Solving Functional Renormalization Group Equations with Neural Networks

Abstract

We employ deep neural networks to represent the field derivative of the scale-dependent effective potential in the functional renormalization group (fRG) framework for nonperturbative quantum field theory. By embedding the fRG flow equations directly into the loss function, the network parameters are determined so as to provide a continuous and differentiable representation of the scale- and field-dependent effective potential without relying on precomputed training data. Focusing on the $O(N)$ scalar field theory within the local potential approximation at finite temperature, we demonstrate that this neural network representation accurately captures the renormalization group flow across symmetric, broken, and critical regimes. A key ingredient is a decomposition of the representation into an analytically known large-$N$ contribution and a learned finite-$N$ correction, which efficiently mitigates numerical stiffness associated with convexity restoration in the broken phase. The physics-driven solutions show excellent agreement with established finite-difference and discontinuous Galerkin methods. We further apply the same strategy to the Wilson--Fisher fixed-point equation in three dimensions, illustrating that neural network representations provide a unified framework for both scale-dependent flows and fixed-point problems. For fixed points, the large-field asymptotic form alone can replace the exact large-$N$ reference, while a composite small- and large-field ansatz further improves accuracy, extending the method to problems without an analytically solvable limit. Our results indicate that physics-driven deep learning offers a robust and flexible numerical tool for functional renormalization group studies.

Accepted for publication in Physical Review D. 18 pages, 12 figures