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$[d\phi] Z^{-1} e^{-S[\phi]}$中对离散场构型$\phi$进行采样的问题，其中$S$是某量子场论连续欧氏作用量$\mathcal S$的格点离散化形式。由于此类密度分布源自底层泛函密度$[\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}$的近似表示，我们将该任务构建为算子学习问题。具体而言，我们提出近似一个时间相关算子$\mathcal V_t$，其时间积分能建立自由场论泛函分布$[\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]}$与目标场论泛函分布$[\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}$之间的映射。当选定特定格点时，算子$\mathcal V_t$可离散化为有限维时间相关向量场$V_t$，该向量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。通过训练，该流可成为离散化自由场论分布$[d\phi] Z_0^{-1} e^{-S_{0}[\phi]}$与目标场论分布$[d\phi] Z^{-1}e^{-S[\phi]}$之间的微分同胚映射。我们针对$\phi^4$理论开展实验，探究此类基于算子的流架构在未经训练的格点尺寸上的泛化能力，并证明在小格点上的预训练可加速仅针对目标格点尺寸的训练过程。