Training oscillator Ising machines to assign the dynamic stability of their equilibrium points

We propose a neural network model, which, with appropriate assignment of the stability of its equilibrium points (EPs), achieves Hopfield-like associative memory. The oscillator Ising machine (OIM), based on Kuramoto-like dynamics, is an ideal candidate for such a model, as all its 0/π binary EPs are structurally stable with their dynamic stability tunable by the coupling weights. Traditional Hopfield-based models store the desired patterns by designing the coupling weights between neurons. The design of coupling weights should simultaneously take into account both the existence and the dynamic stability of the EPs for the storage of the desired patterns. For OIMs, since all 0/π binary EPs are structurally stable, that is, any of these EPs remains an EP regardless of the values of the coupling weights or the system parameters, the design of the coupling weights needs only to focus on assigning appropriate dynamic stability for the 0/π binary EPs according to the desired patterns. In this paper, we establish a connection between the stability and the Hamiltonian energy of EPs for OIMs, and, based on this connection, provide a Hamiltonian-regularized eigenvalue contrastive method (HRECM) to train the coupling weights of OIMs for assigning appropriate stability to their EPs. Finally, numerical experiments are performed to validate the effectiveness of the proposed method.