Scaling function learning: A sparse aerodynamic data reconstruction method for various aircraft shapes

Accurate and complete aerodynamic data is essential for aircraft performance evaluation, yet full-state sampling is costly, and wind tunnel conditions differ from flight. Conventional scaling parameters can extrapolate states but lack shape generalization. To achieve low-cost, full-state nonlinear aerodynamic database construction, this paper proposes the Scaling Function Learning (SFL) method. SFL identifies a unified low-dimensional manifold for aerodynamic forces across shapes and flow states, enabling nonlinear modeling with minimal samples. Using symbolic regression on typical aircraft data, SFL extracts a composite function comprising a shape-universal inner scaling function and an outer weakly nonlinear function, enabling sparse sampling. Thus, new aircraft databases can be sparsely reconstructed and extrapolated across the flight envelope with extremely small samples. Validated on the HB-2 model by learning its axial force coefficient scaling function, SFL demonstrated generalization across diverse geometries (HBS, double ellipsoid, sharp cone, double cone missile, HyTRV waverider). Results show SFL enables nonlinear dimensionality reduction for complex systems, achieving cross-state and cross-configuration generalization. With only 5 state samples, SFL constructed aerodynamic databases for varying Mach number, angle of attack and Reynolds number, demonstrating strong extrapolation (1%–5% error) and reducing required aerodynamic samples by ≥90% compared to Kriging. SFL demonstrates potential for discovering other physical scaling laws, although its capabilities for handling strong nonlinearities, such as flow transition or stall, require further exploration.