Self-learning Monte Carlo with equivariant Transformer

TL;DR

This paper introduces a symmetry-equivariant attention mechanism for self-learning Monte Carlo, improving acceptance rates and scalability in simulating complex physical systems like the spin-fermion model.

Contribution

It presents a novel equivariant Transformer architecture for SLMC that systematically enhances model performance and acceptance rates in physics simulations.

Findings

Overcomes poor acceptance rates of linear models

Shows model quality improves with more layers

Scales similarly to large language models

Abstract

Machine learning and deep learning have revolutionized computational physics, particularly the simulation of complex systems. Equivariance is essential for simulating physical systems because it imposes a strong inductive bias on the probability distribution described by a machine learning model. However, imposing symmetry on the model can sometimes lead to poor acceptance rates in self-learning Monte Carlo (SLMC). Here, we introduce a symmetry equivariant attention mechanism for SLMC, which can be systematically improved. We evaluate our architecture on a spin-fermion model (\textit{i.e.}, double exchange model) on a two-dimensional lattice. Our results show that the proposed method overcomes the poor acceptance rates of linear models and exhibits a similar scaling law to large language models, with model quality monotonically increasing with the number of layers. Our work paves the way for the development of more accurate and efficient Monte Carlo algorithms with machine learning for simulating complex physical systems.