In recent years there has been a growing interest in expanding the scientific applications of machine learning. In this work we will explore two specific examples of how lattice gauge theory stands to benefit from these new developments. Lattice gauge theory is a sub-discipline of high energy physics that makes heavy use of computer simulations to test existing models and generate new results. In order to perform these simulations, spacetime is discretized as a lattice whose behavior can then be controlled using the laws of physics.
We begin by introducing a new technique that is capable of extracting information about a physical system by ‘looking’ at pictures of the system at different temperatures, and provide an analytical framework that explains how this is done behind the scenes. This result demonstrates that our approach is capable of learning non-trivial information about the system without being told explicitly how to do so, and without having to provide any information about the underlying physics.
Next, we look at how machine learning can be used to improve the efficiency of the Hamiltonian Monte Carlo (HMC) algorithm, a widely used technique in lattice gauge theory for generating gauge configurations. These gauge configurations are essentially ‘snapshots’ of the spacetime lattice, and are used to make predictions about quantum theory.
Currently, these configurations are generated via Hamiltonian Monte Carlo simulations, an algorithm that can be summarized as follows:
1. Start with a random initial configuration.
2. Propose a new configuration by (approximately) evolving the current state through time1.
3. Check if this new configuration is better than the previous one. If so, we accept it, otherwise, we retain the current configuration.2
The difference between the current and proposed configurations can be controlled through the step size, a parameter that is (typically) fixed for the duration of the simulation. We can improve the likelihood of accepting a new configuration by using a smaller step size, but this leads to configurations that are highly correlated with each other, which is undesirable. Alternatively, we can take larger steps to try and reduce correlations, but this causes more of the proposed configurations to be rejected. Immediately we see that computational resources are being wasted each time we propose a new configuration that gets rejected, and is a major source of inefficiency in the algorithm.
The approach presented in this work attempts to combat this issue by using machine learning to reduce the number of wasted calculations. Explicitly, this is done by modifying the equations that govern how our system evolves in time, and then training the algorithm to identify those modifications that ultimately produce ‘better’ configurations. In doing so, we are able to reduce the number of unnecessary calculations (which don’t produce new configurations), thereby improving the efficiency of the algorithm as a whole.
1Using Hamilton’s equations, which describe how a system changes in time.
2Technically, configurations which are ‘worse’ will still be accepted occasionally, but this becomes increasingly unlikely as the drop in ‘quality’ increases.