用神经网络模拟渗流相变中的尺度变换，实现小系统训练大系统预测。
Neural Renormalization Group Flow for Percolation
- 设计共享参数的神经网络，递归执行粗粒化与细粒化操作。
- 仅在小网格上训练，即可准确预测大系统临界点处的跨连概率。
- 模型学习到的隐空间具有临界涨落特征，符合重整化群理论结构。
机器学习为数据驱动的实空间重整化提供可能，尤其当相关观测量是非局域且难以显式定义时。本文针对二维位点渗流问题，开发了一种监督式、参数共享的神经架构。该模型在不同尺度上反复应用相同的可学习粗粒化规则，生成潜在表征，并据此预测跨连概率；同时通过对应的细粒化解码器重建最大簇掩码。模型仅在小晶格上训练，却能外推至大幅增加的系统规模，以高保真度恢复跨越簇，并在临界点附近产生符合预期有限尺寸标度规律的物理量。我们发现，实现此性能的关键在于所学的隐空间表征展现出临界涨落与符合渗流重整化群结构的尺度依赖性流动。
原文摘要 · Abstract (English)
Machine learning offers a possible route to data-driven real-space renormalization when the relevant observables are nonlocal and difficult to prescribe explicitly. We explore this idea for two-dimensional site percolation developping a supervised, scale-shared neural architecture. The model recursively applies the same learned coarse-graining rule across scales, producing a latent field from which the crossing probability is predicted, while a corresponding fine-graining decoder reconstructs the largest-cluster mask. Trained only on small lattices, the model extrapolates to substantially larger systems, recovers the spanning cluster with high fidelity, and produces observables obeying the expected finite-size scaling near the critical point. We observe that to get such performance it is key that the learned latent representation exhibits critical fluctuations and scale-dependent flows consistent with the renormalization-group structure of percolation.
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