Approximation of Dynamical Systems by the Kuramoto Model

Analog computing leverages physics to perform computation energy efficiently in hardware. These systems typically have restrictive dynamics, and their expressive power is poorly understood. We address this question of expressivity for the Kuramoto model, an analog compute model that leverages oscillator coupling behavior. We establish a universal dynamical system approximation theorem: An autonomous Kuramoto network can approximate the flow of any continuously differentiable vector field on a torus over a finite time interval, uniformly over initial conditions, without a learned output map or external forcing. The result extends to smooth periodic coupling functions with a nonzero first Fourier harmonic. Experiments on seven dynamical systems validate our trajectory approximation approach, and flow-matching experiments show that Kuramoto networks can approximate learned flows, suggesting a promising use of the Kuramoto model in generative modeling.