研究生: |
李建德 Li, Chien-De |
---|---|
論文名稱: |
人工神經網路在物理上的應用:二維正方形晶格上Potts model 相變之研究 Applications of artificial neural networks in physics : a study of the phase transitions of two dimensional Potts models on the quare lattice |
指導教授: |
江府峻
Jiang, Fu-Jiun |
學位類別: |
博士 Doctor |
系所名稱: |
物理學系 Department of Physics |
論文出版年: | 2018 |
畢業學年度: | 107 |
語文別: | 中文 |
論文頁數: | 44 |
中文關鍵詞: | 蒙地卡羅模擬 、相變 、Potts model 、人工神經網路 |
英文關鍵詞: | Monte Carlo simulations, Potts model, phase transition, artificial neural network |
DOI URL: | http://doi.org/10.6345/DIS.NTNU.DP.022.2018.B04 |
論文種類: | 學術論文 |
相關次數: | 點閱:225 下載:50 |
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這篇論文主要探討了卷積神經網路(convolutional neural network)在二維正方形晶格上的Potts model之應用。我們使用卷積神經網路對蒙地卡羅演算法模擬出的自旋狀態加以分析。不同於相關文獻中常用的方法,在本次研究中,我們使用低溫有序相中的自旋狀態作為訓練集,並以輸出向量O ⃗之長度R做為主要觀測量。藉由此方法,我們得到了和已知文獻上一致的結果。此方法減少了以人工神經網路研究凝態模型時所耗費的計算資源。使用此方式訓練出的卷積神經網路除了可以偵測臨界溫度T_c外,亦可用來辨識相變的類型為一階或二階。
This thesis mainly discusses the application of convolutional neural network to the Potts model on the two-dimensional square lattice. We use the constructed convolution neural network to analyze the spin configurations which were obtained by the Monte-Carlo simulations. Our method is different from those used in the related literature. Here, the spin configurations in the ordered phase are empolyed as the training set. In addition, the norm of the output vectors R is considered as the main observable. With this method, our determined results are consistent with the known ones in the literature. This method dramatically reduces the computational resources needed to study the condensed matter systems using the artificial neural network. Apart from detecting the critical temperature T_c, the convolution neural network built in our study can also be used to identify the nature of phase transition, namely whether they are first order or second order.
[1] Park, D. H., Kim, H. K., Choi, I. Y. and Kim, J. K. A literature review and classification of recommender systems research. Expert Systems with Applications, 39, 11 (2012), 10059-10072.
[2] Rashid, T. Make your own neural network. CreateSpace Independent Publishing Platform, 2016.
[3] Chen, N. Support vector machine in chemistry. World Scientific, 2004.
[4] Polson, N. G. and Sokolov, V. O. Deep learning for short-term traffic flow prediction. Transportation Research Part C: Emerging Technologies, 79 (2017), 1-17.
[5] Chen, S., Ferrenberg, A. M. and Landau, D. Monte Carlo simulation of phase transitions in a two-dimensional random-bond Potts model. Physical Review E, 52, 2 (1995), 1377.
[6] Ferreira, S. J. and Sokal, A. D. Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study. Journal of Statistical Physics, 96, 3 (August 01 1999), 461-530.
[7] Newman, M. and Barkema, G. Monte carlo methods in statistical physics chapter 1-4. Oxford University Press: New York, USA, 1999.
[8] Wang, J.-S., Swendsen, R. H. and Kotecký, R. Antiferromagnetic potts models. Physical review letters, 63, 2 (1989), 109.
[9] Caffarel, M. and Krauth, W. Exact diagonalization approach to correlated fermions in infinite dimensions: Mott transition and superconductivity. Physical Review Letters, 72, 10 (03/07/ 1994), 1545-1548.
[10] Sandvik, A. W. Stochastic series expansion method with operator-loop update. Physical Review B, 59, 22 (06/01/ 1999), R14157-R14160.
[11] Orús, R. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349 (2014/10/01/ 2014), 117-158.
[12] Zhang, W., Liu, J. and Wei, T.-C. Machine learning of phase transitions in the percolation and XY models. arXiv preprint arXiv:1804.02709 (2018).
[13] Carrasquilla, J. and Melko, R. G. Machine learning phases of matter. Nature Physics, 13, 5 (2017), 431.
[14] Li, C.-D., Tan, D.-R. and Jiang, F.-J. Applications of neural networks to the studies of phase transitions of two-dimensional Potts models. Annals of Physics, 391 (2018), 312-331.
[15] Van Nieuwenburg, E. P., Liu, Y.-H. and Huber, S. D. Learning phase transitions by confusion. Nature Physics, 13, 5 (2017), 435.
[16] Wetzel, S. J. Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders. Physical Review E, 96, 2 (2017), 022140.
[17] Wu, F.-Y. The potts model. Reviews of modern physics, 54, 1 (1982), 235.
