Machine Learning Phase Transition: An Iterative Proposal.

We propose an iterative proposal to estimate critical points for statistical models by employing machine-learning tools. Firstly, phase scenarios and rough boundaries of phases are obtained by techniques used in dimensionality-reduction processes. This step not only provides incipient phase boundaries and labelled samples for the subsequent step but also is necessary for its application to unexplored statistical models. Secondly, making use of these samples as training set, neural networks are employed to assign labels to the samples between the phase boundaries in an iterative way. Newly labelled samples would be put in the training set used in subsequent training and the phase boundaries would be updated. The average of the phase boundaries is expected to converge to the critical temperature. In concrete examples, we implement this proposal to estimate the critical temperatures for two q-state Potts models with continuous and first order phase transitions. Techniques used in linear and manifold dimensionality-reduction processes are employed in the first step. Both a convolutional neural network and a bidirectional recurrent neural network with long short-term memory units perform well for the two Potts models in the second step. The convergent behaviors of the estimations correspond to the types of phase transitions. And the results indicate that our proposal has the potential to be used to explore phase transitions for new general statistical models.