\alpha }_{\mathrm{eff}}={\alpha }_f-\arctan \left(V/U\right)\in \left[-20^{\circ },16^{\circ }\right]$. Aerodynamic coefficients were defined in nondimensional form: C_F = 2F/(ρU²A), C_M = 2M/(ρU²Ac), and C_P = 2P/(ρU²), where ρ is the freestream density, U the freestream velocity, A the reference area, and c the chord, with P denoting local surface pressure. The simulations were performed in WaterLily⁵⁹ at chord-based $\mathrm{Re}=1000$. Full numerical and kinematic details appear in “Methods”, while the resolution study is listed in Supplementary Note 2.

To place the wake-impingement configuration within a canonical framework, we adopted the dimensional analysis based on the framework described previously²². The dominant vertical velocity at the rear airfoil is perpendicular to the freestream, allowing the flow to be treated as an equivalent transverse gust. Three nondimensional parameters describe the gust: the gust ratio GR = v/U_∞, encounter time $t_e^{*}=\left(t_{\mathrm{end}}-t_{\mathrm{onset}}\right)U_{\infty }/c$, and development rate ${\mathrm{\Omega }}^{*}={\alpha }^{\circ }\mathrm{eff}c/\left(2U\infty \right)$. Across all cases, GR spanned approximately 0.14–1.08, covering weak to strong gusts, while $t_e^{*}$ and Ω^* followed consistent trends-stronger gusts corresponded to longer encounters and faster angular evolution. Detailed derivations and parameter distributions are provided in Supplementary Note 3.

The simulation dataset comprises 209 gust cases. For each case, the gust-induced lift and sparse surface-pressure signals from nine virtual pressure ports on the rear airfoil were recorded synchronously. The virtual pressure ports were placed non-uniformly, with a higher density near the leading edge to capture the leading-edge suction peak and downstream pressure recovery, as guided by a prior sensitivity analysis¹¹. The nondimensional probe locations are provided in Supplementary Note 2.

Figure 2a summarizes the unsteady aerodynamic response and flow behavior for the 2D gust–airfoil interactions. Panel (a1) shows lift-coefficient histories across all simulated gust cases, with one representative case highlighted in red. Panels (a2) and (a3) illustrate, respectively, the instantaneous vorticity fields and corresponding surface-pressure distributions for that case, demonstrating how the upstream gust modifies the rear airfoil’s local flow and pressure field.

Fig. 2. Datasets used in this study.

a 2D simulation: Gust Response of the Airfoil. (a1) Unsteady lift histories C_L(t) for representative simulation cases across varying gust conditions. (a2) Instantaneous vorticity contours at instants indicated by cyan markers ○ and △ in (a1) from a highlighted case, nondimensionalized by chord length and freestream velocity. (a3) Corresponding spatiotemporal surface-pressure evolution (C_p) on the upper and lower surfaces of the rear airfoil. b 3D Experiment: Gust Response of the Delta Wing. (b1) Time sequence of wing motion for a representative gust case, shown as overlaid outlines (top) and the corresponding surface-pressure field C_p mapped on the wing (bottom). (b2) Same visualization for a second gust case with distinct kinematics; the colormap indicates C_p (legend shown).

The dataset is randomly divided at the case level into training, validation, and testing subsets in a 7:2:1 ratio, ensuring that each time series remains a complete and independent trajectory.

Three-dimensional gusts with high Reynolds numbers

The 2D laminar cases provide a clean baseline to validate core modeling behavior. Real LAE flight, however, operates near $\mathrm{Re}~10^5$ and above, where turbulence, three-dimensionality, separation, transition, and vortex breakdown introduce strong nonlinearities and modeling challenges. Bridging to 3D high-Re flow thus marks a critical step toward a generalizable gust-load predictor.

We investigate three dimensional gust interactions over a non-slender delta wing⁶⁰, a configuration frequently considered for low-altitude UAVs due to vortex-lift benefits and structural simplicity^61–63. Figure 1 centers a delta wing rendering with vortical structures colored by velocity; the left panel shows the baseline (no gust), and the right panel shows a strong gust case. The pronounced increase in vortical intensity on the gust side evidences enhanced separation and altered aerodynamic characteristics, motivating pressure-based sensing and learning in realistic conditions.

