Learning Chern Numbers of Multiband Topological Insulators with Gauge Equivariant Neural Networks
Authors: Longde Huang
Oleksandr Balabanov
Hampus Linander
Mats Granath
Daniel Persson
Jan E Gerken
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Deep Learning
Gauge Equivariant Networks
Condensed-Matter Physics
Primary area: machine_learning_for_sciences

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. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators for the first time. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. A major technical challenge is that the relevant gauge equivariant networks are plagued by instabilities in their training, severely limiting their usefulness. In particular, for larger gauge groups the instabilities make training impossible. We resolve this problem by introducing a novel gauge equivariant normalization layer which stabilizes the training. Furthermore, we prove a universal approximation theorem for our model. We train on samples with trivial Chern number only but show that our model generalizes to samples with non-trivial Chern number and provide various ablations of our setup.

Learning Chern Numbers of Multiband Topological Insulators with Gauge Equivariant Neural Networks
Longde Huang
Oleksandr Balabanov Department of Physics Stockholm University
Department of Mathematical Sciences
Chalmers University of Technology
University of Gothenburg
longde@chalmers,
Hampus Linander VERSES AI Los Angeles, CA, USA hampus linander@verses. ai
Mats Granath
Department of Physics
University of Gothenburg
mats. gra.athpphysics...coooo ge
Daniel Persson
Jan E. Gerken
Department of Mathematical Sciences
Department of Mathematical Sciences
Chalmers University of Technology
Chalmers University of Technology
University of Gothenburg daniel, www.sproooosaaaessssccooo
University of Gothenburg
gorken@chalmers.ss
Abstract
Equivariant network architectures are a well-established tool for predicting invariant equivariant quantities. However, almost all learning problems considered in this context feature global symmetry, i.e. each point of the underlying space is transformed with the same group element, opposed where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. We use gauge equivariant networks predict topological invariants (Chern numbers) of multiband topological insulators for the first time. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. A major technical challenge is that the relevant gauge equivariant networks are plagued by instabilities their training, severely limiting their usefulness. In particular, for larger gauge groups the instabilities make training impossible. We resolve this problem by introducing novel gauge equivariant normalization layer which stabilizes the training. Furthermore, we prove a universal approximation theorem for our model. We train on samples with trivial Chern number only but show that our model generalizes to samples with non-trive con-trivial Chern number and provide various ablations of our setup.
local gauge symmetry,
Introduction
Geometric deep learning is subfield of machine learning that takes advantage of the geometric and topological structures inherent in complex data to construct more efficient neural network architectures [ This approach has been successfully applied in variety of domains, from medical imaging [2) to high-energy physics (3) and quantum chemistry (4). As we show in this paper, this perspective is particularly valuable for studying topological insulators, class of materials which has
been one of the main areas of interest in condensed matter physics over the last two decades [5] 6
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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with broad range of applications, including spintronics and magnetoelectronics quantum devices (9. and quantum computing (1O) T2].
7 photonics
8
The mathematical field of topology studies objects which cannot be deformed continuously into each other. For instance, doughnut is topologically equivalent to coffee cup (the hole in the doughnut becoming the hole in the handle) but not to ball (which does not have a hole). Topological insulators are materials whose interior is insulating yet whose surface or boundary is conducting, with a prime example being materials characterized by a Chern number. The Chern number is an integer topological quantity that originates from the fact that the phase of quantum mechanical save-function is not a physically measurable quantity, thus corresponding to gauge symmetry. The fact that material with finite Chern number cannot be continuously deformed into a material with Chern number O without closing the energy gap (which make the materials insulating) implies that there has to be a zero energy (conducting) state at the boundary.
So far, deep learning models have only been able to predict Chern numbers for materials with the abelian gauge group U(1), corresponding to filled band [13 14). However, these models fail to learn Chern numbers for higher-rank gauge groups U(N), pointing to a fundamental challenge in the multi-band regime N >
We identify the gauge symmetry of the system as the central reason for the failure of traditional approaches in the high-band setting and instead propose to use gauge equivariant network for learning topological invariants such as the Chern number. Consider a neural network W X - Y which maps set of inputs x. (I11 -.. - ... E X (one for each site in a lattice) into a set of outputs y (N .n) (y).... vn) yT E Y. Usual group equivariant networks N satisfy the constraint N N(xx (9)=1..... Px (x)
Agg E G with symmetry group G and representations px,y on the imput- and output spaces, respectively. In contrast, a network is gauge equivariant if it is equivariant under the action of different elements of the symmetry group G on the different inputs, i.e. ((xx (..)
(py (g)v1..... :.).......P..................... PY (g)
........PPPee px (x)))=))) = (py (g) )u,,..., Py (9n)).."
In a discretized input domain X in d dimensions with p points per dimension, a gauge symmetry effectively means that the total symmetry group of the problem consists of pd copies of G. This exponential enlargement of the symmetry group explains why non-equivariant networks are often unable learn tasks which feature a gauge symmetry. We therefore claim that gauge-equivariant networks are necessary to learn high-band Chern numbers of topological insulators. This is novel application for gauge equivariant neural networks, which thus far have mainly found applications within the realm of lattice gauge theories for quantum chromodynamics. In particular, we cast the problem at hand in a form in which we can use an adapted version of the Lattice Gauge Equivariant Convolutional Neural Networks (L.GE-CNNs)
15 to learn multiband Chern numbers.
However, LGE-CNNN have been plagued with instability problems. These networks feature bilinear layers which frequently lead to exploding or vanishing gradients, making the training process unstable. With growing depth or for larger gauge groups, this problem becomes more severe, limiting the applicability of LGE-CNNs. We provide a solution by introducing a new gauge equivariant normalization layer that stabilizes the training and leads to consistent convergence of deep networks.
Our main contributions are as follows:
We resolve the instability issues of LGE-CNNs by introducing a new gauge equivariant normalization layer. By training our purely local network with a novel combinations of loss functions, we obtain a model which generalizes from trivial to non-trivial Chern numbers and to unseen lattice sizes.
We prove universal approximation theorem for our architecture in this context which shows that our model can approximate all U(N) gauge invariants arbitrarily well. We perform ablations over different gauge equivariant architectures motivated by this theoretical investigation.
We provide novel application of U(N) J-gaug equivariant neural networks to the task of learning higher-band Chern numbers. The resulting model can predict Chern numbers in two dimensions for systems with at least N = 7 filled bands. In contrast, previous models could only handle the trivial case of a U11)-symmetry.
Our model also the first to be able to learn higher-dimensional (D = 4) Chern numbers of multi-band topological insulators.
2

