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Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flows

Abstract

We consider the problem of sampling discrete field configurations ϕ\phi from the Boltzmann distribution [dϕ]Z1eS[ϕ][d\phi] Z^{-1} e^{-S[\phi]}, where SS is the lattice-discretization of the continuous Euclidean action S\mathcal S of some quantum field theory. Since such densities arise as the approximation of the underlying functional density [Dϕ(x)]Z1eS[ϕ(x)][\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}, we frame the task as an instance of operator learning. In particular, we propose to approximate a time-dependent operator Vt\mathcal V_t whose time integral provides a mapping between the functional distributions of the free theory [Dϕ(x)]Z01eS0[ϕ(x)][\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]} and of the target theory [Dϕ(x)]Z1eS[ϕ(x)][\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}. Whenever a particular lattice is chosen, the operator Vt\mathcal V_t can be discretized to a finite dimensional, time-dependent vector field VtV_t which in turn induces a continuous normalizing flow between finite dimensional distributions over the chosen lattice. This flow can then be trained to be a diffeormorphism between the discretized free and target theories [dϕ]Z01eS0[ϕ][d\phi] Z_0^{-1} e^{-S_{0}[\phi]}, [dϕ]Z1eS[ϕ][d\phi] Z^{-1}e^{-S[\phi]}. We run experiments on the ϕ4\phi^4-theory to explore to what extent such operator-based flow architectures generalize to lattice sizes they were not trained on and show that pretraining on smaller lattices can lead to speedup over training only a target lattice size.

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