The three-dimensional gust experiments employed a delta-wing model with a NACA 0012 cross-section and a 45° sweep to replicate realistic low-altitude aerodynamic conditions. Tests were performed in a towing-tank environment at chord-based Reynolds numbers between 1.5 × 10⁵ and 4.5 × 10⁵, with gusts generated through controlled towing acceleration and pitching motion. These actuation profiles produced axial, transverse, and combined gusts, enabling systematic exploration of directionally varied unsteady flow interactions. Schematic diagrams of the experimental setups are presented in the “Methods”. Prior to each gust actuation, the model was towed under steady flow to ensure consistent initial conditions. Representative time sequences of the delta-wing motion and the corresponding surface-pressure evolution during gust encounters are shown in Fig. 2b, illustrating the transient response under two distinct gust cases (as shown in panels (b1) and (b2)) and motivating the pressure-based learning formulation.

Gust actuation followed a compact parametric description with 1-cosine transitions for axial and pitch components; full expressions and rates are given in “Methods” and Supplementary Note 4. Building on the pitching-gust analogy proposed in earlier work⁶⁴, a qualification is warranted: decelerating runs include partial gust exposure (traverse distance < c), which weakens the uniform pitching analogy; constant-speed and accelerating runs traverse > c and align more closely (see Supplementary Note 5). Despite the asymmetry, gust forces and surface-pressure profiles remain physically consistent, and the deep learning framework maintains robust performance under these conditions. Since the central aim involves evaluating the framework across diverse aerodynamic regimes, inclusion of partial-exposure cases enhances relevance and supports claims of generalizability.

The 3D gust dataset comprises 32 cases combining pitch and streamwise velocity variations, spanning both pure and coupled gust scenarios. Five transverse and three streamwise gust types form fifteen nominal combinations, with additional cases introduced by varying initial conditions, such as starting AoA. Although moderate in size, the matrix covers a broad aerodynamic envelope with diverse gust directions, strengths, and flow structures, providing a rich basis for validating the prediction framework. Four representative cases—A, B, C, and D (originally cases 25, 13, 20 and 28)—were randomly selected for unbiased evaluation; together they encompass both accelerating and decelerating motions and exhibit distinct surface-pressure patterns. The overall gust parameter space and detailed kinematic conditions are summarized in Supplementary Note 6. Verification of the commanded gust motions was done by comparing the as-performed motion trajectories with the designed profiles. The comparison shows consistently small deviations over the full motion history, supporting accurate reproduction of the designed gust inputs (see Supplementary Note 7).

Gust loads prediction incorporating spatiotemporal dynamics

A hybrid graph-Transformer framework was developed to model gust-induced aerodynamic responses as illustrated in Fig. 3. Sparse surface-pressure measurements were represented as an undirected weighted graph, where nodes correspond to pressure ports and edge weights encode spatial proximity and physical correlation. This graph representation enables the network to capture local and global spatial dependencies inherent in gust flows. Temporal evolution is modeled through a Transformer encoder, whose attention mechanism adaptively emphasizes key gust features—such as onset, strength, direction, and evolving flow structures—across time. The combined architecture jointly resolves spatial and temporal dynamics, allowing a single unified model to predict lift, drag, and moment coefficients from sparse inputs. This integration reduces model fragmentation, improves interpretability, and enhances deployment flexibility for real-world gust environments.

Fig. 3. Framework of the Graph Transformer and Tasks Overview.

The NACA 0012-chord-based delta wing model is shown in the upper left corner, featuring pressure ports on the surface. In the middle row, the three different types of graph links are displayed, alongside the nodes' link weights. A column of pressure ports with green color and one row of pressure ports with red color are highlighted for the streamwise link graph and crosswise link graph, respectively, with the corresponding pressure gradients shown above. The Graph Transformer model is presented at the bottom, along with an overview of the three associated tasks.

To comprehensively address the complexities of gust aerodynamics, the study is organized into three interconnected tasks (see Table 1) that progressively link surface pressure, aerodynamic loads, and flow-state estimation, as illustrated in Fig. 3. Task 1 develops a predictive model mapping sparse surface-pressure histories ${\mathcal{G}}_{1:T}^P$ to key gust loads-lift C_L, drag C_D, and moment C_M. Although pressure and load signals are recorded synchronously, the inherent physical delay caused by flow inertia and vortex dynamics is captured by the Transformer’s temporal attention. Task 2, an inverse task, uses the same pressure inputs to infer the instantaneous geometric angle of attack (AoA) during gust encounters, establishing a mapping between aerodynamic state and surface-pressure response. Task 3 integrates both sources of information—the measured pressure sequence ${\mathcal{G}}_{1:T}^P$ and the inferred AoA trajectory AoA_1:T—to refine predictions and capture coupled spatiotemporal dynamics governing gust-induced loads.

Table 1.

