Scientists have introduced a physics‑constrained data‑driven framework that turns large‑scale nonlinear PDE‑constrained optimization problems into reduced‑order model (ROM) constrained problems. The method maps the optimization onto system dynamics in a Hilbert space, improving numerical stability, lowering memory demands, and cutting computational cost.

Applying this new ROM approach to a 2D flapping wing at a Reynolds number of 2,400, the study achieved a 37 % increase in time‑averaged thrust force and a 40 % rise in propulsive efficiency compared with conventional methods. The same strategy also produced promising results for a 3D wing, hinting at broader applicability in aerospace design.

The gains stem from avoiding the unstable forward sensitivity calculations of traditional ROM techniques, while still capturing the essential physics of the fluid‑structure interaction.