Emma Zhang

Thesis Advisor: Dr. Weiqing Gu (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Daniel Tamayo (Harvey Mudd College - Department of Physics)

Contact: xiazhang@hmc.edu

Probing the Ising Model’s Thermodynamics through Restricted Boltzmann Machines

This thesis explores the connection between physics and machine learning by using Restricted Boltzmann Machines (RBMs) to study the thermodynamic properties of the Ising model. The Ising model is a simple but realistic model that captures the magnetic behavior of a system, where spins occupy a lattice of sites and different spin configurations correspond to different energies. The model exhibits phase transitions between ferromagnetic and paramagnetic phases as a function of temperature. RBMs are two-layered neural networks that can learn probability distributions over binary spins. The study generates 2D Ising model data at different temperatures using Monte Carlo simulations, including the Metropolis algorithm and the Wolff algorithm. RBMs are trained on this data and validated by studying the learned weights and filters. We then use the trained RBMs to generate new Ising configurations. The quality of the RBM-generated configurations is assessed by comparing their probability distributions to those of the original configurations using the Wasserstein distance, a measure from optimal transport theory.

Interestingly, the Wasserstein distance between the generated and original configurations shows an unexpected trend, with lower values around the critical temperature and a sharp dip at T = 2.0. This suggests that the RBM is able to capture important features of the Ising model’s thermodynamics, particularly near the phase transition. The next steps are to further investigate this finding, such as exploring the learned features in the RBM’s hidden layer and generating configurations with more hidden units. Overall, this work demonstrates a promising approach for connecting physics and machine learning to gain insights into complex systems.