Deformations of Boltzmann Distributions
This provides a method for more efficient sampling in computational physics, but it is incremental as it builds on existing normalizing flow techniques.
The paper tackles the problem of sampling from complex Boltzmann distributions by first sampling from a simpler distribution and applying a learned transformation, demonstrating improved performance on a lattice field theory example with normalizing flows.
Consider a one-parameter family of Boltzmann distributions $p_t(x) = \tfrac{1}{Z_t}e^{-S_t(x)}$. This work studies the problem of sampling from $p_{t_0}$ by first sampling from $p_{t_1}$ and then applying a transformation $Ψ_{t_1}^{t_0}$ so that the transformed samples follow $p_{t_0}$. We derive an equation relating $Ψ$ and the corresponding family of unnormalized log-likelihoods $S_t$. The utility of this idea is demonstrated on the $φ^4$ lattice field theory by extending its defining action $S_0$ to a family of actions $S_t$ and finding a $τ$ such that normalizing flows perform better at learning the Boltzmann distribution $p_τ$ than at learning $p_0$.