The team framed the problem as a representative case of what they call partially deterministic data. In many engineering applications, the observable inputs and the target response come from different representations of the same physical process, so they change in a related and physically meaningful way without forming a fully deterministic input-output mapping. Here, the goal was to predict a transient, hotspot-dominated acoustic field from limited steady-state flow variables. Conceptually, the response behaves as a function of the available observables, some unresolved physical information, and numerical or measurement error, where the association denotes physical correlation rather than equality. Changes in the inputs carry information about the expected variation of the target, but no particular input value uniquely determines it. Partial determinism can arise from feature insufficiency, epistemic uncertainty about unobservable details, or a representational gap such as using time-averaged flow variables to predict a transient response. The surrogate is therefore not learning a closed physical mapping, but a structured and physically meaningful correlation between incomplete observables and the target quantity.

The physics of the case study centers on the surface-based, dipole-dominated acoustic response of the landing gear. Under subsonic conditions, Curle’s extension relates the radiated acoustic pressure to the unsteady loading acting on the solid surface, and the local acoustic intensity is proportional to the squared radiated pressure divided by the ambient density and speed of sound. For a rigid body in low-Mach-number flow, the loading term is dominated by pressure, so the acoustic response becomes closely associated with the temporal fluctuation of surface pressure, with radiated acoustic strength scaling with the square of the pressure time derivative. The configuration considered was a simplified landing gear subjected to a horizontal airflow at a low-Mach-number subsonic sea-level condition, with an inlet velocity of 100 meters per second representing a typical aircraft approach scenario, and a porous fairing positioned upstream to partially shield the structure. Multiple fairing configurations enriched the range of acoustic responses observed on the gear surface.

The dataset was generated from acoustic-coupled Computational Fluid Dynamics simulations based on Delayed Detached-Eddy Simulation, a hybrid method capable of resolving the transient vortex shedding and fine-scale turbulence that drive aerodynamic noise. The baseline campaign comprised three DDES acoustic analyses and 36 RANS simulations of the combined fairing-landing-gear system. The prediction objective, denoted surface-dp/dt-RMS, is the root-mean-square magnitude of the temporal derivative of surface pressure, in units of pascals per second, and represents a local measure of the unsteady surface loading associated with dipole-dominated acoustic generation. The final dataset contains 112,452 surface-node samples covering the principal geometric features of the landing gear, including edges, junctions, and narrow gaps. Each sample carries the target response and four local flow variables: dynamic pressure, turbulent kinetic energy, turbulent dissipation rate, and strain-rate magnitude. Node identifiers and spatial coordinates were deliberately excluded as model inputs, so the effects of local surface geometry are represented only indirectly through the deformation, turbulence, and momentum characteristics of the selected flow variables.

Two of the four variables were specifically chosen to capture localized geometric effects that conventional broadband noise models miss. Broadband estimates commonly rely on time-averaged turbulence quantities such as turbulent kinetic energy and a characteristic dissipation variable, but their ability to distinguish spatially complex geometric effects is limited, and they can differ substantially from high-fidelity transient simulations near edges, junctions, narrow gaps, and separated-flow regions. Strain-rate magnitude describes local fluid deformation and shear, indicating rapid flow acceleration, separation, and interactions with complex surface geometry, while dynamic pressure represents the local momentum level and provides a velocity-dependent scale for the aerodynamic loading applied to the surface. A SHAP-based interpretability analysis confirmed that these additions carry real weight in the fitted model. Evaluated on representative samples of 250 test nodes and 250 nodes from the top 10 percent of the reference response, strain-rate magnitude had the highest normalized importance, accounting for 36.7 percent of the global contribution and 39.0 percent within the hotspot region. Turbulent kinetic energy and dissipation rate contributed 26.0 and 21.7 percent globally, with dissipation rate rising to 24.7 percent in hotspots while turbulent kinetic energy fell to 21.2 percent. Dynamic pressure contributed roughly 15 percent in both evaluations. Together, strain-rate magnitude and dynamic pressure accounted for 52.3 percent of global importance and 54.1 percent of hotspot-specific importance. The authors caution that SHAP values quantify sensitivity within the fitted model and should be read as evidence of predictive contribution rather than direct causal attribution.

The study also confronts the hotspot problem, a challenge familiar to anyone modeling acoustic sources, stress concentrations, or thermal failures in electronics. Engineering-critical responses usually occupy only a small fraction of the dataset, so they are easily overwhelmed by the majority of low-to-moderate values during training. A surrogate model can look highly accurate by global metrics while completely missing the high-impact behaviors that dictate design limits. Most conventional models are optimized for the majority of the population, creating a smoothing effect on rare but high-impact responses. In this dataset, high-intensity pressure fluctuation regions concentrate near edges, junctions, narrow gaps, and separation zones, and because these hotspots dominate the global acoustic behavior, identifying and quantifying them matters more than achieving the lowest average error across the full surface.

