Fixed point actions from convolutional neural networks

TL;DR

The paper addresses the challenge of constructing lattice FP actions that are artifact-free and conducive to continuum extrapolations. It combines renormalization-group blocking with lattice gauge-equivariant CNNs (L-CNNs) to parametrize classically perfect FP actions, leveraging a loss that enforces both the FP action values and their gauge-invariant derivatives. The approach shows that L-CNNs achieve substantially higher accuracy than traditional FP parametrizations and exhibit strong locality, enabling efficient simulations on coarser lattices. If validated in full Monte Carlo studies, this method could mitigate critical slowing down and topological freezing, accelerating access to continuum physics in gauge theories.

Abstract

Lattice gauge-equivariant convolutional neural networks (L-CNNs) can be used to form arbitrarily shaped Wilson loops and can approximate any gauge-covariant or gauge-invariant function on the lattice. Here we use L-CNNs to describe fixed point (FP) actions which are based on renormalization group transformations. FP actions are classically perfect, i.e., they have no lattice artifacts on classical gauge-field configurations satisfying the equations of motion, and therefore possess scale invariant instanton solutions. FP actions are tree-level Symanzik-improved to all orders in the lattice spacing and can produce physical predictions with very small lattice artifacts even on coarse lattices. We find that L-CNNs are much more accurate at parametrizing the FP action compared to older approaches. They may therefore provide a way to circumvent critical slowing down and topological freezing towards the continuum limit.

Figures (5)

• Figure 1: Sketch of the procedure of taking the continuum limit: as the gauge coupling is decreased from right to left, $g < g^{'} < g^{"}$, or equivalently $\beta > \beta^{'} > \beta^{"}$ is increased, the lattice spacing decreases, $a < a^{'} < a^{"}$. In the limit $\beta \rightarrow \infty$ the lattice spacing $a\rightarrow 0$, i.e., $\xi/a \rightarrow \infty$, and the continuum limit is reached. Renormalization group transformations map the system from the left to the right side.
• Figure 2: Illustration of the RGT flow in the infinite-dimensional space of couplings. Under repeated RGTs the couplings approach the renormalized trajectory (RT), unless one starts on the critical surface in which case the couplings flow into the FP of the RGT. Figure taken from Holland:2023aaa.
• Figure 3: Illustration of a particular lattice gauge-equivariant convolutional neural network (L-CNN) architecture, cf. text and Ref. Favoni:2020reg for further details.
• Figure 4: Results for the FP action parametrized by a particular L-CNN model as described in the text. We show the invariant loss ( top left plot) and the relative error on the action values ( top right plot) evaluated on the ensembles generated with various values of $\beta$ using the Wilson action. The plots in the lower two rows show the distributions of the difference between true and predicted derivatives. For comparison, we also show the results for the Wilson gauge action and the APE444 and APE431 parametrizations of the FP action.
• Figure 5: Measure of the gauge link couplings as a function of separation in lattice units for one specific L-CNN architecture including three layers.