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2 Literature Review
Equivariance under global matrix groups have been considered [16] and for the orthogonal groups in [17]. Gauge equivariant networks have been considered in two different settings. In the first setting, the gauge symmetry concerns local coordinate changes in the domain of the feature maps [18]
first studied theoretically in (20) and models respecting this symmetry were introduced in [1 22]. Applications of gauge equivariant networks to lattice quantum chromodynamies (QCD) fall into the second setting, where the gauge transformations act on the co-domain of the feature maps. An important problem in lattice QCD is sampling configurations from the lattice action, a problem well gauge equivariant neural-network quantum states (28) have been used. In contrast, our model is based on a gauge
This
for which gauge equivariant normalizing flows [23 24 25
26
27
cquivariant prediction model developed for lattice QCD [15].
Machine learning for quantum physics has seen an explosive development over the last decade with applications in condensed matter physics, materials science, quantum information and quantum computing to name a few (29) 30, 31 32]. In this brief overview we focus on deep learning and applications to topological states of matter. Early breuki-breaking work in this area includes [33) that used supervised learning on small convolutional neural networks for identifying the ground state of the Ising lattice gauge theory, well (34) that developed unsupervised method "learning by confusion" to study phase transitions including topological order of the Kitaev chain. Of particular relevance our work are the papers (35 [3] that used convolutional neural networks and supervised learning to predict U(1) topological invariants. This work was later extended to an unsupervised setting in [14 36) by incorporating the scheme of learning by confusion and augmenting data using ggollog-prresrvingg deformations.
3 Learning Multiband Chern Numbers
In this section, we will introduce the learning task of predicting multiband Chern numbers. We will first outline the features, targets, and relevant symmetries and then demonstrate that two naive approaches of learning in this context fail. The discussion of how the learning task presented here is related to the physics of multiband topological insulators relegated to Appendix A
3.1 Features, targets and symmetries
For most of our experiments, we consider features on a two-dimensional rectangular lattice A with periodic boundary conditions with total of N₂ x Ny Nate grid points. At each grid point (), we have features which are complex unitary x matrices W.,; E U(N), the so-called Wilson loops. These will be the inputs for most of our models. The Wilson loops can be written in terms of Hermitian link matrices U.g U. connected to the grid point (i, The Wilson loops are given by product of links along a 1 x 1 loop of edges,
1. CNNN which are attached to the edges ø- and y directions
W., =
(1)
The learning target is the Chern number Ö defined by
C(WWes (1,5) E
Σ Im Tr log Was- (K.))E
(2)
It can be shown that E defined in this way is an integer (37] The learning task we want to study in this
article is to predict the Chern number C' Z given the Wilson loops (Wi, (i. E A}. Although the Chern number is given by the innocuous-looking equation 2 learning it is not straightforward
as demonstrated in the next subsections. We will show that the main reason for the difficulty of learning 2 lies in the gauge invariance of Ć.
Due to the gauge invariance in the mathematical description of topological insulators, the Chern number Ö invariant under transformations Wi₃ 52 WiyS of the Wilson loops, where crucially the transformation matrices Ω4, U(N) depend on the lattice site. The fact that the symmetry holds even if the Wilson loops at different lattice sites transformed with different
3

