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Physical Review Letters * Highlights * Recent * Accepted * Collections * Authors * Referees * Search * Press * About * Staff * * Open Access * Lattice Gauge Equivariant Convolutional Neural Networks Matteo Favoni, Andreas Ipp, David I. Muller, and Daniel Schuh Phys. Rev. Lett. 128, 032003 - Published 20 January 2022 [icon-twitt] [icon-faceb] [sharethis]More x * Article * References * No Citing Articles * Supplemental Material [Article ] PDFHTMLExport Citation [Abstract ] [chevron-ri] Abstract Authors Article Text ACKNOWLEDGMENTS Supplemental Material References [chevron-le] Abstract We propose lattice gauge equivariant convolutional neural networks (L-CNNs) for generic machine learning applications on lattice gauge theoretical problems. At the heart of this network structure is a novel convolutional layer that preserves gauge equivariance while forming arbitrarily shaped Wilson loops in successive bilinear layers. Together with topological information, for example, from Polyakov loops, such a network can, in principle, approximate any gauge covariant function on the lattice. We demonstrate that L-CNNs can learn and generalize gauge invariant quantities that traditional convolutional neural networks are incapable of finding. * Figure * Figure * Figure * Figure * Figure * Received 23 December 2020 * Revised 6 August 2021 * Accepted 1 December 2021 DOI:https://doi.org/10.1103/PhysRevLett.128.032003 [creativeco] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP^3. Published by the American Physical Society Physics Subject Headings (PhySH) 1. Research Areas Lattice QCDLattice field theory 1. Physical Systems Artificial neural networks 1. Techniques Gauge symmetriesMachine learning Particles & Fields Authors & Affiliations Matteo Favoni ^+, Andreas Ipp ^++, David I. Muller ^*, and Daniel Schuh ^SS * ^Institute for Theoretical Physics, TU Wien, A-1040 Wien, Austria * ^*Corresponding author. dmueller@hep.itp.tuwien.ac.at * ^+favoni@hep.itp.tuwien.ac.at * ^++ipp@hep.itp.tuwien.ac.at * ^SSschuh@hep.itp.tuwien.ac.at Click to Expand Article Text Click to Expand Supplemental Material Click to Expand References Click to Expand Issue Vol. 128, Iss. 3 -- 21 January 2022 [CROSSMARK_Color_horizo]Reuse & Permissions[prx-energy][PRX] APS and the Physical Review Editorial Office Continue to Support Researchers COVID-19 has impacted many institutions and organizations around the world, disrupting the progress of research. 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Authorization Required --------------------------------------------------------------------- [ ][ ]Log In Other Options * Buy Article >> * Find an Institution with the Article >> x Download & Share --------------------------------------------------------------------- PDFExportReuse & Permissions * * Tweet x Images * Figure 1 A possible realization of a L-CNN. Lattice data in the form of U links are first preprocessed by Plaq. and Poly. in order to generate elementary locally transforming W objects. A L-Conv is used to parallel transport nearby W objects (green dots) along the coordinate axes to a particular lattice site (red dot). A L-Bilin combines two layers by forming products of locally transforming objects, which are stored in an increasing number of channels (indicated by stacked lattices). The second input layer (blue) for this operation can be a duplicate of the original layer (red). An additional L-Act (L-Exp) can modify W (U) in a gauge equivariant way (green layer). A trace layer generates gauge invariant output that can be further processed by a traditional CNN. The example depicts a 1+1D lattice but applies to higher dimensions as well. The basic layers presented can be combined to form other deeper network architectures. Reuse & Permissions * Figure 2 Sketch of the proof that L-CNNs can generate arbitrary Wilson loops. (a) An arbitrary contractible Wilson loop (depicted here in three dimensions) surrounds a surface that can be tessellated into n tiles of 1x1 unit lattice area. The blue dot indicates the starting point of the untraced Wilson loop. (b) A Wilson loop with n tiles can be composed of an untraced Wilson loop with n-1 tiles and a path along the boundary to the missing tile using a L-Bilin. (c) An arbitrary return path to and from a 1x1 plaquette is obtained by successive applications of L-Convs after an initial Plaq. Reuse & Permissions * Figure 3 Scatter plots comparing best L-CNN models to baseline CNN models for Wilson loops of various sizes for 1+1D. For each example in the NsNt=8x8 test dataset, we plot the true value vs the model prediction. Perfect agreement is indicated by the dashed 45deg line. As the size of the traced Wilson loops grows, the performance of the baseline CNN models worsens quickly. On the other hand, L-CNN models achieve high agreement in all cases. The values in the upper left corner denote the MSEs of each plot. Reuse & Permissions * Figure 4 Prediction uncertainty in W(1x2) due to breaking of gauge symmetry for our best baseline CNN and L-CNN models on 8x8 test data. Black crosses (MC) denote the calculated true value of the Wilson loop. The red bands show the effects of random gauge transformations and transformations obtained from adversarial attacks. Predictions by the L-CNN models are invariant by construction. Reuse & Permissions * Figure 5 Predictions of our best L-CNN for the topological charge on a Wilson flowed 8x243 lattice configuration at 2/g2=0.2. 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