Application of deep neural networks for computing the renormalization group flow of the two-dimensional phi^4 field theory

TL;DR

The paper addresses automating real-space RG for continuum scalar field theories by introducing RGFlow, a bijective, neural-network based framework that learns coarse-graining rules from data using a RealNVP-based latent representation and a Fisher-divergence objective. It preserves information through a bijective mapping and treats discarded degrees of freedom as Gaussian noise, enabling accurate RG flows without preselected order parameters. The method is validated analytically in 1D where decimation is recovered exactly, and numerically in 2D $\phi^4$ theory where a Wilson-Fisher-like fixed point is identified and a correlation-length exponent $\nu \approx 0.885 \pm 0.015$ is estimated (within ~10% of the exact value). These results demonstrate automatic learning of RG transformations across parameter space and potential applicability to strongly coupled field theories, with avenues for improvement via architectural and training enhancements.

Abstract

We introduce RGFlow, a deep neural network-based real-space renormalization group (RG) framework tailored for continuum scalar field theories. Leveraging generative capabilities of flow-based neural networks, RGFlow autonomously learns real-space RG transformations from data without prior knowledge of the underlying model. In contrast to conventional approaches, RGFlow is bijective (information-preserving) and is optimized based on the principle of minimal mutual information. We demonstrate the method on two examples. The first one is a one-dimensional Gaussian model, where RGFlow is shown to learn the classical decimation rule. The second is the two-dimensional phi^4 theory, where the network successfully identifies a Wilson-Fisher-like critical point and provides an estimate of the correlation-length critical exponent.

Figures (5)

• Figure 1: A cartoon drawing of the Wilson-Fisher fixed point of the $\phi^4$ model in $d = 4 - \epsilon$ dimension. The arrows indicate the directions of the RG flow. For the definition of the model and its parameters $r$, $u$, see Eq. \ref{['eqn:8']}.
• Figure 2: An example of the proposed RGFlow scheme in 2D. Coarse-grained configurations $\psi$ (in orange) and irrelevant features $\xi$ (in greens) are sampled from corresponding expressions before zero-padded and reshaped into size $4\times 4$. The expanded configurations are consisted of four identical RG cells of size $2\times 2$ (The dashed black boxes). There are one coarse-grained site and three irrelevant features in one cell. $\psi$ and $\xi$ make up the latent configuration $z$. They are then treated as two different channels and inputted to the RGFlow network, which generates the predicted fine-grained configurations $\phi'$. The colors in $\phi'$ represent field strengths.
• Figure 3: The predicted RG flow diagram for the 2D $\phi^4$ model using the RGFlow algorihthm. The blue arrows are vectors $\Delta \textbf{K} = \textbf{K}_{\text{IR}}-\textbf{K}_{\text{UV}}$. The purple star pinpoints the fixed point of the saddle-point structure. The red solid line represents the critical line predicted by RGFlow. As a comparison, we graph equation \ref{['eqn:critical_small']} in solid black and present the Monte Carlo results reported in schaich_improved_2009 and De_investigations_2005 as brown and yellow dots respectively. The corresponding correlation maps at points A (pink) and B (green) are provided in Fig. \ref{['fig:maps']}. The red dashed line indicates the relevant direction of the flow. The inset figure provides a detailed view of the RG flow around the saddle point. Here, the red vector field represents the RGFlow predicted flow while the blue vector field is the result from equation \ref{['eqn:11']}. Other features remain the same as the outer graph.
• Figure 4: Correlation maps between sites in the RG unit cell and the fine-grained configuration. In panel (a), the unit cell of interest is highlighted by the dashed box. The correlation maps (\ref{['eqn:14']}) between four cell sites $\xi_1$, $\xi_2$, $\xi_3$, and $\psi_1$ and all fine-grained sites $\phi_j$ are computed at two specific locations, labeled A and B, in the flow diagram (see Fig. \ref{['fig:RG']}). The corresponding results are demonstrated in panels (b) and (c), respectively.
• Figure 5: Comparisons between the gradients computed by PyTorch and the ideal gradients obtained via finite difference (F.D.) at one training step. The left and right panels illustrate the gradient values corresponding to the parameters $r$ and $u$, respectively, within the 2D $\phi^4$ model. The gradients produced by PyTorch exhibit excellent agreement with the finite-difference estimates, and both approaches lead to convergence toward the same local minimum. This consistency is maintained throughout the entire training process.