Far Eastern Mathematical Journal

To content of the issue

Neural Network for Prediction of Curie Temperature of Two-Dimensional Ising Model

Korol A.O., Kapitan V.Yu.

2021, issue 1, Ñ. 51–60
DOI: https://doi.org/10.47910/FEMJ202105

The authors describe a method for determining the critical point of a second-order phase transitions using a convolutional neural network based on the Ising model on a square lattice. Data for training were obtained using Metropolis algorithm for different temperatures. The neural network was trained on the data corresponding to the low-temperature phase, that is a ferromagnetic one and high-temperature phase, that is a paramagnetic one, respectively. After training, the neural network analyzed input data from the entire temperature range: from 0.1 to 5.0 (in dimensionless units) and determined (the Curie temperature T_c). The accuracy of the obtained results was estimated relative to the Onsager solution for a flat lattice of Ising spins.

Ising model, Curie temperature, Monte Carlo method, Convolutional neural network

Download the article (PDF-file)


[1] A.M. Turing, “Computing Machinery and Intelligence”, Mind, 59:236 (1950), 433–460.
[2] A.G. Makarov i dr., “K chislennomu raschetu frustratsii v modeli Izinga”, Pis'ma v Zhurnal eksperimental'noi i teoreticheskoi fiziki, 110:10 (2019), 700–705.
[3] J. Carrasquilla and R.G. Melko, “Machine learning phases of matter”, Nature Physics, 13:5 (2017), 431–434.
[4] S. Kenta, et al., “Machine-Learning Studies on Spin Models”, Scientific Reports, 10:1 (2020).
[5] K. V. Shapovalova i dr., “Metody kanonicheskogo i mul'tikanonicheskogo semplirovaniia prostranstva sostoianii vektornykh modelei”, Dal'nevostochnyi matematicheskii zhurnal, 17:1 (2017), 124–130.
[6] V.Iu. Kapitan i dr., “Termodinamicheskie svoistva sistem spinov Geizenberga na kvadratnoi reshetke s vzaimodeistviem Dzialoshinskogo–Moriia”, Dal'nevostochnyi matematicheskii zhurnal, 20:1 (2020), 63–73.
[7] I. Goodfellow, Y. Bengio, A. Courville, Deep learning, MIT press, 2016.
[8] M. Abadi, et al., “TensorFlow: Large-scale machine learning on heterogeneous systems”, 2015, TensorFlow.

To content of the issue