Solving Functional Renormalization Group Equations with Neural Networks

TL;DR

This paper introduces a neural network approach to solve functional renormalization group equations, accurately capturing the flow of effective potentials in quantum field theory without precomputed data, and demonstrating its effectiveness across various regimes.

Contribution

The authors develop a physics-informed neural network method that directly embeds fRG flow equations into the loss function, enabling continuous, differentiable solutions without prior training data.

Findings

Accurately captures RG flow in $O(N)$ scalar theories across regimes.

Effectively mitigates numerical stiffness in broken phase.

Provides a unified neural network framework for flow and fixed-point equations.

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. Our results indicate that physics-driven deep learning offers a robust and flexible numerical tool for functional renormalization group studies.