方正晶格上二維五態和二態鐵磁性帕茲模型的神經網絡研究

A neural network study of two-dimensional 5-state and 2-state ferromagnetic Potts models on the square lattice

曾云萱(Tseng Yun-Hsuan)

指導教授 ： 江府峻

國立臺灣師範大學/理學院/物理學系/碩士(2022年)

https://doi.org/10.6345/NTNU202201275

摘要

本論文分為兩個主題，首先，(1)利用簡單且通用的監督式神經網路來研究二維五態和二態鐵磁性帕茲模型在正方晶格上的相變行為，有別於一般的訓練方法[1]，神經網路的訓練集是由一維200個
晶格點上以人工產生的兩種(0和1)組態所構成的，並將預測結果繪製成輸出向量長度|R ⃗|的直方圖，從圖中看出是否為雙峰分布，進而得知是一階相變或是二階相變。利用這樣簡單的神經網絡模型來探討大型的自旋系統(含有數百萬個自旋)，可以得到五態鐵磁性帕茲模型的相變為微弱一階相變。如此龐大的系統，如果是用一般的訓練方法期計算量是普通電腦無法負荷的。
　　另外，(2)使用長短期記憶模型(LSTM)來產生由蒙地卡羅演算法計算出來的能量密度，結果顯示利用少量的訓練集就可以得到相近的平均值。
　　本論文部分章節已發表於arXiv:2111.14063。

關鍵字

帕茲模型 ； 相變 ； 多層感知器 ； 長短期記憶模型

並列摘要

This thesis is divided into two topics. First, use a simple and general supervised neural network, we study the phase transition of the two-dimensional 5-state and 2-state ferromagnetic Potts models on the square lattice. The training employed in our investigation is different from the general training methods [1]. The training set of the neural network is composed of two configurations of one-dimensional 200 lattices, and the histogram of the output vector length |R ⃗| are considered as the predicted results. By examining the histograms, we can determine whether the phase transition is a first-order or second-order. Using such a simple neural network model to investigate large spin systems (containing millions of spins), we find that the phase transition of the 5-state Potts model is weakly first-order. If the conventional training method is used to study such a huge system, then the amount of calculations is beyond the load of ordinary computers. In addition, with a long short-term memory model(LSTM), we generate the energy density using the data calculated by Monte Carlo algorithm. The results show that a similar average value can be obtained with a small training set. Parts of this thesis have been published on arXiv:2111.14063.

並列關鍵字

Potts model ； phase transition ； MLP ； LSTM

參考文獻

[1] Alexandou, C., Athenodorou, A., Chrysostomou, C., & Paul, S. (2020). The critical temperature of the 2D-Ising model through deep learning autoencoders. The European Physical Journal B, 93(12), 1-15

[2] A. M. TURING, I.—COMPUTING MACHINERY AND INTELLIGENCE, Mind, Volume LIX, Issue 236, October 1950

[3] Rashid, T. (2016). Make your own neural network (p. 222). CreateSpace Independent Publishing Platform.

[4] R.B. Potts, Some Generalized Order-Disorder Transformations. Mathematical Proceedings of the Cambridge Philosophical Society, 48, 106.

[5] F.-Y. Wu, The potts model, Reviews of modern physics, 54, 235 -Published 1 January(1982).