錯亂方案在以類神經網絡研究相變上的有效性之檢驗

Abstract

本研究旨在探討類神經網路在物理系統相變偵測中的應用成效，以二維二態Ising模型以及三態與四態的反鐵磁 Potts 模型[1]研究，透過 Nieuwenburg 等人於 2017 年提出的錯亂方案（Confusion Scheme）方法[2]，結合卷積神經網路（Convolutional Neural Network, CNN）[3]，預測系統中潛在的分類邊界與相變行為。設定一系列假設相變點 T'c 並重新訓練分類器，透過繪製出機率隨 T'c 變化的 W 型曲線，以此辨識出神經網路最能正確分類樣本的相變點位置。 方法上，我們對每一假設相變點T'c 訓練一組分類模型，並繪製該模型在所有溫度樣本上的分類準確率。透過準確率與T'c 的關係圖，我們觀察到在 Ising 模型中出現典型的 W 字型結構，高峰值對應系統實際相變溫度 Tc，顯示錯亂方案能正確預測相變行為。然而，在 Q=3（Tc=0）與 Q=4（無相變點）模型中，我們觀察到的準確率曲線皆呈現 V 字型，表示分類邊界並未位於樣本數據範圍之內，與理論相符，卻無法由錯亂方案直接推得 Tc 為零或不存在的結論。 本研究將在結論中探討原始的confusion scheme是否可應用於有多重相變的未知系統中為未來研究方向之一，也可以結合無監督學習[4]、貝葉斯神經網路[5]與拓樸相變模型[6]等發展方向，驗證深度學習於物理相變辨識中的潛力與挑戰。本研究的計算是基於Tensorflow寫成[7] 。This study explores the application of neural networks in detecting phase transitions within physical systems. Focusing on the two-dimensional Ising model , three-state and four-state antiferromagnetic Potts models, we employ the confusion scheme proposed by Nieuwenburg et al. (2017), in combination with convolutional neural networks (CNNs), to predict latent classification boundaries and critical behaviors. By setting a range of hypothetical critical temperatures T'c and retraining the classifier for each, we construct a characteristic W-shaped curve of prediction accuracy versus T'c . The peak of this curve indicates the point at which the neural network most effectively distinguishes between phases, serving as an estimate of the system's critical point. In terms of methodology, we train a separate classifier for each hypothetical critical point T'c , and compute its classification accuracy over all temperature samples. From the accuracy vs. T'c plots, we observe a typical W-shaped structure in the Ising model, where the central peak corresponds to the true critical temperature Tc , indicating that the confusion scheme successfully predicts the critical behavior. However, in the Q=3 model (Tc=0) and the Q=4 model (which has no true critical point), the resulting accuracy curves exhibit a V-shaped structure, implying that the decision boundary does not lie within the sampled data range. While this matches theoretical expectations, the confusion scheme itself fails to explicitly infer that Tc =0 or that no critical point exists.In the conclusion, we further discuss whether the original confusion scheme can be applied to systems with multiple or unknown phase transitions, suggesting this as a direction for future research. Moreover, potential extensions may involve integrating unsupervised learning methods, Bayesian neural networks, and models with topological phase transitions to evaluate the potential and challenges of deep learning in phase transition recognition in physics.The calculations in this thesis are based on TensorFlow [7].