Airborne capture net devices mounted on unmanned aerial vehicles (UAVs) provide an effective physical countermeasure against rogue drones. However, conventional deterministic designs neglect parameter uncertainties such as manufacturing tolerances and assembly errors that degrade flexible net deployment dynamics. To resolve this, this study proposes a multi-objective robust design optimization framework integrating a feedforward neural network (FNN) with uncertainty quantification. A lumped-mass dynamic model is formulated; static deployment experiments validate its high fidelity during the initial phase (t ≤ 0.08 s), while subsequent full-deployment and contraction phases are captured via experimentally calibrated numerical extrapolation. To mitigate computational costs, an adaptive FNN surrogate model is developed, reducing single-evaluation times from hours to seconds. Monte Carlo simulations coupled with the non-dominated sorting genetic algorithm II (NSGA-II) are then employed to optimize the mean and standard deviation of deployment and effective distances. Compared with deterministic optimization, the robust solution yields a 23.8% increase in the 95th percentile of deployment distance with only a 1.5% decrease in effective distance. Crucially, the standard deviations of these metrics decrease by 22.5% and 38.5%, respectively, demonstrating significantly attenuated sensitivity to parameter variations and offering a valuable engineering framework for robust airborne interception.