在這次的研究中，我們使用蒙地卡羅模擬以及人工神經網路技術來研究二 維正方形晶格上之Q-state Potts model。由上一章節的數值結果中可以得到結 論：以卷積神經網路之輸出向量長度R 做為主要觀測量和使用傳統觀測量|m|一 樣可偵測臨界溫度Tc以及辨別相變為一階或二階相變。
在其他文獻中使用T>Tc以及T<Tc的自旋狀態作為訓練集的方式需要較多 的計算資源，使得計算尺度無法和傳統方式相比擬。藉由改用低溫自旋狀態作 為訓練集，訓練時所需的資源較先前的方式大幅減少，強化了人工神經網路應 用於凝態系統的能力。
除了使用在第三章中提及的卷積神經網路架構之外，我們也嘗試了改一些 人工神經網路的超參數，以確保數值結果並非使用人工「微調」出來的。例 如：改變卷積核之大小，由原本的的3×3 改變為 2×2、改變卷積核心之初始值 等等。調整後的模型之能力與原先的版本並無顯著差異（見 Fig. 5.1），足見 在本次研究中使用的方法並不會過度依賴人工神經網路模型的超參數。
值得一提的是，對於弱一階相變Q=5，由於其 correlation length 極大，使用 傳統序参数（order parameter）或本次研究中使用的觀測量 R 均難以偵測出其一 階相變之特徵。最近一年的文獻中曾提出一些新的分析方法[48]，希望能在晶 格大小L 不大的情況下偵測到一階段相變，但仍有改善的空間。人工神經網路 作為研究凝態系統的新工具，在小尺度下偵測弱一階相變將會是我們未來的一 個研究重點之一。
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除此之外，使用人工神經網路研究凝態系統仍有許多問題值得探索的，例
如檢驗R 計算二階相變之 critical exponent 的能力等等，這些相關的應用仍需要 更多詳細的數值研究。
Fig. 5.1 用大小為 2x2 的卷積核後的數值結果。
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附錄
表格 2 常用的單變數活化函數
函數名稱 方程式 函數圖形 輸出區間
linear f(x) = x (−∞, +∞)
step f(x) = {0, x < 0
1, x ≥ 0 {0, 1}
sigmoid f(x) = 1
1 + 𝑒−𝑥 (0, 1)
tanh f(x) = tanh(x) (−1, 1)
41 ReLU f(x) = {0, x < 0
x, x ≥ 0 [0, ∞)
Leaky ReLU f(x) = {λx, x < 0
x, x ≥ 0 (−∞, ∞)
表格 3 常用的多變數活化函數
函數名稱 方程式 輸出區間
Softmax f(𝑧 )𝑗= 𝑒𝑧𝑗
∑𝐾𝑘=1𝑒𝑍𝑘 for 𝑗 = 1, … , 𝐾 (0, 1)
Maxout f(𝐱) = max (xi) (−∞, ∞)
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