Summary of the three tasks in the study and their applicability to different datasets

Task	Input	Output	Datasets
T1	${\mathcal{G}}_{1:T}^P$	CL (2D); ${\left[C_L,C_D,C_M\right]}_{1:T}$ (3D)	2D, 3D
T2	${\mathcal{G}}_{1:T}^P$	AoA1:T	3D
T3	${\mathcal{G}}_{1:T}^P$ and AoA1:T	${\left[C_L,C_D,C_M\right]}_{1:T}$	3D

For both Task 2 and Task 3, the geometric AoA—defined by the wing’s pitch angle—is used as the ground truth or model input. In the experiments, the wing performs pitching and axial surging motions without plunging or lateral translation. Because the gust-induced perturbation modifies only the pitch angle over a short duration (0.3 s) and does not generate vertical velocity, the geometric AoA remains a valid and sufficient descriptor of the aerodynamic state, with minimal deviation from the effective AoA. For the two-dimensional dataset, only Task 1 is applied, focusing on lift C_L prediction as a baseline due to the absence of spanwise effects and dynamic AoA variations. Together, the three tasks form a cohesive framework that unifies direct and inverse learning for robust gust-load modeling.

The model was first trained on the respective datasets with hyperparameters optimized using Wandb⁶⁵. Detailed settings are listed in the Appendix. Code optimization and efficient GPU utilization reduced training time for complex experimental datasets to approximately two minutes, making it well-suited for real-world applications involving large and intricate aerodynamic datasets.

We begin with the two-dimensional airfoil-gust interaction dataset. Two representative test cases were selected to evaluate the predictive performance of the Graph Transformer (Task 1). Results of the forward prediction and corresponding flow structures are shown in Fig. 4a. In the left case, the front and rear airfoils have geometric angles of attack of 18° and 2°, respectively. Upward heaving of the front airfoil (V = U/4) reduces its effective angle to roughly 4°, below the ~ 8° separation threshold for $\mathrm{Re}=1000$^66,67, yielding a laminar wake and a single sharp lift peak on the rear airfoil. In contrast, the right case combines front and rear angles of −4° and 12°, respectively, with upward heaving that increases the effective front angle to 22°, well within the separated-flow regime. The resulting vortex impinges on the rear airfoil, inducing multiple unsteady lift peaks. These contrasting behaviors demonstrate the framework’s ability to capture both mild and strongly separated gust interactions with high temporal fidelity.

Fig. 4. Task 1 (2D): gust–airfoil load prediction and quantitative error analysis.

a Representative prediction results for two simulated gust cases. The left panels correspond to a mild gust case (${{\alpha }_f}_{eff}=4^{\circ }$), showing a moderate, single-peak lift response (ΔC_L ≈ 0.3). The right panels illustrate a strong gust case (${{\alpha }_f}_{eff}=-18^{\circ }$), exhibiting complex separated flow and a large lift response (ΔC_L ≈ 1.0). The annotated geometric and effective AoA for each case are shown in the insets. Solid lines show simulation results, and dashed lines represent Transformer predictions. The background maps indicate attention weights over time, while the rightmost panels show vortex field visualizations at time instances denoted by multiple cyan markers in the lift histories, highlighting the key flow structures. b Quantitative error statistics over the full 2D test set. Left: each bubble represents a test case, in which the bubble size and color indicate the MSE of predicted C_L using the Graph Transformer model. The two representative cases, a mild gust and a strong gust, are annotated. Right: comparison of C_L between simulation and prediction for the case with the largest MSE.

The left panel of Fig. 4a represents a mild gust case with attached flow and a modest lift increase (ΔC_L ≈ 0.3), corresponding to an effective angle of attack α_eff = 4°, below the separation threshold (α_eff > 8°). The right panel shows a strong gust case with large-scale separation and an amplified lift response (ΔC_L ≈ 1.0), where α_eff = −18° places the flow deep in the vortex-dominated regime. These two contrasting examples span different aerodynamic complexities, demonstrating that the Graph Transformer accurately predicts gust-induced lift across both attached and separated conditions.

Quantitatively, Fig. 4b shows the mean-squared error (MSE) distribution for all test cases, visualized as colored bubbles. Most cases achieve low errors (MSE < 0.003), confirming high predictive accuracy. The mild and strong gust cases yield MSE = 6.9 × 10⁻⁵ and 4.7 × 10⁻⁴, respectively. The largest error occurs near the edge of the training domain (α_r = 16°, ${{\alpha }_f}_{\mathrm{eff}}=10^{\circ }$), where both airfoils operate at high angles, producing phase and amplitude discrepancies typical of data-driven limits. These results emphasize the need for broader data coverage to further enhance model generalization.

