学习在最短时间内实现同步
Learning to Synchronize in Minimum Time
AI总结：
针对耦合振子最短时间同步反馈律未知的问题，基于动态规划场训练三次谐波策略，性能优于贪婪策略，还提炼出跨规模适用的闭式同步控制律。
AI中文摘要：
驱动耦合振子群体达到同步的最短时间反馈律目前尚属未知。本文针对瞬时功率约束下的全同Kuramoto振子（仓本振子）解决了这一问题。在每个时刻最大化$\boldsymbol{\r}$的贪婪控制在$N=2$时是严格最优的，当振子数更多时则是次优的，这一点已通过$N=3、4$时的动态规划（dynamic programming，DP）得到验证。其阻碍源于几何层面：贪婪闭环是$r$的一种重参数化梯度流，其路径不受功率预算的影响，并且作为一次谐波强迫，它无法脱离单一的莫比乌斯轨道。我们基于动态规划场（轨迹专家）训练的三次谐波策略，搭配平滑的首达时间目标，在$N=10$到$100$的场景下性能优于贪婪策略$10$--$14\\%$，在存在基准真值的情况下与动态规划结果的误差在$0.3\\%$以内。对该策略进行提炼而非直接部署，可将其简化为含两个常数的定律：$u_i\propto-\sin\phi_i+a_2\sin2\phi_i+a_3\sin3\phi_i$，该形式可恢复网络$84$--$99\\%$的性能优势，且在$N=3$到$N=1000$的范围内均适用。机器学习发现了一个直接解析方法无法得到的闭式定律。
英文摘要：
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 Möbius 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, $u_i\propto-\sinϕ_i+a_2\sin2ϕ_i+a_3\sin3ϕ_i$, 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.
