Deep Learning for Phase Transitions

Authors

• Conte William, william.conte@studenti.unipd.it
• D'Amore Edoardo, edoardo.damore@studenti.unipd.it
• Gasparotto Giacomo, giacomo.gasparotto@studenti.unipd.it

Abstract

Statistical mechanics provides a framework for understanding macroscopic phenomena through the microscopic behavior of systems. One of its central challenges is the detection of phase transitions, which traditionally relies on analytical or numerical techniques. In this report, we investigate the potential of deep learning methods, in particular Convolutional Neural Networks (CNNs) and Variational Autoencoders (VAEs), to identify phase transitions in the two-dimensional Ising model.

We begin by generating spin configurations using MCMC methods across a temperature range that includes the analytically known critical temperature $T_c$ . Our dataset includes both labeled and unlabeled data, enabling exploration of both supervised and unsupervised approaches. [see generate_data.ipynb]

The CNN is trained to classify phases, and its softmax outputs are used as a proxy for magnetization to estimate $T_c$ via a sigmoid fit. The model achieves a classification accuracy of 99.36% on the test set and estimates $T_c = 2.2483 \pm 0.0007$, in good agreement with the theoretical value. [see cnn.ipynb]

In the unsupervised setting, we use a VAE to learn latent representations of spin configurations. We show that the latent variable $z$ strongly correlates with the average magnetization and separates the phases effectively.

Fitting a sigmoid to the rescaled latent variable yields $T_c = 2.322 \pm 0.005$, demonstrating that VAEs can infer phase transitions without explicit labels. [see vae.ipynb]

Our results confirm the feasibility of deep learning for identifying critical phenomena and highlight the strengths and limitations of supervised versus unsupervised approaches in small-system regimes.