Learning continuous spin models with real-valued restricted Boltzmann machines

Statistical systems with continuous spin or vector variables, such as the XY model, the Heisenberg model, and liquid crystals, have long been a challenge for machine learning methods due to their large number of degrees of freedom (compared with discretized Ising-like models). We propose novel real-valued restricted Boltzmann machines with nonlinear cos/sin activations that can effectively learn the underlying Boltzmann distribution of these systems and generate configurations that capture the thermodynamics and topological order of physical systems. We find that phase transition information can be extracted from the weight parameters of our machine learning models without prior physics knowledge.