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group elements of U(N) makes this into a gauge symmetry which is exponentially large in N In summary, C satisfies
Č( ((Wis)( E E)) = C ((1)) Way (4.5) E A)) A (Ω., E U(N)}
(3)
In the following two sections, will demonstrate that two naive approaches to learn 2 both fail, motivating the use of LGE-CNNs which are by construction equivariant with respect to local gauge transformations.
3.2 Learning Chern numbers using ResNets
Using the fact that log( log (X), follows that the Cher number defined in (2) can be written as a sum over Im log det(W), As warmup to predicting Chern numbers, we start by considering the simpler problem predicting determinants of N N real matrices A.
Table I: Relative error 8 for linear MLP and bilinear residual architectures pre- dicting the determinant of real 4 ma- trices with uniform random elements in [0, 1]. Standard MLP architecture fails to learn the determinant relation.
10-3
10a3
Matrix
Architecture
Layers
8
Figure 1: Best relative error of predicted matrix determinants for polynomial architectures as function of increasing matrix size. The ablations include layers up to order 4. Dashed line indicates relative error of mean predictor. Architectures considered include layers of order < 4, and depth < 4, containing terms of up order 16 by composition.
MLP
2
1.02
3
1.02
Bilinear
2
0.01
3
0.01
We construct a dataset containing N x N matrices with elements sampled from a uniform distribution on the unit interval [0, 1]. As baseline, we naive multilayer perceptron (MLP) with residual connections f N TR, taking the matrix elements as input and predicting the determinant value.
The determinant of an N x N matrix can be expressed as an order N polynomial in the matrix elements. Inspired by this, we consider higher order layers with structure
A
Σ
N2...2R Ain
Akn-1kann
kE....keR
where R the number of factors of A in the layer, and of....aan are learnable parameters. An architec- ture containing layers with R = 2 will be referred to as bilinear. As the number of parameters grows layers of order R < 4 and construct higher order terms by composition
quickly with order R, of multiple layers and residual connections. Predicted determinants f(A) are evaluated against the target determinant value det(A) (A) using absolute relative error 5 = (f( det(A)) [det(A)).
Table shows the failure of a residual MLP architecture to learn the determinant matrices with uniformly distributed elements on [0, 1], whereas bilinear architecture with layers of order 2 can achieve low relative error. Even though these higher order layers provide an architecture that is expressive enough for determinants in small dimensions, they quickly run into issues for larger matrices. Figure shows that for matrix sizes corresponding to band size > 4, learning the determinant becomes prohibitively hard without better model priors. See Appendix Foror more details on the architecture ablation.
x real
3.3 DeepSpec: Baseline Equivariant Model
The gauge symmetry of the system implies the equivalence relation
(5) on the set of Wilson loops at each grid point. According to the spectral theorem, there is exactly one diagonal matrix in each equivalence class (group orbit) with elements in U(1), up to reordering
W SQ? JΩ U(N)

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Sum
Iaput: W
GEBL
GEAct
TrNorm
CEBL
GEAs
TrNorm
Trace
Dense
Figure 2: Architecture of GEBLNet. In this figure, the rectangles represent the spatial grid, and the number of layers (Nect) represents the number of channels (γ). Each circle represents site the grid, and quantities on different sites do not interact with each other, until the last summation on grids.
of the diagonal elements. In other words, the set of equivalence classes for this relation is given by U(1) /sy- This fact has been used in the construction of gauge equivariant spectral flows (24) Based on this, we construct baseline invariant model DeepSpec.
DeepSpec preprocesses the flux data with a diagonalization, and then feeds the eigenvalues into DeepSet-like model, which is invariant under permutation of the input data, calculating locally the output as ((1)) o (E, ((x))) where p. & are arbitrary neural networks, set MLPs.
We train DeepSpec to predict Chern numbers on x 5 grids while gradually increasing the number of bands. The data generation scheme is consistent with other experiments, formally introduced in Section 6 The model successfully learns Chern numbers for the trivial case (N = 1), but fails as the number of bands increases and the system complicates. Specifically, the accuracy evaluation drops drastically to 44.7% for N = 2, and 7.8% for N = 4.
Rather than enforcing group equivariance layer-by-layer, DeepSpec predicts gauge- invariint quantities by restricting its input to invariant features only. We attribute its failure to this structural characteristic and thus aim to develop models that manifestly respect gauge equivariance.
4 Network Architecture
As demonstrated in the previous section, even hand-crafted polynomial architectures fail to learn simplified version of the Chern number. Furthermore, naive equivariant models that operate solely on invariant features, such as DeepSpec, also struggle to effectively learn the target quantity. Motivated by the performance of our gauge equivariant networks, we propose that this due the size of the gauge symmetry present in this problem. Since the Wilson loops each site can be transformed independently, the total symmetry group is exponentially larger than the symmetry groups of more traditional group equivariant networks.
The input data in our network is the set of discretized Wilson loops W E U(N) and all equivariant layers in our setup operator on tensors of this form. Here, we use the shorthand & = (i.j) for the lattice point. The index 7 counts the number of different orientations of the Wilson loops per site (in 2D, there is only one) for the input and serves as a general channel index in deeper layers. Hence, our layers operate on complex tensors of the shape Noh Nates x N N.
4.1 Gauge equivariant layers
Our model is composed of the following equivariant layers, which were introduced in 15 as well as our new gauge equivariant normalization layer.
GEBL (Gauge Equivariant Bilinear Layers) Given input tensor W the layer computes local quantity per site as Why = E,,, ('y where W has Nour channels and Otyyu
Na..NN. are trainable parameters. This the primary feature mixing layer, since not only extends the feature space, and also captures higher order terms of the original flux data W, which is demonstrated be crucial for the model's 's expressivity Section can easily be checked that this layer is equivariant. In practice, GEBL includes also linear and bias term which obtained by
5

