Bounded Channel-Adaptive Spectral Learning for Forward-Consistent Inverse Flapping-Wing Aerodynamics

Experimental investigation and modeling of flapping-wing aerial vehicles are limited by the scarcity of large-scale records that temporally align wing kinematics, aerodynamic responses, and flight states. Existing datasets are often limited in scale and affected by measurement noise and temporal misalignment between rapidly varying wing motion and the associated dynamic response, particularly during high-frequency flapping. We introduce FlapKAD, an episode-structured simulation dataset comprising 2,000 rigid-wing flight episodes and 720,152 valid time steps, with bilateral wing kinematics, aerodynamic responses, and flight states recorded synchronously within a common clock. FlapKAD supports a unified bidirectional sequence-prediction benchmark constructed from the same temporally aligned episodes. The forward task predicts future vertical force coefficients and body vertical velocity from histories of realized flap and twist angles, whereas the inverse task reconstructs future flap- and twist-angle trajectories from the corresponding response histories. A benchmark of eight representative time-series architectures across multiple prediction horizons reveals direction- and horizon-dependent model behavior, systematically higher reconstruction errors for twist angle than for flap angle, and no consistent advantage from increased architectural complexity. FlapKAD provides a reproducible dataset and benchmark for studying coupled wing-kinematic, aerodynamic, and flight-state dynamics in flapping-wing aerial vehicles.

Bidirectional thrust grants quadrotors a second equilibrium condition and increased control authority, expanding the envelope of possible aggressive maneuvers and enabling inverted flight, perching, and sensing. Prior geometric control approaches extend differential flatness through Hopf fibration-based attitude representations to support bidirectional thrust, but struggle with actuator saturation and motor reversal delay during inversions, requiring heuristic thrust posture scheduling and waypoint tuning. We propose a learning-based framework that modulates a constant reference trajectory to perform compact, position-constrained quadrotor inversions while remaining compatible with traditional trajectory generation and tracking across flight regimes. Separate policies are trained via reinforcement learning for nominal-to-inverted and inverted-to-nominal transitions. In JAX-based simulation, the proposed method achieves the lowest position deviation and settling time across all evaluated baselines, reducing position root mean square error (RMSE) by 32% and settling time by 57% relative to the strongest optimization-based baseline. Hardware experiments demonstrate successful inversion across multiple yaw configurations with position RMSE below 0.35m, and compatibility with downstream trajectory generation and control through circular flight in both regimes. Additionally, we provide an open-source implementation of the proposed framework.

High-fidelity simulations of complex physical systems are often too expensive for repeated prediction, design, and control. Reduced-order models address this computational cost by compressing high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they often lose accuracy when online dynamics drift beyond the offline training data. Adaptive methods address this limitation by updating the reduced subspace online using external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. To exploit this channel, we continually update the reduced representation using the model's own predictions through an incremental singular-value decomposition with a forgetting factor. This produces a trajectory-informed spectral preconditioner in which the reduced subspace remains unchanged, while the basis is reweighted and realigned according to the directions visited by the evolving dynamics. This internal reorganization prepares the basis to absorb future out-of-span corrections more effectively, improving adaptation without requiring additional external information. We expose the mechanism through a three-dimensional spiral example and demonstrate its benefits on nonlinear partial differential equations, including viscous Burgers and Fisher--KPP dynamics. We also discuss how in-span learning can be interpreted as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.