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.

abstract i
1 Introduction 1
2 Schrödinger equation 4
3 Related works 8
3.1 Physics-informed neural networks . . . . . . . . . . . . . . . . . . . 9
3.2 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Fully-connect Neural Network and output layer . . . . . . . . . . . 12
3.4 Loss function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Method 15
4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Loss function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Summary of proposed method . . . . . . . . . . . . . . . . . . . . . 19
5 Results 21
5.1 Example 1. 1D Simple harmonic oscillators . . . . . . . . . . . . . . 22
5.2 Example 2. 2D decoupled harmonic oscillators . . . . . . . . . . . . 25
5.3 Example 3. 2D coupled harmonic oscillators . . . . . . . . . . . . . 28
5.4 Example 4. 3D decoupled Simple harmonic oscillators . . . . . . . . 30
6 Conclusion 31
Reference 32

[1] Uhlich, S., Giron, F., Mitsufuji, Y. Deep neural network based instrument
extraction from music. In 2015 IEEE International Conference on Acoustics,
Speech and Signal Processing (ICASSP) IEEE pp. 2135-2139, (2015).
[2] Yang, W., Jiachun, Z. . Real-time face detection based on YOLO. In 2018
1st IEEE international conference on knowledge innovation and invention
(ICKII) IEEE pp. 221-224, (2018).
[3] Chen, T. E., Yang, S. I., Ho, L. T., Tsai, K. H., Chen, Y. H., Chang, Y. F.,
Wu, C. C..... S1 and S2 heart sound recognition using deep neural networks.
IEEE Transactions on Biomedical Engineering, 64(2) pp. 372-380. (2016).
[4] Li,H., Zhai,Q. Chen,J.Z. Neural-network-based multistate solver for a static
Schr¨odinger equation. Phys. Rev. A 103, (2021) 032405.
[5] Kapetanovi´c AL, Poljak D. Numerical Solution of the Schrodinger Equation
Using a Neural Network Approach. IEEE pp.1-5, (2020).
[6] Raissi.M, Perdikaris.P, and Karniadakis.G.E. Physics-informed neural networks:
A deep learning framework for solving forward and inverse problems
involving nonlinear partial differential equations Journal of Computational
physics. p.686-707, (2019) 378.
[7] Foulkes, W. M. C., Mitas, L., Needs, R. J., Rajagopal, G. Quantum Monte
Carlo simulations of solids. Reviews of Modern Physics, 73(1), (2001) 33.
[8] Ceperley, D. Ground state of the fermion one-component plasma: A Monte
Carlo study in two and three dimensions. Physical Review B, 18(7), (1978)
3126.
[9] Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D., Verstraete, F.
Quantum metropolis sampling. Nature, 471(7336) pp. 87-90, (2011).
[10] Odaka, K., Kishi, T., Kamefuchi, S. On quantization of simple harmonic
oscillators. Journal of Physics A: Mathematical and General, 24(11), (1991)
L591.
[11] Jin, H., Mattheakis, M., Protopapas, P. Physics-informed neural networks for
quantum eigenvalue problems. In International Joint Conference on Neural
Networks (IJCNN) IEEE pp. 1-8, (2022).
[12] Razakh, T. M., Wang, B., Jackson, S., Kalia, R. K., Nakano, A., Nomura, K.
I., Vashishta, P. PND: Physics-informed neural-network software for molecular
dynamics applications. SoftwareX, 15, (2021) 100789.
[13] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., Yang,
L. Physics-informed machine learning. Nature Reviews Physics, 3(6) pp. 422-
440, (2021).
[14] J. Han, A. Jentzen, and E. Weinan, Solving high-dimensional partial differential
equations using deep learning, Proceedings of the National Academy of
Sciences of the United States of America, vol. 115 34 pp. 8505–8510, (2017).
[15] Wang, Z., Bovik, A. C. Mean squared error: Love it or leave it? A new look at
signal fidelity measures. IEEE signal processing magazine, 26(1) pp. 98-117.
(2009).
[16] Karlik, B., Olgac, A. V. Performance analysis of various activation functions
in generalized MLP architectures of neural networks. International Journal of
Artificial Intelligence and Expert Systems, 1(4) pp. 111-122. (2011).
[17] Reddi, S. J., Kale, S., Kumar, S. On the convergence of adam and beyond.
arXiv preprint arXiv:1904. (2019) 09237.
[18] Moritz, P., Nishihara, R., Jordan, M. A Linearly-Convergent Stochastic LBFGS
Algorithm. PMLR. Artificial Intelligence and Statistics pp. 249-258,
(2016).
[19] Kumar, S. K. On weight initialization in deep neural networks. arXiv preprint
arXiv:1704. (2017) 08863.
[20] Messiah, A. Quantum mechanics. Courier Corporation. (2014).
[21] Berezin, F. A., Shubin, M. The Schr¨odinger Equation (Vol. 66). Springer
Science Business Media. (2012)
[22] Smith, Gordon D., Gordon D. Smith, and Gordon Dennis Smith Smith. Numerical
solution of partial differential equations: finite difference methods.
Oxford university press, (1985).
[23] Diedrich, F., Bergquist, J. C., Itano, W. M., Wineland, D. J. Laser cooling
to the zero-point energy of motion. Physical review letters 62.4, (1989) 403.
[24] Hu, B., Li, B., Liu, J., Gu, Y. Quantum chaos of a kicked particle in an
infinite potential well. Physical review letters, 82(21), (1999) 4224.
[25] Howe, R. Quantum mechanics and partial differential equations. Journal of
Functional Analysis, 38(2) pp. 188-254. (1980).
[26] Teng, P. Machine-learning quantum mechanics: Solving quantum mechanics
problems using radial basis function networks. Physical Review E, 98(3),
(2018) 033305.
[27] https://pytorch.org/