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

Figures (9)

• Figure 1: Left panel: The dimensionless sigma and pion propagators $1/(1+\bar{m}_{\sigma}^2)$ (solid lines) and $1/(1+\bar{m}_{\pi}^2)$ (dashed lines) as functions of the rescaled field $\hat{\rho}=\rho/(\Lambda T_c)$ at different RG scales $k=1000, 100, 20\,\mathrm{MeV}$. Right panel: The RG flow of the effective potential $\partial_t V(\rho)$ (dashed lines) and its field derivative $\partial_t V'(\rho)$ (solid lines) at the same RG scales. Both panels correspond to the $O(4)$ model at $T=100\,\mathrm{MeV}$ in the broken phase.
• Figure 2: Neural network architecture for learning the fRG flow. The network takes as input the RG scale parameter $t$ and the rescaled field variable $\hat{\rho}$, and outputs the finite-$N$ correction to the field derivative of the effective potential, $\Delta\hat{V}'(t,\hat{\rho}) = \hat{V}'(t,\hat{\rho}) - \hat{V}'_{\mathrm{LN}}(t,\hat{\rho})$. The architecture consists of two sub-networks, a field encoding network and an output network with concatenated inputs.
• Figure 3: Training loss convergence for the $O(4)$ model. Left panel: Loss evolution at $T=10\,\mathrm{MeV}$ in the deeply broken phase over $4 \times 10^6$ epochs, displaying characteristic staircase-like multi-stage descent followed by a fluctuating refinement phase with loss around $10^{-9}$. Right panel: Loss evolution for transfer learning at $T=100, 150, 200\,\mathrm{MeV}$ using pre-trained weights from a model at $T=120\,\mathrm{MeV}$ as initialization. The transfer learning dramatically accelerates convergence, with all temperatures reaching loss values around $10^{-9}$ to $10^{-10}$ within $\sim 5\times 10^5$ epochs.
• Figure 4: Field derivative of the rescaled effective potential $\hat{V}'_k(\hat{\rho})$ as a function of the rescaled field $\hat{\rho}$ for the $O(4)$ model at $T=100\,\mathrm{MeV}$. Left panel: Evolution at high RG scales ($k = 1000, 800, 500, 100\,\mathrm{MeV}$). Right panel: Evolution at low RG scales ($k = 100, 80, 50, 20\,\mathrm{MeV}$), where convexity restoration flattens $\hat{V}'_k$ in the small-field region. The neural network results (solid gray lines) are compared with the Local Discontinuous Galerkin (LDG) method (dotted lines) and Finite Difference BDF solver (dashed lines), showing excellent agreement across all RG scales.
• Figure 5: Field derivative of the rescaled effective potential $\hat{V}'_k(\hat{\rho})$ as a function of the rescaled field $\hat{\rho}$ for the $O(4)$ model in the deeply broken phase at $T=10\,\mathrm{MeV}$. Left panel: Evolution at high RG scales ($k = 1000, 800, 500, 100\,\mathrm{MeV}$). Right panel: Evolution at low RG scales ($k = 100, 80, 50, 20\,\mathrm{MeV}$), showing the convexity restoration in the small-field region. The neural network results (solid gray lines) are compared with the Local Discontinuous Galerkin method (dotted lines) and Finite Difference BDF solver (dashed lines), demonstrating excellent agreement across all RG scales.