Numerical optimization using flow equations

TL;DR

This paper introduces a novel multidimensional optimization method combining flow equations, homotopy continuation, and a dynamic maximum entropy approach, applicable to complex problems in condensed matter physics.

Contribution

It presents a new optimization technique that updates priors continuously during the flow, improving upon traditional Bayesian methods.

Findings

Effective for numerical analytic continuation from imaginary to real frequencies.

Successfully finds ground states of frustrated quantum Ising models.

Demonstrates applicability to complex condensed matter problems.

Abstract

We develop a method for multidimensional optimisation using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimising functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the crucial step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of our optimisation method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground-states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.