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enlarging W with its Hermitian conjugate and the identity matrix. In order to merge two branches of the network, two different W can also be used on the right-hand side.
GEAct (Gauge Equivariant Activation Layers) Given a tensor W the layer maintains a channel
size of Ni and serves as equivariant nonlinearity defined by W = a (TrW3) W where a is usual activation function. Section 5 we prove a universal approximation theorem for a certain type of a. In practice, we use (2) = (Rez) to avoid gradient vanishing; hence, we also refer to this layer as GEReLU.
GECony (Gauge Equivariant Convolution Layers) Given a tuple (U WQ), the layer performs convolution as
W ΣΣ p.d
Wudra (U
W Wk+da ktdp) k+da
(6)
where U = U U
()j and då are the length-d vectors in the μ direction in the lattice. Mathematically, this layer is capable of simulating the integral of the flux over any closed area, therefore, it is the only layer in our setup that uses link variables Ug and introduces interactions between neighboring points.
The output W of this layer has the shape Nowr x Nate N x N and trainable weights of shape Ndi x x Nout x Nn- Note that this layer does not update the links. Using the transformations (15) of the link variables and the transformation of the Wilson loops (3). one verify that this layer is equivariant as well. When we take a zero convolution kernel size, the layer degenerates into an equivariant linear layer that is completely local.
Trace Layer Given a tensor W this layer maintains the channel size and takes the trace of the fluxes T TrW Since the trace is invariant under the transformations 3), this layer renders the features gauge invariant. The output has the shape Nin x Naie-
Dense Layer After the trace layer, we perform a real valued linear layer on Ta as T tike Re(T))) + Wim Im(Ts) + for the final prediction, where WRe. Wim and are trainable parameters. The output features have shape Nout Note- Note that we only transform gauge invariant features with this layer since it does not respect the gauge symmetry.
TrNorm (Trace Normalization Layers) The bilinear GEBL-layers introduced above quickly lead training instabilities when stacked deeply, as demonstrated in Section 6 below. To solve this problem, we introduce novel gauge-equivariant normalization layer which we insert after the nonlinearities. Given an input tensor W7, this layer maintains the channel size and performs channel-wise normalization as
/ mean, (Tr W
This operation gauge equivariant since the prefactor gauge invariant. After the normalization, the output features W' satisfy mean,,
('rr,""Tr'"" [TrW""} = et for some dh E IR.
4.2 Network Architecture
We now use the layers introduced in the previous section to construct three different equivariant network architectures.
GEBLNet (Gauge Equivariant Bilinear Network) GEBLNet is model that only operates locally, iee. the inputs are the fluxes W alone, and features at different sites only interact with each other in the final sum. processes the fluxes through repeated blocks of GEBL, GEAct, and TrNorm layers. The outputs are then aggregated through a Trace layer, followed by dense and summation over sites to produce the prediction on the Chern number. This is our primary model to study. An example structure is shown in Figure 2

