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.

中文翻译：

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用最优分析预测器替换神经网络以检测相变

识别相变和物质相分类对于理解广泛的材料系统的特性和行为至关重要。近年来，机器学习 (ML) 技术已成功应用于以数据驱动的方式执行此类任务。然而，尽管这种方法取得了成功，但我们仍然缺乏对用于检测相变的机器学习方法的清晰理解，尤其是那些利用神经网络 (NN) 的方法。在这项工作中，我们推导了三种广泛使用的基于 NN 的相变检测方法的最佳输出的解析表达式。这些最优预测对应于在高模型容量限制下获得的结果。因此，在实践中，例如，可以使用足够大、训练有素的神经网络来恢复它们。通过优化输出对输入数据的显式依赖性揭示了所考虑方法的内部工作原理。通过评估解析表达式，我们可以直接从实验可访问的数据中识别相变，而无需训练神经网络，这使得该过程在计算时间方面是有利的。我们的理论结果得到了广泛的数值模拟的支持，例如拓扑、量子和多体定位相变。我们期望类似的分析能够更深入地理解凝聚态物理中的其他分类任务。这使得这个过程在计算时间方面是有利的。我们的理论结果得到了广泛的数值模拟的支持，例如拓扑、量子和多体定位相变。我们期望类似的分析能够更深入地理解凝聚态物理中的其他分类任务。这使得这个过程在计算时间方面是有利的。我们的理论结果得到了广泛的数值模拟的支持，例如拓扑、量子和多体定位相变。我们期望类似的分析能够更深入地理解凝聚态物理中的其他分类任务。