Data-Driven Discovery and Formulation Refines the Quasi-Steady Model of Flapping-Wing Aerodynamics

Insects control unsteady aerodynamic forces on flapping wings to navigate complex environments. While understanding these forces is vital for biology, physics, and engineering, existing evaluation methods face trade-offs: high-fidelity simulations are computationally or experimentally expensive and lack explanatory power, whereas theoretical models based on quasi-steady assumptions offer insights but exhibit low accuracy. To overcome these limitations and thus enhance the accuracy of quasi-steady aerodynamic models, we applied a data-driven approach involving discovery and formulation of previously overlooked critical mechanisms. Through selection from 5,000 candidate kinematic functions, we identified mathematical expressions for three key additional mechanisms -- the effect of advance ratio, effect of spanwise kinematic velocity, and rotational Wagner effect -- which had been qualitatively recognized but were not formulated. Incorporating these mechanisms considerably reduced the prediction errors of the quasi-steady model using the computational fluid dynamics results as the ground truth, both in hawkmoth forward flight (at high Reynolds numbers) and fruit fly maneuvers (at low Reynolds numbers). The data-driven quasi-steady model enables rapid aerodynamic analysis, serving as a practical tool for understanding evolutionary adaptations in insect flight and developing bio-inspired flying robots.