[18] Gottlob, A. P. and Hasenbusch, M. TheXY model and the three-state antiferromagnetic Potts model in three dimensions: Critical properties from fluctuating boundary conditions. Journal of Statistical Physics, 77, 3-4 (1994), 919-930.
[19] Wang, J.-S., Swendsen, R. H. and Kotecký, R. Three-state antiferromagnetic Potts models: a Monte Carlo study. Physical Review B, 42, 4 (1990), 2465.
[20] Billoire, A. First order phase transitions of spin systems. arXiv preprint hep-lat/9501003 (1995).
[21] Kihara, T., Midzuno, Y. and Shizume, T. Statistics of Two-Dimensional Lattices with Many Components. Journal of the Physical Society of Japan, 9, 5 (1954/09/15 1954), 681-687.
[22] Baxter, R. J. Potts model at the critical temperature. Journal of Physics C: Solid State Physics, 6, 23 (1973), L445.
[23] Binder, K. Static and dynamic critical phenomena of the two-dimensionalq-state Potts model. Journal of Statistical Physics, 24, 1 (1981), 69-86.
[24] Rudnick, J. expansion for the free energy of the continuous three-state Potts model: evidence for a first-order transition. Journal of Physics A: Mathematical and General, 8, 7 (1975), 1125.
[25] Herrmann, H. J. Monte Carlo simulation of the three-dimensional Potts model. Zeitschrift für Physik B Condensed Matter, 35, 2 (June 01 1979), 171-175.
[26] Jensen, S. J. K., Mouritsen, O. G., Hansen, E. K. and Bak, P. Crossover from first-order to second-order phase transitions in a symmetry-breaking field: Monte Carlo, high-temperature series, and renormalization-group calculations. Physical Review B, 19, 11 (06/01/ 1979), 5886-5901.
[27] Blöte, H. W. J. and Swendsen, R. H. First-Order Phase Transitions and the Three-State Potts Model. Physical Review Letters, 43, 11 (09/10/ 1979), 799-802.
[28] Andelman, D. and Berker, A. N. q-state Potts models in d dimensions: Migdal-Kadanoff approximation. Journal of Physics A: Mathematical and General, 14, 4 (1981), L91.
[29] Salas, J. and Sokal, A. D. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Journal of Statistical Physics, 86, 3 (February 01 1997), 551-579.
[30] Georgii, H.-O. Gibbs measures and phase transitions. Walter de Gruyter, 2011.
[31] Baxter, R. J. Critical antiferromagnetic square-lattice Potts model. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 383, 1784 (1982), 43-54.
[32] Nightingale, M. P. and Schick, M. Three-state square lattice Potts antiferromagnet. Journal of Physics A: Mathematical and General, 15, 1 (1982), L39.
[33] Ono, I. Phase Transitions of Antiferromagnetic Potts Models. Progress of Theoretical Physics Supplement, 87 (1986), 102-111.
[34] Chen, S., Ferrenberg, A. M. and Landau, D. P. Monte Carlo simulation of phase transitions in a two-dimensional random-bond Potts model. Physical Review E, 52, 2 (08/01/ 1995), 1377-1386.
[35] Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F. Potts model on complex networks. The European Physical Journal B, 38, 2 (March 01 2004), 177-182.
[36] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. Equation of state calculations by fast computing machines. The journal of chemical physics, 21, 6 (1953), 1087-1092.
[37] Swendsen, R. H. and Wang, J.-S. Nonuniversal critical dynamics in Monte Carlo simulations. Physical review letters, 58, 2 (1987), 86.
[38] Luijten, E. Introduction to cluster Monte Carlo algorithms. Springer, City, 2006.
[39] Landau, D. P. and Binder, K. A guide to Monte Carlo simulations in statistical physics. Cambridge university press, 2014.
[40] Mehta, P., Bukov, M., Wang, C.-H., Day, A. G., Richardson, C., Fisher, C. K. and Schwab, D. J. A high-bias, low-variance introduction to machine learning for physicists. arXiv preprint arXiv:1803.08823 (2018).
[41] Bottou, L. Stochastic Gradient Descent Tricks. Springer Berlin Heidelberg, City, 2012.
[42] Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014).
[43] Qian, N. On the momentum term in gradient descent learning algorithms. Neural networks, 12, 1 (1999), 145-151.
[44] Duchi, J., Hazan, E. and Singer, Y. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12, Jul (2011), 2121-2159.
[45] Russell, P., Hertz, P. and McMillan, B. Biology: The Dynamic Science, Volume 1 (Units 1 & 2). Nelson Education, 2013.
[46] Chollet, F. c. c. o. a. o. Keras (2015).
[47] Peczak, P. and Landau, D. P. Monte Carlo study of finite-size effects at a weakly first-order phase transition. Physical Review B, 39, 16 (06/01/ 1989), 11932-11942.
[48] Iino, S., Morita, S., Sandvik, A. W. and Kawashima, N. Detecting signals of weakly first-order phase transitions in two-dimensional Potts models. City, 2018.