Learning Phase Transitions from Regression Uncertainty

For performing regression tasks involved in various physics problems, enhancing the precision, or equivalently reducing the uncertainty of the regression results is undoubtedly one of the central goals. Here, somewhat surprisingly, we find that the unfavorable regression uncertainty in performing regression tasks of inverse statistical problems actually contains "hidden" information concerning phase transitions of the system under consideration. By utilizing this "hidden" information, we develop a new unsupervised machine learning approach dubbed "learning from regression uncertainty" for automated detection of phases of matter, with the core working horse being a neural network performing regression tasks instead of classification tasks as in various related machine learning approaches developed so far. This is achieved by revealing an intrinsic connection between regression uncertainty and response properties of the system under consideration, thus making the results of this machine learning approach directly interpretable. We demonstrate the approach by identifying the critical point of the ferromagnetic Ising model and revealing the existence of the intermediate phase in the six-state and seven-state clock models. Our approach paves the way towards intriguing possibilities for unveiling new physics via machine learning in an interpretable manner.