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 $[d\phi] Z^{-1} e^{-S[\phi]}$ 中采样离散场位形 $\phi$ 的问题，其中 $S$ 为某量子场论连续欧几里得作用量 $\mathcal S$ 的格点离散化形式。由于此类密度分布源于底层泛函密度 $[\mathcal D\phi(x)] \mathcal Z^{-1} e^{-\mathcal S[\phi(x)]}$ 的近似，我们将该任务建模为算子学习实例。具体而言，我们提出近似一个依赖于时间的算子 $\mathcal V_t$，其时间积分能建立自由理论泛函分布 $[\mathcal D\phi(x)] \mathcal Z_0^{-1} e^{-\mathcal S_{0}[\phi(x)]}$ 与目标理论泛函分布 $[\mathcal D\phi(x)]\mathcal Z^{-1}e^{-\mathcal S[\phi(x)]}$ 之间的映射。当选定特定格点时，该算子 $\mathcal V_t$ 可离散化为有限维时变向量场 $V_t$，进而诱导出选定格点上有限维分布之间的连续标准化流。该流经训练后可成为离散化自由理论与目标理论之间 $[d\phi] Z_0^{-1} e^{-S_{0}[\phi]}$, $[d\phi] Z^{-1}e^{-S[\phi]}$ 的微分同胚映射。我们在 $\phi^4$ 理论上开展实验，探究此类基于算子的流架构在未经训练的格点规模上的泛化能力，并证明在较小格点上的预训练可加速仅针对目标格点规模的训练过程。