Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks

• URL: http://arxiv.org/abs/2502.15376v1
• Date: Fri, 21 Feb 2025 11:00:34 GMT
• Title: Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networks
• Authors: Longde Huang, Oleksandr Balabanov, Hampus Linander, Mats Granath, Daniel Persson, Jan E. Gerken,
• Abstract summary: We introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics.<n>We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators.<n>We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number.
• Score: 14.739095503241849
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.

Related papers

• A Characterization Theorem for Equivariant Networks with Point-wise Activations [13.00676132572457]
  We prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. We show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.
  arXiv  Detail & Related papers  (2024-01-17T14:30:46Z)
• Geometrical aspects of lattice gauge equivariant convolutional neural networks [0.0]
  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.
  arXiv  Detail & Related papers  (2023-03-20T20:49:08Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism [12.008737454250463]
  We propose using fusion diagrams, a technique widely employed in simulating SU($2$)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.
  arXiv  Detail & Related papers  (2022-11-14T16:06:59Z)
• Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces [52.424621227687894]
  We introduce a unified framework for group equivariant networks on homogeneous spaces. We take advantage of the sparsity of Fourier coefficients of the lifted feature fields. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation.
  arXiv  Detail & Related papers  (2022-06-16T17:59:01Z)
• Generalization capabilities of neural networks in lattice applications [0.0]
  We investigate the advantages of adopting translationally equivariant neural networks in favor of non-equivariant ones. We show that our best equivariant architectures can perform and generalize significantly better than their non-equivariant counterparts.
  arXiv  Detail & Related papers  (2021-12-23T11:48:06Z)
• Frame Averaging for Invariant and Equivariant Network Design [50.87023773850824]
  We introduce Frame Averaging (FA), a framework for adapting known (backbone) architectures to become invariant or equivariant to new symmetry types. We show that FA-based models have maximal expressive power in a broad setting. We propose a new class of universal Graph Neural Networks (GNNs), universal Euclidean motion invariant point cloud networks, and Euclidean motion invariant Message Passing (MP) GNNs.
  arXiv  Detail & Related papers  (2021-10-07T11:05:23Z)
• Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds [70.32518963244466]
  A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant.
  arXiv  Detail & Related papers  (2021-06-10T19:54:19Z)
• Geometric Deep Learning and Equivariant Neural Networks [0.9381376621526817]
  We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifold $mathcalM$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We analyze several applications of this formalism, including semantic segmentation and object detection networks.
  arXiv  Detail & Related papers  (2021-05-28T15:41:52Z)
• A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups [115.58550697886987]
  We provide a completely general algorithm for solving for the equivariant layers of matrix groups. In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before. Our approach outperforms non-equivariant baselines, with applications to particle physics and dynamical systems.
  arXiv  Detail & Related papers  (2021-04-19T17:21:54Z)
• Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs [81.12344211998635]
  A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs) We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.
  arXiv  Detail & Related papers  (2020-03-11T17:21:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.