Renormalization-Inspired Effective Field Neural Networks for Scalable Modeling of Classical and Quantum Many-Body Systems

We introduce effective field neural networks (EFNNs), a new architecture based on continued functions-mathematical tools used in renormalization to handle divergent perturbative series. Our key insight is that neural networks can implement these continued functions directly, providing a principled approach to many-body interactions. Testing on three systems (a classical three-spin infinite- range model, a continuous classical Heisenberg spin system, and a quantum double exchange model), we find that EFNN outperforms standard deep networks, ResNet, and DenseNet. Most striking is EFNN's generalization: Trained on 10×10 lattices, it accurately predicts behavior on systems up to 40×40 with no additional training-and the accuracy improves with system size, with a computational time speedup of 10^{3} compared to ED for 40×40 lattice. This demonstrates that EFNN captures the underlying physics rather than merely fitting data, making it valuable beyond many-body problems to any field where renormalization ideas apply.