我們針對薛丁格方程式的四種不同位能做神經網路的訓練，包含Simple
harmonic oscillator，Double-well inverted Gaussians，Infinite well和Random。
在Kyle Mills(2017) 文中，他所提出的16層CNN對於單輸出有不錯的結果，但對
於多輸出訓練卻做得很差，而在Jyun-Wei Chen(2020)文中，他對其架構加入了
兩層雙向的LSTM，使得多輸出也能保持著不錯的結果，但兩種架構共同的缺
點是訓練時間太久，這些促使我們提出一種替代的有效神經網絡架構來解決此
類問題。
我們使用了Inception block來構建我們的模型，並參考了stem的架構來進行
設計。訓練資料來源是文章中四種不同位能的方程，藉由有限差分法的方式去
計算特徵值。每筆資料都是256x256的解析度對應10個能階(其中包含一個基態
和9個激發態)
我們的模型特別的地方在於，將二維位能當作是一張圖片(input)，每一像素
值相當於是該位置的位能值(output)，每一張圖片(即位能)，對應多個標籤(即
能量)。而特點在於其非常的輕，對於硬體要求降低許多，參數量相比於上述兩
篇文章的模型分別減少了47 % 以及85 % ，在訓練時間大幅減少的情況下卻能保
持差不多的結果甚至更好，也呈現了Jyun-Wei Chen(2020)文中缺少的Random
potential。

We propose a neural network model with Inception structure to solve the Schrodinger equation under four different potentials, including simple harmonic oscillators (SHOs), infinite wells (IWs), double-well inverted Gaussians (DIG), and random potentials. We used Python, TensorFlow and Keras to implement the code. The paper by Kyle Mills et al proposed 16-layer CNN which has good results for a single output value, but the same architecture has no good performance for multiple output training. In Jyun-Wei Chen’s thesis, he added two different architectures. The bidirectional LSTM can maintain good results for multiple output values, but the common disadvantage of these two architectures is that the amount of calculation is too large and the time for training process is too long. These motivates us to propose an alternative efficient neural network architecture to solve such problem. The training data are solving numerically by using finite difference method. Our proposed model are constructed by several neural network techniques, including stem, Inception convolution, and pooling. We use the discretization values of potential which has 256×256 grid points to as the input data. The features of our model are that it has a much reduced number for training parameters, but produces the comparable accuracy results. We also show the results for case of the random potentials which did not be presented in Jyun-Wei Chen’s thesis.

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