Learning to Synchronize in Minimum Time

Abstract

The minimum-time feedback law for driving a population of coupled oscillators into synchrony is unknown. Here we settle it for identical Kuramoto oscillators under an instantaneous power constraint. Greedy control, maximizing $\dot r$ at each instant, is exactly optimal at $N=2$ and suboptimal above, as dynamic programming confirms at $N=3,4$. The obstruction is geometric: the greedy closed loop is a reparametrized gradient flow of $r$, fixing its path independently of the power budget, and as a first-harmonic forcing it cannot leave a single M\"obius orbit. A three-harmonic policy trained on the DP fields, a trajectory expert, and a smooth first-hitting-time objective beats greedy by $10$--$14\%$ at $N=10$--$100$ and matches DP to within $0.3\%$ where ground truth exists. Reading the policy rather than deploying it collapses it to a two-constant law, $u_i\propto-\sin\phi_i+a_2\sin2\phi_i+a_3\sin3\phi_i$, which recovers $84$--$99\%$ of the network's advantage, with the same functional form holding from $N=3$ to $N=1000$. Machine learning discovered a closed-form law beyond the reach of direct analytical methods.