我們針對薛丁格方程式的三種不同位能做機器學習的訓練，包含Simple harmonic oscillator，Double-well inverted Gaussians和Infinite well。
程式語言方面選擇的是Python撰寫，並且利用Tensorflow與Keras的套件包去對神經網路做修改。
在Kyle Mills(2017) 文中，他所提出的16層CNN對於單個輸出值有不錯的展現，但是同一種架構做多輸出訓練效果非常差，所以我們對其做修改及延伸。

我們在16層CNN架構後面加入了兩層雙向的LSTM，讓輸出欄位從一個增加至十個。
而訓練資料來源是藉由文章中所提出三種不同位能的方程，藉由有限差分法的方式做特徵值的計算。
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)。
在文獻提出的16層卷積層中，主要是以壓縮圖片解析度以達到逐層加密與壓縮的動作。
讓遞迴層對卷積層所傳下的值，做解密提取值的動作。

我們的模型特別的地方在於，
將二維位能看成是一張圖片，每一像素值即代表該位置的位能值，而每一張圖片(即位能)，能對應出多個標籤(即能量)，
而且這個標籤並不是明確的分類問題，所以並不是像手寫辨識那樣0跟1的差別。
而原文中所使用的優化器AdaDelta，我們也替換成現在所常見的Adam，而每層的輸出激活函數皆為ReLU。
新的模型架構可以使得訓練資料從原本的20萬降到最多6萬筆，但是預測能力與原本的CNN模型一樣甚至更好。
在所需資料降成三分之一，對於硬體計算量而言減少了許多，總訓練時間也大幅降低，對於硬體要求也同樣降低許多。

We construct a neural network model which can simultaneously predict the ground-state and many excited-state energies of the Schrodinger equation.
We consider three different kinds of potentials, including simple harmonic oscillators, double-well inverted Gaussians and infinite wells.
The tool used is Python, TensorFlow and Keras.
The training data is constructed by solving the eigenvalue problem of Schrodinger equation through the finite difference method.
The model architecture has multilayer convolutional neural network and multilayer bidirectional Long Short-Term Memory.
Image resolution is 256x256 grid points, and each grid point is a double precision number, which represents the value of potential at position.
The effect of convolutional neural network is encrypting and compressing layer by layer.
And we use recurrent layer to decrypt the features obtained from the convolutional neural network.
The main characteristic features of our model is that it is capable to capture multiple energy levels for a given potential.
In addition, we use the most common optimizer Adam of model and the output activation function of all layers is ReLU.
From our numerical experiments, it observes that we only need the fewer training data (20,000-60,000) to achieve
the required accuracy, while a larger data (200,000-400,000) for convolutional neural network.
The results show that the proposed neural network model has better accuracy of all computed energy levels compared with
the method in literature.

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