Machine Learning Unveils the Power Law of Finite-Volume Energy Shifts

Finite-volume extrapolation is an important step for extracting physical observables from lattice calculations. However, it is a significant challenge for the system with long-range interactions. We employ symbolic regression to regress finite-volume extrapolation formula for both short-range and long-range interactions. The regressed formula still holds the exponential form with a factor Lⁿ in front of it. The power decreases with the decreasing range of the force. When the range of the force becomes sufficiently small, the power converges to −1, recovering the short-range formula as expected. Our work represents a significant advancement in leveraging machine learning to probe uncharted territories within particle physics.