An initial screening using second-order polynomial regression, a multilayer perceptron, Random Forest, and XGBoost identified the tree ensembles as the most promising direction, so the comprehensive benchmark retained Random Forest as the representative bagging method alongside LightGBM and CatBoost as gradient-boosting approaches, with TabNet and Kolmogorov-Arnold Networks added to test whether alternative neural architectures could better recover localized input-response relationships. Gaussian Process Regression, though attractive for its probabilistic predictions, was not computationally feasible for a node-level dataset exceeding 112,000 samples because of the covariance-matrix operations required, so support vector regression was selected as the practical kernel-based alternative. After iterative hyperparameter optimization, every candidate was tasked with predicting surface pressure fluctuation values across essentially the full landing-gear surface, and the predictions were compared against the high-fidelity simulation results through global metrics and node-wise overlap with the reference field. The final Random Forest consisted of 500 trees with a maximum depth of 20, a minimum of two samples for node splitting, and one sample per terminal leaf.

The benchmark delivered a verdict that cuts against the prevailing assumption that deeper architectures win by default. Random Forest achieved the strongest overall result, followed by CatBoost and LightGBM, while TabNet performed the weakest. Support vector regression and Kolmogorov-Arnold Networks failed to produce effective models, both yielding an R-squared below 0.1 and were excluded from the main comparison. The authors attribute Random Forest’s advantage to its bagging-based ensemble structure: because the trees are trained independently and aggregated afterward, the model is generally more robust to noisy data, hidden feature interactions, and locally inconsistent patterns. Gradient boosting learns sequentially by correcting the residuals of earlier trees, which improves fitting efficiency in more deterministic problems but can reduce robustness when the available features do not fully resolve the target response. Boosting methods are designed to uncover deeper interactions, but such interactions may not exist in a stable or learnable form within the selected feature set. TabNet likely combines higher sensitivity to noisy or insufficient features with a stronger tendency toward smooth global fitting, making it less capable of preserving the localized hotspot behavior. Even the best mean-prediction model, however, still underestimated the magnitude of the strongest peaks.

The correction came from the prediction distribution itself. Rather than estimating only the conditional mean, the team collected the outputs of the individual trees for each input and calculated an empirical ensemble percentile, training the forest only once with the conventional regression objective. This differs from standard Quantile Regression Forest estimation, which constructs a conditional distribution from training observations and forest-derived weights. The quantile level introduces an engineering trade-off: lower quantiles stay closer to the mean and preserve global accuracy, while higher quantiles improve peak sensitivity at the cost of upward bias and degraded accuracy in low- and medium-intensity regions. Beyond approximately Q75, the penalty became more significant than the gain in hotspot capture, and extreme upper quantiles produced less stable, spike-like overprediction. Q75 provided the best overall balance, raising R-squared from 0.4926 for the mean prediction to 0.6607, an absolute increase of 0.1681 and a relative improvement of roughly 34.1 percent, while the F1-score evaluated at the 90th-percentile threshold rose from 0.604 to 0.810. Repeated independent runs confirmed repeatability, with standard deviations of 0.00213, 2082.5, and 879.4 for R-squared, RMSE, and MAE respectively. A spatially withheld test, in which the model was trained on selected surface regions and evaluated on unseen nodes to reduce information leakage, supported Q75 as the best overall choice, with Pearson’s r and Spearman’s rho varying by no more than about plus or minus 0.02 between Q50 and Q80, showing that the adjustment primarily modifies prediction scale while preserving the spatial response relationship.

Crucially, the quantile trick is not universally applicable. When LightGBM and CatBoost were trained with quantile-specific loss objectives, both models degraded, failing to recover previously unidentified hotspot peaks while increasing error in low- and medium-intensity regions. The explanation is that upper-quantile adjustment cannot recover spatial features absent from the underlying prediction; it primarily inflates existing outputs, including errors, leading to overprediction without improving hotspot identification. The adjustment is most useful when the baseline model already represents the principal response structure and hotspot locations but underestimates their magnitudes. The authors emphasize that the specific model rankings, performance values, and the Q75 setting remain case-dependent, and that future work will expand the transient simulation dataset, pursue more rigorous quantile selection through grouped cross-validation, conduct experimental validation when facilities permit, and investigate physically informed machine learning to incorporate aerodynamic and aeroacoustic relationships more directly into the learning process. For now, the message for engineers wrestling with incomplete, hotspot-dominated data is clear: match the model to the information structure of the data, and stability can beat sophistication.

Subject of Research: Machine learning surrogate modeling with quantile adjustment for landing gear aeroacoustic surface pressure fluctuation prediction

Article Title: Machine learning surrogate modeling with quantile adjustment for landing gear surface pressure fluctuation prediction

Article References: Cheng, Z., & Behdinan, K. (2026). Machine learning surrogate modeling with quantile adjustment for landing gear surface pressure fluctuation prediction. Discover Artificial Intelligence, 6(1), Article 1635. https://doi.org/10.1007/s44163-026-02600-6

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02600-6

Keywords: machine learning, surrogate modeling, landing gear, aeroacoustics, Random Forest, quantile regression, hotspot prediction, CFD, DDES, pressure fluctuations, SHAP, bias adjustment