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GEConvNet (Gauge Equivariant Convolutional Network) GEConvNet is model that features GEConv layers, therefore takes both the links (U*) and the fluxes (W7) as input. Each GEBL- GEAct-TrNorm block, similar to that of GEBL.Net is paralleled
block, with their outputs combined through subsequent GEBL layer. This process is repeated through several iterations, after which the resulting output is passed to the Trace- Dense- and sum sequence to produce the final prediction. An example structure is shown Figure in Appendix
5 Theoretical Foundations of the Model
In this section will present universal approximation theorem for models. We focus GEBLNet, whose inputs are solely WA. Recalling the equivalence relation (3), the local quantity is in fact a class function on the gauge group (n). which is defined as follows:
- C such that Definition 5.1. (complex) class function on Lie group is a function f(g fh), for all h G.
We denote the closed subspace formed with square integrable class functions as Li (G).
We now present main theoretical result, which shows our model's capability learn arbitrary class function.
Theorem 5.2 (Universal Approximation Theorem). For a compact Lie group G, and with the nonlinearity in GEAct laking the form could approximate any class function on G.
Re, where is bounded and non-decreusing, GEBLNet
The full proof is presented Section D in the Appendix. Here we present a brief sketch of it.
Sketch of Proof. Consider network with M GEBL layers and sufficient width. One can show that the output such a network can approximate any function the form
2M
W:o Re
Ois Try
oayTry Trg') + b.
1=0
Trg?"*. Therefore 2 }} LG) By showing that the space Unrlf(Trg,... " }} n (G) is dense in L (G), we conclude that (8) can approximate any function in L2tase (G).
However, this is equivalent to one-layer MLP on the polynomials "rrg...,TTT"",, can approximate any function in {f(Trg..., Trg=*
The argument above be generalized to the following theorem in higher dimensions if restrict the group G to be the unitary group, and we discuss this further in Section Din in the Appendix.
Theorem 5.3
UAT for GEBLNET). Denote the unitary group as G. Under the same assumptions on the nonlinearity, GEBLNET can approximate arbitrarily well any square- integrable function f (G* such that
flg..... 9K) = f(ighh "en.h hgghh Vhe G.
(9)
Experiments
To assess the performance of our gauge-oquivariant neural network, we conducted extensive numerical
38 39,. we train on synthetically generated data. In particular, we train on uniformly distributed link variables and on link variables whose distribution was adjusted for a specific distribution of Chern numbers. These allow us to test our model on input configurations spanning wide range of Chern numbers with many filled bands, going beyond the Hamiltonian models common in the literature. Our datasets have varying grid sizes and distributions and are based on 2D grids, unless otherwise stated. Data samples have the form (UÉ, U.V. Wx)k-
experiments. As is standard in the literature in this domain [13] 34
35|
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6.1 Data Generation
General Dataset The data generation pipeline consists of two main steps. Given a grid size
N₂ x Ny. we first generate random link variables using the following algorithm:
1. For μ y, draw AK ~ (O 1)NxNN
2. Perform QR dccomposition on Af, decomposing it into the product of a unitary matrix
UK and semi-definite matrix 2g A = U/22%
3. Use Uh as the link variable for site l: in direction μr.
It can be shown that the distribution of the links generated in this way is uniform on U(N), i.e. the random variables U and gt are identically distributed for all g E U(N). See Appendix C for details. In the next step, we compute the fluxes W' using ( and the discrete Chern number C using (2) Ultimately, this yields dataset of data-value pairs ((U W2), ; C). We generate the training samples continuously during training to avoid overfitting.
Diagonal Dataset For some of our experiments, require control over the distribution of the Chern numbers in our training data. To this end, we employ a different data generation strategy. Due to the invariance of our model under gauge transformations, training samples which lie in the same gauge orbit are equivalent in the sense that the parameter updates they induce are the same. This implies that Section 3.3, there is always a diagonal matrix with U(1 >-valued components in the orbit. Therefore, by training these matrices in U(1) and manipulating the distribution of the diagonal values, we can generate datasets with different distributions of Chern numbers. Note that networks trained on these datasets of course generalize non-diagonal Wilson loops since they are gauge invariant. Detailed discussions are provided in Appendix C
select arbitrary element along the gauge orbit for training. As discussed in
6.2 Training and Evaluation
We adopt two loss functions for our training: the global loss Lg and the standard deviation loss Lytd- Specifically, given the network f(W) and the Chern number Ć, Lyg calculates [l ČlI Meanwhile, Ledd evaluates the entrywise standard deviation of the network output after the dense layer, denoted as (W()))), namely max( (WWa))))) set to 0.5 by default. The standard deviation loss is necessary to prevent the model from collapsing to only zero outputs, since it forces the model to output locally different quantities. This is particularly relevant for the training on trivial topologies only, as described below. The total loss function Lictal adds these two terms, Lioual Lg + Lgtd-
6|] where 8 is hyperparameter,
For evaluation, we compute the accuracy by rounding the network output f(W) to the nearest integer and comparing it with the Chern number Č, unless otherwise stated. For the main GEBLNet, we set representative model configuration, whose GEBL layers and hyperparameters are listed in Table in Appendix L
6.3 Experimental Results
Model Comparison For a basic model comparison, we train GEBLNet, GEConvNet, and TrMLP on 2D grids to learn Chern numbers by training on non-diagonal data. We find that GEBLNet outperforms the other models in both accuracy and robustness, see Figure in the Appendix.
Using a benchmark grid size of 5 x 5 and N = 4 filled bands, GEBLNet achieved approximately 95% accuracy across different seeds, demonstrating strong robustness. In contrast, GEConvNet struggled to learn correct results with positive kernel sizes, likely due to redundant information. We also tested a degenerate GEConvNet with kernel size (a local network) which performed better than its non-local counterpart, but it remained less robust than GEBLNet. A complexity accurrcy comparison in Figure Appendix Appendix([demonstrall [emoosttratss the balance achieved by the representative model, which performed consistently well while maintaining efficiency. Additionally, tests increasing band sizes (Table 2 show that the representative model effectively learns Chern numbers up to bands, retaining high accuracy and robustness.
8

