The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks

ArXi:2604.10272v1 Announce Type: new We prove that in a coupled Kuramoto oscillator network at stable equilibrium, the physical phase displacement under weak output nudging is the gradient of the loss with respect to natural frequencies, with equality as the nudging strength beta tends to zero. Prior oscillator equilibrium propagation work explicitly set aside natural frequency as a learnable parameter; we show that on sparse layered architectures, frequency learning outperforms coupling-weight learning among converged seeds (96.0% vs. 83.3% at matched parameter counts, p = 1.8e-12.