The Transformer attention maps in Fig. 4a highlight temporal regions associated with gust encounters. In the mild case, attention peaks coincide with gust onset (t ≈ 13) and the start of lift rise, while in the strong case, the attention tracks all peaks and troughs of the unsteady lift signal. This alignment confirms that the temporal-attention mechanism identifies key flow events and underpins the model’s accurate predictions across distinct gust regimes.

Having established the Graph Transformer’s ability to predict gust-induced lift in the two-dimensional dataset, we next examine its performance under the more complex three-dimensional gust conditions. In the 2D case, temporal dynamics dominated the learning process, as spatial relationships were limited to streamwise connections among pressure ports. The 3D experiments, however, involve crosswise and secondary flow interactions at high Reynolds numbers (~ 10⁵), where turbulent, separated, and vortex-dominated structures emerge. Under these conditions, the graph representation must capture richer spatial relationships—linking pressure ports in both streamwise and crosswise directions—to model the coupled spatiotemporal behavior of gust aerodynamics.

Figure 5a presents the predictions of lift C_L, drag C_D, and moment C_M from the full-link Graph Transformer model, together with the corresponding attention maps for four representative cases. The unified model accurately reproduces gust-onset dynamics and subsequent fluctuations across all load components, reflecting the complex vortex-wing interactions. Attention maps highlight the temporal regions associated with gust encounters, particularly the onset and peak activity phases, demonstrating that the Transformer effectively identifies critical unsteady events. All three aerodynamic loads are predicted simultaneously by a single model, underscoring its versatility and internal consistency. By leveraging shared spatial and temporal representations, the Graph Transformer captures interdependencies among lift, drag, and moment, providing coherent and reliable predictions across diverse 3D gust scenarios.

Fig. 5. Task 1 (3D): gust-load prediction for the delta wing and quantitative error analysis.

a Representative prediction results for four experimental test cases (A–D). The comparison between the gust loads and the ground truth of the delta wing model is shown together, including C_L, C_D, and C_M. The corresponding attention weights from the Transformer model are presented beneath. Above the attention map, the schematics of the delta wing models and the corresponding surface pressure maps are displayed for time slots marked by cyan markers ○, △ and □ in the lift histories. b MSE of gust load predictions for 3D experimental test cases (A–D). The prediction errors are shown separately for C_L, C_D, and C_M. Each curve connects the MSE values across the four test cases. The model maintains consistent accuracy across all outputs, with slightly elevated errors in C_L and C_D for case B, highlighting challenging floe conditions in those scenarios.

While global trends are well captured, sharp extrema exhibit case-dependent bias: case A shows an overpredicted C_D peak; case B shows an underpredicted C_D peak and a deeper C_M trough; case C aligns closely; case D shows an underpredicted C_M peak and a shallower trough. Two factors likely contribute: limited representation of brief high-intensity transients in the experimental training set (32 cases in total, 28 used for training) and an MSE-weighted objective that favors average fit and can smooth isolated spikes. Additional influences, such as sensor noise and modest timing offsets, may also contribute.

To quantify model accuracy under realistic 3D conditions, per-coefficient MSEs were computed for lift, drag, and moment (C_L, C_D, C_M) across the four representative test cases (A–D), as shown in Fig. 5b. The Graph Transformer maintains low prediction errors for all coefficients, with most MSE values below 0.005. Slightly larger deviations in C_L and C_D, notably in case B, likely arise from sharp vortex formation or asymmetric separation, whereas C_M remains consistently accurate, indicating robust generalization for moment estimation. These results confirm that the unified model reliably predicts multiple unsteady aerodynamic loads across diverse gust profiles.

To assess whether full spatial connectivity is required, three graph structures were evaluated (Fig. 3): a full-link graph connecting all pressure ports, and two sparse baselines with only streamwise or only crosswise links. Figure 6 reports gust-load MSEs as percentages relative to the full-link model. Both sparse graphs incur substantially larger errors across most cases and coefficients (frequently > 100% and in some instances > 200%), demonstrating that gust interactions create coupled dependencies that cannot be captured by one-directional connectivity alone. A full-link surface-pressure graph is therefore essential for accurate multi-load prediction during gust encounters. (Qualitative differences in temporal focus under streamwise-only and crosswise-only connectivity are illustrated by the corresponding attention maps in Supplementary Note 8.)

Fig. 6. MSE comparison of Graph Transformer models with different graph structures for delta wing gust loads prediction.

