Replacing neural networks by optimal analytical predictors for the detection of phase transitions

TL;DR

This paper derives analytical expressions for optimal neural network predictions in phase transition detection, enabling direct identification from data without training, thus enhancing understanding and efficiency in condensed matter physics.

Contribution

It provides the first analytical framework for understanding NN-based phase transition detection, revealing the optimal predictions and their dependence on data.

Findings

Analytical expressions for optimal NN predictions derived

Optimal predictions can be obtained without training NNs

Numerical simulations confirm the theoretical results

Abstract

Identifying phase transitions and classifying phases of matter is central to understanding the properties and behavior of a broad range of material systems. In recent years, machine-learning (ML) techniques have been successfully applied to perform such tasks in a data-driven manner. However, the success of this approach notwithstanding, we still lack a clear understanding of ML methods for detecting phase transitions, particularly of those that utilize neural networks (NNs). In this work, we derive analytical expressions for the optimal output of three widely used NN-based methods for detecting phase transitions. These optimal predictions correspond to the results obtained in the limit of high model capacity. Therefore, in practice they can, for example, be recovered using sufficiently large, well-trained NNs. The inner workings of the considered methods are revealed through the explicit dependence of the optimal output on the input data. By evaluating the analytical expressions, we can identify phase transitions directly from experimentally accessible data without training NNs, which makes this procedure favorable in terms of computation time. Our theoretical results are supported by extensive numerical simulations covering, e.g., topological, quantum, and many-body localization phase transitions. We expect similar analyses to provide a deeper understanding of other classification tasks in condensed matter physics.