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Table Accuracy of GEBLNet trained and evaluated on a 5 grid.
Bands
8
Accuracy 95.9%
94.0%
93.8%
91.7%
52.5%
0.4
Global Loss
Global Loss
0.498
Sid Loos
0.05
Global Los fg
0.2
Global Loss by
00.2
Sundend Desiation
0.496
Stonkol Devistion l.o
0.00
0.0
-0.0
0%
Ik
2k
0k
Ik
2k
3k
Epocts
Epochs
Figure 3: Comparison of global and standard deviation loss on validation data between the same two runs learning only zero Chern numbers, shown in Figure The former, without TrNorm layers, collapses to zero local quantities everywhere, hence having a lower Lg on trivial samples, yet could generalize to nontrivial cases. In contrast, the latter, with TrNorm layers, succeeds in learning global quantities and local differences simultaneously.
Training on Trivial Topologies In order to test the generalization properties of our model, we train exclusively on topologically trivial samples. To achieve this, non-trivial samples were manually filtered out during data generation, turning lgg effectively into [[(W))
100.0%
Mon Volue [T-)] Motel withon TWNer Valienee of (T Moddl witou T/Nbnn Layes Value [Ts Model T.Nai
10
10"
Accurncy
95.0%
-90-
Varance of (leng Model with tchorn Layer
Volue
10"
10a
90.0%
GEBLNN Tramed 5
Trained
10
GEBLNa Traned on 10' Gold
10
100
200
300
400
500
Grid for evaluation
10
GBBL
GBAst
TaNem
GEBL
GEAd
TiNorm
GRBL
GEAst
TiNasm
Trac
Datse
Figure 5: Comparison of model accuracy across different grid sizes. Each run, represented by markers of the same color, has identical configu- rations, but is trained on grids of a different size. Each line represents linear regression the corresponding run.
Layes
Figure 4: Comparison of statistics of [Tww]]) across each layer, between two training runs on a 5 x 5 grid, with filled bands, with without TrNorm Layers.
naive GEBLNet model without TrNorm layers can learn the Chern number for up bands but fails for 4 bands and above, mostly outputting zero local quantities except for a few random seeds. Analyzing the statisties of output traces (Figure reveals that variance accumulates across layers, reaching 10 after the final GEAct layer, causing numerical instability and vanishing gradients.
Introducing TrNorm layers mitigates this issue, stabilizing the variance and enabling the model to learn Chern number of systems with more than 4 bands. Figure compares the loss curves between models with and without TrNorm layers, demonstrating the effectiveness of this modification.
Furthermore, training solely on trivial samples limits the model to learning the Chern number up to a global, sample independent, rescaling factor, i.c. f(W) ss &C for some k. To evaluate on general samples, we compute the rescaling factor Rwale = mean( (C) CC)/mean(f(())) set of non-trivial samples and scale the output as R The training results demonstrated prediction accuracy comparable to that of training on general datasets, as detailed in Table 3 in the Appendix, obtaín accuracy of approximately 94.1% four bands, comparable 95.9% for training on data which includes non-trivial Chern numbers. Meanwhile, Figure 6a shows the model's ability capture local quantities accurately.
using a large
Larger grids Due its local structure, GEBLNet can process samples of arbitrary grid sizes. We evaluate its generalization abilities by testing samples larger grids using representative

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Rescaled Local Quanityy
0.2
Local Quanityy Sid
2
0.1
0.023
Ourpuis
Outputs
-2
-2.5
0.0
2.5
-0.2
0.0 True Labels
0.2
True Labels
(a) 2D grids
(b) 4D grids
Figure 6: Comparison of rescaled local outputs with local true values. Points closer to the reference
line y = x indicate higher accuracy in capturing local quantities.
model trained on smaller grids. The results, shown in Figure 5 indicate that our models show excellent generalization ability to larger grid sizes. The moderate accuracy decrease we observe is approximately linearly with the number of grid sites, likely due to accumulating errors. Training on larger grids slightly improves performance but comes at a considerable computational cost.
Learning higher-dimensionall Chern numbers We extended the task learning Cher numbers to 4D grids, whose definitions are significantly more complex than the 2D case (see Appendix [B. Furthermore, instead of a single flux Wa, there are C2 Wilson loops per site in 4D. Since for higher dimensions, the discrete approximation to the Chern number is not necessarily we cannot use the accuracy for evaluation. Therefore, we use the MAE (global loss I.g) instead, see Figure 10 the Appendix. With an MAE of around 0.25, our models can predict these higher- dimensional Chern numbers well within rounding errors. Figure 6b demonstrates the predictions of local quantities, showing good agreement with the targets.
integer,
7 Conclusions and Limitations
In this paper we have introduced a gauge equivariant model that can learn Chern numbers certain simplistic topological insulators. A limitation of our work is that only consider Chern numbers and not other topological invariants. We expect that our model can be adapted to more general setting, and thereby deal with other interesting invariants. In fact, construction may be viewed as a toy model for more interesting physical systems. Since even learning the Chern number is challenging, this is a stepping stone towards more sophisticated condensed matter systems exhibiting richer topological properties.
Acknowledgments
We want to thank David Müller and Daniel Schuh for inspiring discussions. The work of J.G. and D.P. is supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg (KAW) Foundation. M.G. acknowledges support from KAW through the Wallenberg Centre for Quantum Technology (WACQT), The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the Swedish Research Council through grant agreement no. 2022-06725.
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A Physical Background
In this section, we give a brief overview of the physics of topological insulators and show how the expression 22 arises as discretization of the contiuous Chern number.
The band structure of the topological insulators we want to consider here is described by so-called Bloch Hamiltonians H (k) which are maps from the Brillouin zone to the space of M x .M complex Hermitian matrices. The Brillouin zone the space of momenta of the electrons which is periodic in each dimension. At each point k in the Brillouin zone, we consider the eigenvectors un (k) E CM ...... N, of the Bloch Hamiltonian with negative eigenvalues since these correspond to the bands occupied by electrons in the material.
A nontrivial Chern number means that it is impossible to find smoothly varying eigenvectors over the entire Brillouin zone. In two dimensions, it is defined via
1
/
C
[r
(10)
2Ti BZ
where F an N x N-matrix known as the non-abelian Berry curvature defined by
F Ay ax. Ax() + A,,
(11)
Here, (k),), 2, ys the non-abelian Berry connection, another N N N-matrix whose components are -vetoos given by
[A,(k)]),m U, (k) 0x, (k)
(12)
in terms of the eigenvectors of the Bloch Hamiltonian with negative eigenvalues.
It can be shown that the Chern number defined by (TO) is an integer and there are generalizations to higher dimensional Brillouin zones, on which we performed experiments, see Section 6.3
The link matrices U UN, C in terms of which the Wilson loops are constructed according to ① capture the overlap between the eigenvectors of the Bloch Hamiltonians of neighboring grid points and have components
U UNylmm = Um (ki..) T Un (ki- 1.j)
(13)
[ em (ka.j)))) Tun (ki.j1))
(14)
The links are discrete analogous of the operator exp(iA(k)ddk). Under gauge transformations, the links transform according
UH - UT,
UH, - oi
(15)
The Wilson loops correspond to closed loops of the link variables. Their gauge transformation W5, 2. W.yS follows from the transformation (5) of the links. In higher dimensions, there are several Wilson loops W per grid point which are aligned with different directions y in the lattice.
B
Higher Order Chern Numbers
In
3.1
we defined in 2 for two dimensional Brillouin zones the non-abelian Berry curvature
F O - Az, + (A),
For 2n dimensional Brillouin zones, which topologically equivalent to R²n /gan, there different oriented planes, i.e. for every two directions k,,, ki, there is planar flux
W UH (UN) (UF)).
(16)
It is easy to verify that Wax (W Similarly, there is a planar curvature
F Fu, ar A. (k) Ox Ax) (A.) (0))))(()))))(40)))))
(17)
21