The MSEs of the two graph models, which utilize streamwise and crosswise links between surface pressure ports, respectively, are presented relative to the graph model, which employs a comprehensive full link capturing all surface pressure interconnections.

Building upon the gust-load predictions, Task 2 addresses the inverse problem of estimating the time-varying aerodynamic orientation (AoA) directly from sparse surface-pressure measurements. The Graph Transformer models the AoA trajectory as a function of the pressure sequence ${\mathcal{G}}_{1:T}^P$, with final prediction results shown in Fig. 7. The model achieves an average MSE of about 8, accurately recovering the temporal evolution of AoA for most cases. Performance is strongest for cases B, C and D (MSE  ≈ 1.1–2.3), where the predicted curves closely follow the measured profiles. Case A shows underestimation after the gust event due to the non-unique mapping between pressure and AoA: similar geometric angles can correspond to distinct unsteady pressure patterns depending on gust strength, timing, or flow history. The performance degradation in case A likely reflects limited training coverage and sensitivity of pressure signals to small timing offsets. Although geometric AoA is used here, incorporating effective AoA (including vertical velocity effects) could further improve inverse estimation in more complex motion regimes. Attention maps reveal increased focus near gust onset and recovery, confirming that the Transformer effectively identifies key temporal segments during the inverse-learning process.

Fig. 7. Inverse problem (Task 2) predictions of instantaneous AoA for the four experimental test cases (A–D).

Top: Comparison of predicted AoA and experimental reference, with background color indicating attention weights from the Transformer; the delta-wing schematics above illustrate the corresponding AoA at time instants indicated by cyan markers ○ and □ in the lift histories. Bottom: Bar plot of the MSE for each test case.

Task 3 integrates the inverse and forward frameworks by first estimating the time-varying AoA from sparse surface-pressure data (Task 2) and then feeding it into the gust-load predictor (Task 1). This joint application evaluates the framework’s overall capability to combine physical state inference and force prediction. As shown in Fig. 8a, the Graph Transformer achieves performance comparable to Task 1, with similar accuracy and attention-map structures. In Case A, incorporating AoA reduces the overestimation of the instantaneous drag peak by about 30%, illustrating the benefit of adding flow-state information. Figure 8b compares the MSEs of Tasks 1 and 3: while overall errors remain similar, improvements occur during specific unsteady phases. Because AoA estimation accuracy varies across cases, the benefit of Task 3 is condition-dependent-accurate AoA inputs enhance predictions, whereas noisy estimates can introduce additional variance. These results demonstrate the potential of integrating physically informed variables, while highlighting the need for improved inverse modeling to fully exploit this advantage. Even with 10% noise, MSEs remain on the order of 10⁻³ (see Supplementary Note 9), demonstrating that the framework maintains robust performance under moderate sensing uncertainty.

Fig. 8. Task 3: joint application on the 3D experimental dataset.

a Predicted gust-induced loads and the corresponding attention maps for four representative test cases (A–D), shown from left to right. b Quantitative comparison of prediction errors between Task 1 and Task 3 across the four experimental test cases. MSEs are reported for C_L, C_D, and C_M. Hollow markers denote results from Task 1, while solid markers denote results from Task 3.

To contextualize the framework’s performance, Table 2 compares inference efficiency and prediction accuracy with existing state-of-the-art models. Although the Graph Transformer runs slightly slower than a pure Transformer, it achieves an MSE of 2.53 × 10⁻³, outperforming standalone graph-based networks such as TAGConv⁶⁸ and ChebConv⁶⁹. This improvement confirms that coupling graph-based spatial representation with temporal attention effectively captures nonlinear aerodynamic interactions and yields high predictive fidelity across gust conditions.

Table 2.

Comparative analysis of model performance and inference efficiency in the forward task with experimental 3D dataset

Modelsa	MSE (10−3)	Parameters (108)	Inference time (ms)b
GTc	2.53/2.76/3.00	1.2	30
Transformer	12.4	1.4	12
ChebConv	20.2	0.8	150
TAGConv	40.7	0.7	320

^aCompared in the same training hyperparameters.

^bTested on the Tesla V100.

^cCompared in three graph structure.

Discussion

The emerging low-altitude economy (LAE) offers new opportunities for urban transport and logistics but also faces severe aerodynamic challenges from highly unsteady gust environments that threaten flight safety. Accurate modeling of such nonlinear and unpredictable flow conditions is critical for reliable LAE operations. The proposed Graph Transformer framework addresses this need by combining physically grounded graph representations of pressure-sensor relationships with temporal attention for unsteady flow learning. Unlike traditional simulation-based or case-specific models, it provides a unified data-driven approach that generalizes across both numerical and experimental gust datasets. This study thus bridges the gap between simulation accuracy and real-world applicability, offering a scalable pathway for understanding and mitigating gust-induced aerodynamic hazards in low-altitude flight.

