Local Learning Rules for Out-of-Equilibrium Physical Generative Models

AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands (Dated: August 28, 2025) We show that the out-of-equilibrium driving protocol of score-based generative models (SGMs) can be learned via local learning rules. The gradient with respect to the parameters of the driving protocol is computed directly from force measurements or from observed system dynamics. As a demonstration, we implement an SGM in a network of driven, nonlinear, overdamped oscillators coupled to a thermal bath. We first apply it to the problem of sampling from a mixture of two Gaussians in 2D. Finally, we train a 12 12 oscillator network on the MNIST dataset to generate images of handwritten digits "0" and "1".