Machine learning generated configurations in presence of a conserved quantity: a cautionary tale

TL;DR

This paper examines the capabilities and limitations of machine learning models, specifically CNNs and RBMs, in analyzing the two-dimensional Ising model with conserved magnetization, highlighting the risks of misrepresenting physical constraints.

Contribution

It demonstrates that CNNs can accurately locate phase transitions and exhibit critical scaling, while RBMs fail to respect conservation laws and produce unphysical configurations.

Findings

CNNs accurately identify phase transition points.

RBMs generate configurations violating conservation laws.

RBMs produce incorrect energy and correlation distributions.

Abstract

We investigate the performance of machine learning algorithms trained exclusively with configurations obtained from importance sampling Monte Carlo simulations of the two-dimensional Ising model with conserved magnetization. For supervised machine learning, we use convolutional neural networks and find that the corresponding output not only allows to locate the phase transition point with high precision, it also displays a finite-size scaling characterized by an Ising critical exponent. For unsupervised learning, restricted Boltzmann machines (RBM) are trained to generate new configurations that are then used to compute various quantities. We find that RBM is incapable of recognizing the conserved quantity and generates configurations with magnetizations and energies forbidden in the original physical system. The RBM generated configurations result in energy density probability distributions with incorrect weights as well as in wrong spatial correlations. We show that shortcomings are also encountered when training RBM with configurations obtained from the non-conserved Ising model.