The results confirm that graph-based data structures effectively capture spatial dependencies among surface pressures, providing a representation consistent with physical principles. Such graph formulations facilitate generalization across varying sensor configurations by embedding geometric and relational information directly into the learning process. The Transformer’s attention mechanism complements this spatial modeling by maintaining temporal consistency and computational efficiency, as evidenced by the attention-weight maps. These maps reveal that optimal attention should focus on localized but not overly narrow temporal regions, emphasizing gust onset and recovery without diluting critical unsteady features. The unified framework’s ability to predict lift, drag, and moment simultaneously underscores its robustness and internal consistency across multiple aerodynamic loads. Minor peak-value discrepancies observed in Fig. 5a—including transient under and overshoots—likely stem from limited representation of high-intensity gust events and the smoothing bias of an MSE objective. Future improvements will involve expanding the experimental dataset and employing peak-sensitive loss functions with targeted augmentation of strong-gust segments to enhance fidelity at sharp transients while maintaining global accuracy.

Practical deployment limits the number of surface-pressure sensors due to weight, cost, and integration constraints, making sensor reduction desirable. Although optimization was not the primary focus here, our prior gradient-based study¹¹ showed that leading-edge sensors dominate gust-load estimation and that as few as four taps can suffice for accurate lift prediction. In this work, we emphasize spatiotemporal learning with the Graph Transformer, whose internal structures—node weights and attention maps—implicitly reflect sensor importance. Together with the observed nonlinear error scaling in node-deletion tests described in the “Methods”, these results suggest that the framework naturally supports practical sensor reduction. Future work will explore autonomous selection of sensor locations and quantities for gust-load prediction.

Moving forward, the investigation will target flow-control and maneuverability studies under strongly nonlinear gusts. Experimental observations show that gust passages induce pronounced nonlinearity, which complicates the inverse problem; with limited training data, standalone AoA estimators recover trends but can miss large excursions. Substantial AoA changes also perturb the attention patterns, motivating improved noise handling and attention regularization. To expand robustness and generalization, we will fuse multi-source data—combining high-fidelity simulations with experiments via domain adaptation—to train a single model that spans gust types, Reynolds numbers, and flow complexities. Taken together, these directions extend the present framework toward practical guidance for aerodynamic design and LAE operations under real-world unsteady conditions.

Methods

Numerical setup for the 2D gust–airfoil simulations

The simulations used WaterLily⁵⁹, an incompressible viscous solver, at a chord-based Reynolds number of $\mathrm{Re}=1000$. Two NACA 0012 airfoils were positioned two chord lengths apart and centered in the domain.

In each case, the upstream airfoil heaved from y/c = − 3.3 to +3.3 at constant velocity V = U/4, generating a symmetric velocity disturbance relative to the freestream. The streamwise spacing x/c = 2 provided wake development prior to interaction with the downstream airfoil.

The front-airfoil geometric angle of attack α varied from −6° to 30° in 2° steps, producing effective angles ${\alpha }_{\mathrm{eff}}=\alpha -\arctan \left(V/U\right)$ ranging from − 20° to 16°. The nearly symmetric range simplifies control and enables balanced sampling across positive/negative vortex strengths. The downstream airfoil remained fixed with α_r ∈ [0°, 20°] in 2° steps.

Pressure signals were sampled at nine discrete chordwise locations on the rear airfoil. One tap was placed at the leading edge (x/c = 0), while the remaining eight taps were distributed symmetrically about the mid-chord. The pressure taps were placed at nondimensional streamwise positions x/c = {0, 0.045, 0.171, 0.380, 0.669}, measured relative to the leading edge with a unit chord length (c = 1). A schematic of the numerical configuration and tap layout is provided in Supplementary Note 2.

Changes in α_eff modify wake-vortex size and roll-up rate, shifting the onset of gust–airfoil interaction. The induced temporal variability offers an effective testbed for attention-based temporal modeling within the proposed Graph Transformer framework.

Experimental setup for 3D gust-delta wing interactions

Experiments were conducted using a non-slender delta wing featuring a NACA 0012 cross-section and a 45° leading-edge sweep. The wing had a center-chord length of 0.3 m and was equipped with 15 surface-pressure taps distributed on one side and a six-axis force/torque transducer at the root. Tests were carried out in a 15 m towing-tank facility using a dual-actuator traverse that enabled controlled axial (u) and vertical (v) motions, allowing the generation of streamwise and transverse gust components. The towing speed was varied between 0.5 m s⁻¹ and 1.5 m s⁻¹, corresponding to chord-based Reynolds numbers of 1.5 × 10⁵–4.5 × 10⁵. Each run began with steady towing to establish a baseline flow before applying prescribed gust profiles. A schematic of the instrumentation and facility is provided in Supplementary Note 4.

