Nonperturbative functional renormalization group for Higgs-singlet models with physics-informed neural networks

Abstract

We develop a nonperturbative functional renormalization group framework within the LPA' to solve the Wetterich flow equation for the $Z_2$-symmetric real singlet extension of the Standard Model at finite temperature, without a low-order polynomial truncation of the loop corrections to the effective potential, using a physics-informed neural network (PINN) representation. In contrast to conventional truncations based on low-order field expansions, our hybrid tree-level-plus-neural-network ansatz yields a continuous, mesh-free description of the effective potential over the full field and scale range. As a proof of concept, we apply the framework to the finite-temperature effective potential relevant to the electroweak phase transition: one-dimensional field-space slices at two benchmark temperatures, and a two-dimensional reconstruction at T=100 GeV yielding a two-step bounce action. The flow is implemented in a multi-domain setup in scale and field space with derivative matching conditions ensuring smoothness and numerical stability. Gauge and Yukawa sectors are incorporated via independently computed perturbative two-loop running couplings, and anomalous dimensions are included in the flow at the LPA' level. We benchmark the network against resummed perturbation theory and against a grid-based relaxation solver of the same equation. Finally, we introduce a soft consistency constraint that keeps the solution close, in sign and order of magnitude, to perturbation theory across the field-space domain. We find this constraint necessary, rather than merely helpful, for selecting a physically sensible solution of the flow equation. The converged result nevertheless retains a residual dependence on this guidance -- through hand-tuned weights and an analytic thermal target -- which we identify as the central open problem for mesh-free FRG treatments of this kind.