Given the rapidly growing scale and resource requirements of machine learning applications, the idea of building more efficient learning machines much closer to the laws of physics is an attractive proposition. One central question for identifying promising candidates for such neuromorphic platforms is whether not only inference but also training can exploit the physical dynamics. In this work, we show that it is possible to successfully train a system of coupled phase oscillators—one of the most widely investigated nonlinear dynamical systems with a multitude of physical implementations, comprising laser arrays, coupled mechanical limit cycles, superfluids, and exciton-polaritons. To this end, we apply the approach of equilibrium propagation, which permits to extract training gradients via a physical realization of backpropagation, based only on local interactions. The complex energy landscape of the XY/Kuramoto model leads to multistability, and we show how to address this challenge. Our study identifies coupled phase oscillators as a new general-purpose neuromorphic platform and opens the door towards future experimental implementations.

The increasing complexity of neural networks and the energy consumption associated with training and inference create a need for alternative neuromorphic approaches, e.g. using optics. Current proposals and implementations rely on physical non-linearities or opto-electronic conversion to realise the required non-linear activation function. However, there are significant challenges with these approaches related to power levels, control, energy-efficiency, and delays. Here, we present a scheme for a neuromorphic system that relies on linear wave scattering and yet achieves non-linear processing with a high expressivity. The key idea is to inject the input via physical parameters that affect the scattering processes. Moreover, we show that gradients needed for training can be directly measured in scattering experiments. We predict classification accuracies on par with results obtained by standard artificial neural networks. Our proposal can be readily implemented with existing state-of-the-art, scalable platforms, e.g. in optics, microwave and electrical circuits, and we propose an integrated-photonics implementation based on racetrack resonators that achieves high connectivity with a minimal number of waveguide crossings.

The fermionic Kitaev chain is a canonical model featuring topological Majorana zero modes. We report the experimental realization of its bosonic analogue in a nanooptomechanical network, in which the parametric interactions induce beam-splitter coupling and two-mode squeezing among the nanomechanical modes, analogous to hopping and p-wave pairing in the fermionic case, respectively. This specific structure gives rise to a set of extraordinary phenomena in the bosonic dynamics and transport. We observe quadrature-dependent chiral amplification, exponential scaling of the gain with system size and strong sensitivity to boundary conditions. All these are linked to the unique non-Hermitian topological nature of the bosonic Kitaev chain.<br>We probe the topological phase transition and uncover a rich dynamical phase diagram by controlling interaction phases and amplitudes. Finally, we present an experimental demonstration of an exponentially enhanced response to a small perturbation. These results represent the demonstration of a new synthetic phase of matter whose bosonic dynamics do not have fermionic parallels, and we have established a powerful system for studying non-Hermitian topology and its applications for signal manipulation and sensing.