Gust loads were measured using a six-axis force/torque transducer mounted at the wing’s center, with signals sampled at 1000 Hz and resolutions of 0.125 N and 0.0015 N ⋅ m. Surface pressures were recorded by OMEGA PX419 absolute transducers (response time ≈ 1 ms, precision 0.08%), connected to 16 channels: 15 surface taps on one wing side and one upstream stagnation probe (probe 0). The tap layout, shown in Fig. 3, spans leading-edge, vortex-core, and reattachment regions. Leading-edge ports capture low-pressure peaks linked to separated flow, while downstream taps track vortex reattachment and decay. A mid-chord tap was replaced by a stagnation probe to provide freestream reference data without compromising spatial coverage. Each condition was repeated ten times and ensemble-averaged, with an 8-min interval between runs to ensure stable initial flow conditions.

Graph data and algorithms

Spatial-temporal data can be interpreted as a discrete-time graph set. Consider a set of sensors indexed by i for i = 1, …, n, where each sensor records pressure over time.

This paper defines the structure of a dynamic, undirected weighted graph set to represent data. Each edge e_ij ∈ E connects the vertices v_i and v_j, which means that every node is connected. The adjacency matrix A of the graph is defined such that each element a_ij is a measurement of the normalization distance d_ij, which denotes the weight of the edge between nodes.

UWG (Undirected weighted Graph):

$$e_{ij}=\mathrm{Normalization}\mathrm{distance}\mathrm{between}\mathrm{node}i\mathrm{and}j$$

In the dynamic graph set ${\mathcal{G}}_{1:T}^p=\left\{{\mathcal{G}}_1,{\mathcal{G}}_2,\cdots ,{\mathcal{G}}_T\right\}$, each graph ${\mathcal{G}}_t$ consists of a pair $\left(\mathbf{A},{\mathbf{X}}_t\right)$, where $\mathbf{A}\in {\mathbb{R}}^{N\times N}$ represents the adjacency matrix and ${\mathbf{X}}_t\in {\mathbb{R}}^{N\times D}$ denotes the attribute matrix at time t. N corresponds to the number of sensors, and D = 1 specifies the dimensionality of the pressure data captured by each sensor.

Preliminary: A general GNN learns node representations through two primary functions: Aggregate ( ⋅ ) and combine ( ⋅ ). The Aggregate ( ⋅ ) function computes and aggregates messages from neighboring nodes, while the combine ( ⋅ ) function merges the aggregated and previous states to transform node embeddings. Formally, the k-th layer in a GNN is defined by the extended

$${\mathbf{a}}_{\mathbf{i}}^{\left(\mathbf{k}\right)}={\mathrm{AGGREGATE}}^{\left(k\right)}\left(\left\{{\mathbf{h}}_{\mathbf{j}}^{\left(\mathbf{k}-\mathbf{1}\right)}:v_j\in \mathbf{N}\left(v_i\right)\right\}\right),{\mathbf{h}}_{\mathbf{i}}^{\left(\mathbf{k}\right)}={\mathrm{COMBINE}}^{\left(k\right)}\left({\mathbf{h}}_{\mathbf{i}}^{\left(\mathbf{k}-\mathbf{1}\right)},{\mathbf{a}}_{\mathbf{i}}^{\left(\mathbf{k}\right)}\right),$$

where, ${\mathbf{a}}_i^{\left(k\right)}$ and ${\mathbf{h}}_i^{\left(k\right)}$ represent the aggregated message from neighbors and the embedding of the transformed node v_i in the k-th layer, respectively.

