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Unlike a conventional phase transition, a crossover is not identified by a singularity in the free energy. A conventional phase transition can be identified in two ways, with the first being a singularity in a derivative of the free energy as proposed by Ehrenfest and the second being a broken symmetry exhibited by an order parameter as proposed by Landau. Other systems may not even exhibit a true phase transition, but rather a crossover region where there is no singularity across different phases that can be difficult to characterize with conventional methods. This is not a hypothetical situation, indeed hidden orderings for some interesting materials, such as heavy fermion materials and cuprate superconductors, have been proposed for long time 6, 7, 8. Perhaps more importantly, for some sufficiently complex systems, their phase transitions do not have obvious order parameters, often prohibiting the detection of such pattern changes using conventional methods. Similar form of pattern changes in the positions of the constituent atoms are often present in molecular systems in general. The widely adopted Lindemann parameter, for example, is essentially a measure of the deviations of atomic positions in the system from equilibrium positions and is often used to characterize the melting of a crystal structure 5.

Fortunately, this is in fact exactly what happens in most phase transitions.

In order to utilize ML approaches for studying phase transitions, one must assume that there is some pattern change in the measured data across the phase transition. Outstanding problems involving the predictions of transition points and phase diagrams are also of great interest for treatment with ML methods. There is a unique opportunity to take advantage of the advances in ML algorithms and implementations to provide interesting new approaches to understanding physical data and even perhaps improve upon existing numerical methods 4. Some ML methods such as inference methods have been routinely applied to certain physical problems, such as the maximum likelihood method and the maximum entropy method 1, 2, but applications which utilizing ML methods have only recently attracted attention in the physical sciences, particularly for the study of interacting systems on both classical and quantum scales 3. Conventional approaches often neglect possible nuance in the structure of the data in favor of rather simple measurements that are often untenable for sufficiently complex problems. This presents a colossal opportunity for modern scientific investigations, particularly numerical studies, as they naturally involve large data sets and complex systems where obvious explicit instructions for analysis can be elusive. Conceptually, the ML approach can be regarded as a data modeling approach employing algorithms that eschew explicit instructions in favor of strategies based around pattern extraction and inference driven by statistical analysis. Machine learning (ML) and consequently data science as a whole have seen rapid development over the last decade or so, due largely to considerable advances in implementations and hardware that have made computations more accessible.

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