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This study introduces a supervised, scale-shared neural architecture for data-driven real-space renormalization in two-dimensional site percolation, addressing the challenge of nonlocal observables. By recursively applying a learned coarse-graining rule, the model effectively predicts crossing probabilities and reconstructs the largest-cluster mask, demonstrating its ability to extrapolate from small to large lattice systems. The results reveal that the model's performance hinges on the learned latent representation capturing critical fluctuations and scale-dependent flows, aligning with the renormalization-group structure of percolation.
Achieving high-fidelity predictions for spanning clusters in percolation, this model leverages learned critical fluctuations to extrapolate from small to large systems effectively.
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.