Algorithm for the forward problem

1: Input is ${\mathcal{G}}_{1:T}^p$

2: while time step t < T do

3:   info  ← AoA(t)            ⊳ selectable [Batch, T] 4:   Process pressure by GNN:

5:     ${\mathcal{G}}_t\leftarrow \sigma \left(\mathrm{GNN}\left({\mathcal{G}}_t^p\mid \mathrm{info}\left(t\right)\right)\right)$

6:     ${\mathcal{G}}_t^h\leftarrow \mathrm{PoolGraph}\left({\mathcal{G}}_t\right)$

7: end while

8: ${\mathcal{G}}_{1:T}^h\leftarrow \left\{{\mathcal{G}}_1^h,{\mathcal{G}}_2^h,\dots ,{\mathcal{G}}_T^h\right\}$

9: Output by transformer:

10: ${\mathcal{G}}_{1:T}^f\leftarrow \left\{{\mathcal{G}}_1^f,{\mathcal{G}}_2^f,\dots ,{\mathcal{G}}_T^f\right\}$

11: return Reshape to [Batch, T, 3]        ⊳ Out: CL, CD, CM

Algorithm for the inverse problem

1: Input is ${\mathcal{G}}_{1:T}^p$

2: while time step t < T do

3:   Process pressure by GNN:

4:     ${\mathcal{G}}_t\leftarrow \sigma \left(\mathrm{GNN}\left({\mathcal{G}}_t^p\right)\right)$

5:     ${\mathcal{G}}_t^h\leftarrow \mathrm{PoolGraph}\left({\mathcal{G}}_t\right)$

6: end while

7: ${\mathcal{G}}_{1:T}^h\leftarrow \left\{{\mathcal{G}}_1^h,{\mathcal{G}}_2^h,\dots ,{\mathcal{G}}_T^h\right\}$

8: Output by transformer:

9: ${\mathcal{G}}_{1:T}^A\leftarrow \left\{{\mathcal{G}}_1^A,{\mathcal{G}}_2^A,\dots ,{\mathcal{G}}_T^A\right\}$

10: return Reshape to [Batch, T]           ⊳ Out: AoA

Training setting

As for the learning objective in the training, we employ the root mean squared error (RMSE) as the learning objective and test performance for training our model, defined as follows:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2},$$

where n represents the number of samples, y_i is the actual value of the i-th observation, and ${\widehat{y}}_i$ is the predicted value from the model.

The model was trained using fixed hyperparameters, which are summarized in Supplementary Note 10.

Analyzing attention maps in inverse problems

As inverse problems directly output AOA, which is the most straightforward representation of gust information, and given that their predictive performance is relatively poor, analyzing the attention maps for these problems can provide valuable insights into the attention mechanism.

Comparing the initial attention map shown in Supplementary Note 11 with the final attention map illustrated in Fig. 7, it can be observed that the attention map is quite dispersed at the beginning of training, resulting in poor prediction. As training progresses, the model’s focus gradually narrows, showing a trend towards consolidation. The self-attention mechanism can facilitate the concentration of the focus area on gusts. However, both overly narrow and overly broad highlighted areas (attention scores) can lead to deviations in prediction results. This phenomenon may be due to the model encountering overly complex data patterns. In contrast, with simpler patterns such as simulation results in Fig. 4a, attention tends to highlight along columns, indicating that the underlying data patterns are relatively simple and capable of capturing gust information.

Test with deleting nodes in the joint application

A robustness test was conducted to assess model performance under incomplete pressure sensing by randomly removing a subset of pressure-tap nodes and reconstructing the missing values prior to inference. In this test, between one and five pressure nodes were randomly removed, and the missing values were reconstructed using inverse distance weighting (IDW) interpolation before being fed into the trained model. The resulting prediction errors were evaluated in terms of the MSE, averaged across the aerodynamic coefficients (C_L, C_D, and C_M) in Task 3.

The results indicate a gradual degradation in model performance as the number of missing nodes increases, demonstrating graceful performance decay rather than abrupt failure. The dominant source of error arises when missing nodes are located near the boundaries, where interpolation accuracy is reduced. Detailed procedures, statistical analysis, and quantitative results are provided in Supplementary Note 12.

Author contributions

D.C. contributed to the conception of the study, conducted experiments, developed artificial intelligence models, handled the visualization and coding aspects, and was responsible for writing the manuscript. A.L. was involved in developing artificial intelligence models and coding. B.S. contributed to both the coding and the conception of the study. D.E.R. and D.F. provided supervision and secured funding for this paper.

Peer review

Peer review information

Communications Engineering thanks the anonymous reviewers for their contribution to the peer review of this work. Primary Handling Editors: [Philip Coatsworth]. A peer review file is available.

Data availability

The two-dimensional simulation dataset generated for the study is publicly available at 10.17605/OSF.IO/DE9CP. The three-dimensional experimental datasets form part of ongoing research and are available under restricted access; requests for access should be directed to the corresponding author.

Code availability

The code implementing the Graph Transformer framework for gust-load prediction is available at https://github.com/liangaomng/gust_ce. The repository includes model definitions, training scripts, and simulation datasets for reproducing key results in this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s44172-026-00612-9.