We consider the problem of sampling discrete field configurations $\phi$ from the Boltzmann distribution $[d\phi] Z^{-1} e^{-S[\phi]}$, where $S$ is the lattice-discretization of the continuous Euclidean action $\mathcal S$ of some quantum field theory. Since such densities arise as the approximation of the underlying functional density $[\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}$, we frame the task as an instance of operator learning. In particular, we propose to approximate a time-dependent operator $\mathcal V_t$ whose time integral provides a mapping between the functional distributions of the free theory $[\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]}$ and of the target theory $[\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}$. Whenever a particular lattice is chosen, the operator $\mathcal V_t$ can be discretized to a finite dimensional, time-dependent vector field $V_t$ 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 $[d\phi] Z_0^{-1} e^{-S_{0}[\phi]}$, $[d\phi] Z^{-1}e^{-S[\phi]}$. We run experiments on the $\phi^4$-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.

翻译：我们考虑从玻尔兹曼分布 $[d\phi] Z^{-1} e^{-S[\phi]}$ 中采样离散场构型 $\phi$ 的问题，其中 $S$ 是某个量子场论连续欧几里得作用量 $\mathcal S$ 的格点离散化。由于此类密度是对底层泛函密度 $[\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}$ 的近似逼近，我们将该任务视为算子学习的一个实例。具体而言，我们提出近似一个时间依赖算子 $\mathcal V_t$，其时间积分提供了自由理论 $[\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]}$ 与目标理论 $[\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}$ 之间泛函分布的映射。当选定特定格点时，算子 $\mathcal V_t$ 可离散化为有限维时间依赖向量场 $V_t$，该场进而诱导出选定格点上有限维分布之间的连续归一化流。该流可被训练为离散化自由理论与目标理论 $[d\phi] Z_0^{-1} e^{-S_{0}[\phi]}$、$[d\phi] Z^{-1}e^{-S[\phi]}$ 之间的微分同胚。我们在 $\phi^4$ 理论框架下进行实验，探究此类基于算子的流架构在多大程度上可泛化至未经训练的格点尺寸，并表明在较小格点上的预训练相比仅训练目标格点尺寸可带来加速效果。