---

Where A, is similarly defined as in (12). We showed in 3.1 the definition of Chern numbers 2D Brillouin zone in (TP.. For a 2n dimensional Brillouin zone, a nihh order Chern number defined as
1 2Tii
/
C =
d=
(18)
B2
Here, F(k) represents wedge product of differential forms F (ke(k) dkp dhiv, which could be written equivalently as
2" m (2n)
Σ
Epi 11
Fuz 1.12 (k).
(19)
P1.022mm/2an 1.f22
It could be shown that C' is always an integer, Vn 2 1.
In practice, since the fluxes Whe an approximation of exp((F (F,,,, calculate the discrete version of higher order Chern numbers with the following equation
Cr (Ti)
n!!
Σ
Σ
Tre
log W P2-1-122
(20)
When taking = 1, Equation (20) coincides with (2. Since log function is analytical, which means could be represented by power series, and (I) w W()"" Q W"Ω, we have
)(W!) C( VΩ U(N)
This discretized Chern number is integer only in the continuum limit, therefore we use the MAE (global loss Lgg instead for evaluation.
C Data Generation
C.1 Uniform Distribution on (N) with QR Decomposition
matrix A NNN whose entries have i..dd _N 1) real and imaginary parts. We assume the algorithm to generate U from is single-valued, i.c. U = (AA) for some function f NN (N). We show the "left invariance" of the random variable U.
In the experiments generate UNV) with QR Decomposition
Proposition C.1. and gU are identically distributed, Vg E G.
Proof. By definition, gU = fggA). Then it suffices to show gAA and A are identically distributed.
Re.A
For complex matrices, we consider the two bijections. The first one p
Then
ImA
Reg
Img'
A(gA)=
= 5p( A). is easy to verify ğ orthogonal. We then flatten the matrix
Img
Reg
with vec operator
vec(A) (A11, A12..... .I.. AIN..... .......MM.......... . .M.....A))))
The following property is well known.
Proposition C.2 (Vec Operator Identity). vec(gA) = (g 69 IN)vec(A)
Where @ is the Kronecker product. Then vec(p(gA)) (g In (eccp((A)) A)). However, vec(p(A)) just (ReAij. ImAij), which follows the distribution (1, I2N=). and 9 @ IN still orthogonal, follows that
vec(p(gA)) ~ No0, (g TN) IN)² = /O I2N=)
Therefore vec(p(gA)) vec(p(A)). By bijectivity, gA A.
22

