本篇論文使用基於內嵌物理知識神經網路的方法，解決了在耦合與去耦合位能
下的薛定格方程中，基態和多個激發態的能量及其相應的波函數。模型中使用
全連接神經網絡，同時輸出多個需要的不同狀態的波函數。損失函數由每個狀
態的能量和、波函數的歸一化條件、兩兩狀態之間的正交條件以及薛定格方程
在所有網格點上的均方誤差組成。值得注意的是，邊界條件並不包含在損失函
數中，而是模型在邊界處輸出接近零的值。為了加快收斂速度，損失計算在訓
練一定數量的epoch後進行更改，從而加快損失值的收斂速度。

This thesis solves the energies and corresponding wave functions of ground and
multiple excite states of Schr¨odinger equation under the decoupled and coupled
harmonic potentials by using the method based on the Physics-Informed neural
network (PINN). The fully connected neural network (FCNN) is utilized in the
model and outputs the wave functions of multiple desired states simultaneously.
The loss function comprises the sum of energies for each state, normalization
conditions of wave functions, orthogonality conditions of pairwise states, and the
mean square error of the Schr¨odinger equation over all grid points. It is worth
noting that boundary conditions are not included in the loss function; instead, the
model outputs values close to zero at the boundaries. To speed up convergence,
the calculation for the loss is changed after training for a certain number of epochs,
resulting in faster convergence of the loss value.

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