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.
Chalmers University of Technology（查尔姆斯理工大学）；TU Wien（维也纳工业大学）
Lattice gauge equivariant convolutional neural networks (L-CNNs) are a framework for convolutional neural networks that can be applied to non-Abelian lattice gauge theories without violating gauge symmetry. We demonstrate how L-CNNs can be equipped with global group equivariance. This allows us to extend the formulation to be equivariant not just under translations but under global lattice symmetries such as rotations and reflections. Additionally, we provide a geometric formulation of L-CNNs and show how convolutions in L-CNNs arise as a special case of gauge equivariant neural networks on SU…
Institute for Theoretical Physics, TU Wien（维也纳工业大学理论物理研究所）
In these proceedings we present lattice gauge equivariant convolutional neural networks (L-CNNs) which are able to process data from lattice gauge theory simulations while exactly preserving gauge symmetry. We review aspects of the architecture and show how L-CNNs can represent a large class of gauge invariant and equivariant functions on the lattice. We compare the performance of L-CNNs and non-equivariant networks using a non-linear regression problem and demonstrate how gauge invariance is broken for non-equivariant models.
We review a novel neural network architecture called lattice gauge equivariant convolutional neural networks (L-CNNs), which can be applied to generic machine learning problems in lattice gauge theory while exactly preserving gauge symmetry. We discuss the concept of gauge equivariance which we use to explicitly construct a gauge equivariant convolutional layer and a bilinear layer. The performance of L-CNNs and non-equivariant CNNs is compared using seemingly simple non-linear regression tasks, where L-CNNs demonstrate generalizability and achieve a high degree of accuracy in their prediction…
TU Wien（维也纳工业大学）；Institute for Theoretical Physics（理论物理研究所）
The introduction of relevant physical information into neural network architectures has become a widely used and successful strategy for improving their performance. In lattice gauge theories, such information can be identified with gauge symmetries, which are incorporated into the network layers of our recently proposed Lattice Gauge Equivariant Convolutional Neural Networks (L-CNNs). L-CNNs can generalize better to differently sized lattices than traditional neural networks and are by construction equivariant under lattice gauge transformations. In these proceedings, we present our progress …
Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network
University of the Pacific（太平洋大学）；Institute for Theoretical Physics, TU Wien（维也纳技术大学理论物理研究所）；Institute for Theoretical Physics, University of Bern（伯尔尼大学理论物理研究所）
Massachusetts Institute of Technology（麻省理工学院）；DeepMind；New York University（纽约大学）
We develop a flow-based sampling algorithm for $SU(N)$ lattice gauge theories that is gauge-invariant by construction. Our key contribution is constructing a class of flows on an $SU(N)$ variable (or on a $U(N)$ variable by a simple alternative) that respect matrix conjugation symmetry. We apply this technique to sample distributions of single $SU(N)$ variables and to construct flow-based samplers for $SU(2)$ and $SU(3)$ lattice gauge theory in two dimensions.
We demonstrate that a state-of-the art multi-grid preconditioner can be learned efficiently by gauge-equivariant neural networks. We show that the models require minimal re-training on different gauge configurations of the same gauge ensemble and to a large extent remain efficient under modest modifications of ensemble parameters. We also demonstrate that important paradigms such as communication avoidance are straightforward to implement in this framework.
Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA（弗吉尼亚大学物理系）；Department of Computer Science, Old Dominion University, Norfolk, VA 22904, USA（老 Dominion 大学计算机科学系）
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Flatiron Institute（弗莱提荣研究所）
Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with Zd gauge group on different geometries. Focusing on the special case of Z2 gauge group on a periodically identified square la…
Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）；California Institute of Technology（加州理工学院）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
Gauge Theory plays a crucial role in many areas in science, including high energy physics, condensed matter physics and quantum information science. In quantum simulations of lattice gauge theory, an important step is to construct a wave function that obeys gauge symmetry. In this paper, we have developed gauge equivariant neural network wave function techniques for simulating continuous-variable quantum lattice gauge theories in the Hamiltonian formulation. We have applied the gauge equivariant neural network approach to find the ground state of 2+1-dimensional lattice gauge theory with U(1) …
Given a gauge theory with gauge group G, it is sometimes useful to find an equivalent formulation in terms of a non-trivial gauge subgroup H of G. This amounts to fixing the gauge partially from G down to H. We study this problem systematically, both from the algebraic and the path integral points of view. We find that the usual BRST cohomology must be replaced by an equivariant version and that the ghost Lagrangian must always include quartic ghost terms, even at tree level. Both the Cartan and Weil models for equivariant cohomology play a role and find natural interpretations within the phys…
Massachusetts Institute of Technology（麻省理工学院）；New York University（纽约大学）；DeepMind（深度思维）
We define a class of machine-learned flow-based sampling algorithms for lattice gauge theories that are gauge-invariant by construction. We demonstrate the application of this framework to U(1) gauge theory in two spacetime dimensions, and find that near critical points in parameter space the approach is orders of magnitude more efficient at sampling topological quantities than more traditional sampling procedures such as Hybrid Monte Carlo and Heat Bath.
I construct a Lattice Gauge Theory (LGT) with discrete Z_2 structure group and an equivariant BRST symmetry that is physically equivalent to the standard SU(2)-LGT. The measure of this Z_2-LGT is invariant under all the discrete symmetries of the lattice and its partition function does not vanish. The Topological Lattice Theories (TLT) that localize on the moduli spaces are explicitly constructed and their BRST symmetry is exhibited. The ghosts of the Z_2-invariant local LGT are integrated in favor of a nonlocal bosonic measure. In addition to the SU(2) link variables and the coupling g^2, thi…
Generalizable Equivariant Diffusion Models for Non-Abelian Lattice Gauge Theory
Centre for Quantum Fields and Gravity, Department of Physics, Swansea University（量子场与引力中心，物理系，萨瑟兰大学）；Institute for Theoretical Physics, TU Wien（理论物理研究所，维也纳技术大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论和数学科学中心（iTHEMS））；Institute for Physics of Intelligence, Graduate School of Science, The University of Tokyo（智能物理研究所，科学研究生院，东京大学）；State Key Laboratory of Dark Matter Physics, Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University（暗物质物理国家重点实验室，粒子天体物理与宇宙学重点实验室（MOE），粒子物理与宇宙学上海重点实验室，上海交通大学）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（南方核科学理论中心（SCNT），现代物理研究所，中国科学院）
We extend the construction of lattice chiral gauge theories based on non-perturbative gauge fixing to the non-abelian case. A key ingredient is that fermion doublers can be avoided at a novel type of critical point which is only accessible through gauge fixing, as we have shown before in the abelian case. The new ingredient allowing us to deal with the non-abelian case as well is the use of equivariant gauge fixing, which handles Gribov copies correctly, and avoids Neuberger's no-go theorem. We use this method in order to gauge fix the non-abelian group (which we will take to be SU(N)) down to…
Machine learning methods based on normalizing flows have been shown to address important challenges, such as critical slowing-down and topological freezing, in the sampling of gauge field configurations in simple lattice field theories. A critical question is whether this success will translate to studies of QCD. This Proceedings presents a status update on advances in this area. In particular, it is illustrated how recently developed algorithmic components may be combined to construct flow-based sampling algorithms for QCD in four dimensions. The prospects and challenges for future use of thi…
We propose a new method for simulating lattice gauge theories in the presence of fermions. The method combines flow-based generative models for local gauge field updates and hierarchical updates of the factorized fermion determinant. The flow-based generative models are restricted to proposing updates to gauge-fields within subdomains, thus keeping training times moderate while increasing the global volume. We apply our method performs to the 2-dimensional (2D) Schwinger model with $N_f=2$ Wilson Dirac fermions and show that no critical slowing down is observed in the sampling of topological s…