---

C.2
Diagonal Dataset
There is equivalance relation among samples WE: W,. ~ Ω D.Wa AZ, E U()), vk. Namely the equivalent classes of fluxes is a subset of (n)) Max, By the isomorphism U(n)/Ad U(L)"/Sn,
(21)
could generate plaquettes Wi diagonal matrices, i.e. WA diagfe c Notice that each link appears exactly twice in all plaquettes, once in itself, and once inversed. For example, U. appears in itself in Wi and inversed in Wk-s- Then we have:
de det W = II
det U (det - = 1e,
Σx Im Im(log(de) (det Wi)) Imog ([l det Wx)) mod 2m, the discrete Chern num- ber Ö is an integer.
Specifically, since
Proposition C.3. = 2 -. = =
Then the necessary condition for a set of plaquettes to be generated from some links is:
IIIl
E. Ex
(22)
On the other hand, given any W that is diagonal per site, suppose it is generated by diagonal links U di t ki }. Then for each index À have the following equations:
II ohtteee -i e 1, Vk:
(23)
This implies necessary condition for W* to be generated from diagonal links is that, for any X, E = O. We omit the subscript À for now.
Recall that & the flattened index of (i.j), which could have the possible form = Na + j. If we further flatten the index (k:, 14) as k for μ k Nsite for μ y. then the equations become linear:
Tk + T[k++NNa + mod 2N
T(+++N, mod Nat
T = 0g. Vk
(24)
Which is just:
Oo 01
Ti
T2N
1
-1
(25)
The coefficient matrix has rank Nate and it is solvable iff. Σe 0g. O, and that is exactly what the necessary condition specifies. Therefore, the fluxes WA can be generated from diagonal U and only
aA, II o &
(26)
This determines a submanifold M' in {m E U(I)NxN : m satisfies (22) } with codimension
N With the natural metric on (N) Mate M, defined as d(g. h) 1 12. where eiy are phase
angles of eigenvalues gh- M is a V
N
-ne M. For each channel X, suppose Ee 0g Φx.
Nus
Φx E T). Let the new be 0% 0º + / Naite Then
d(W. W)
Σ A.A
<
Natte
Nate
As the number of sites gets larger (the grid gets more refined), the net gets denser. We can further extend the sufficient condition by considering the permutations, since the permutation matrices also unitary and their actions on fluxes are adjoint.
We now propose the diagonal data generation scheme:
23

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0.4
00.
Global Loss
Sid Loss
0.2-
0.2
Global Loss ba
Standard Deviation Loss Lus
0.0
0.0
0k
1k
2k
3k
Epochs
Figure Global Loss and Standard Deviation Loss curve of the baseline model, trained diagonal, trivial dataset, to learn the Chern number on a 5" grid, with 4 filled bands.
Table Accuracy of the same run in Figure evaluated on non-diagonal, non-trivial data on 5 grid, with 4 filled bands.
Seeds
No.1
No.2
No.3
No.4
Accuracy
92.7%
94.3%
95.4%
93.8%
1. Generate label F2 E [- R), such that > Fx 2xn.
2. If only zero samples: check > P's = O.
3. For every & but the last one, generate (0k)) such that Σx ob Fk.
4. For every & but the last one, let Wi. be diag(e"') ,...,, e000)
5. Let the last Wg be He W
The last product will not cause confusion since diagonal matrix multiplication is commutative.
It could also go the other way around: generate the fluxes first, then find a solution to 25 to get the links. This way, we could operate directly on the distribution of eigenvalues, thus customizing the data generation process. Furthermore, the diagonal dataset reduces the computation cost significantly for training.
For validation, we show in Figure 7 the loss curves and in Table 3 accuracies of evaluation on nontrivial, general (non-diagonal) datasets, of training diagonal, trivial dataset.
24

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D Proof of the Universal Approximation Theorem
Here give the complete proof of Theorem 5.2 or, rather its stronger form in Theorem D.3 We begin with some properties of the class functions defined in [.]
Theorem D.1. The space of symmetric polynomials { over eigenvalues {Ax} forms an orthonor- mal basis of Li (G).
Proof. Follows directly from the Peter- Peter-Weyl theorem and the expansion of class functions in terms of irreducible characters.
Furthermore, consider the of polynomials over eigenvalues g: {Px (AI...., ,,) -, our setting these polynomials can be identified with the set of traces of group elements of the form Trg". Using Newton's 's identities for symmetric polynomials:
ke A
(-11)-- PP-jPj EK
II K
(27)
En, =
one may deduce the following Corollary D.2. The space
n (G)) is dense in Las
Now, consider the network architecture GEBLNet. Given the flux tensor Wk. stack the identity and its Herimitian conjugate to a second channel as
- (Wa,0, WK,1, Wa,-1) (7, W W
Afterwards we put through several blocks, each containing three layers: GEBL, GEAct and TrNorm. In this section, we ignore TrNorm Layers, since they are introduced to boost training results. We call each block a "packed layer".
After several packed layers we calculate the trace per-channel and add a linear layer (the "Dense layer") in the end. The Dense layer sum over the site index (to calculate the topological invariant).
the real and imaginary parts separately. Then we take the
So the outputs have the following form:
(EBL,(We)) b. Where (W?) (TrW W? We denote the set of these functions by B.N,(G) where the subscript a indicates the choice activation function. We further denote by BCN (G) the subset of BLN, (G) with & packed layers.
WE
GEBL
Since we attempted learn local quantities F(Wx), we omit the subscript k. Furthermore, we treat the flux W as an abstract element in the Lie group G, denoted as In this case, where the input channel size is one, we propose the main result:
Theorem D.3 (Universal Approximation Theorem). For any activation function 5 Re, where is bounded and non decrressesnn BLN, (G) is dense in Li (G).
The proof of this will require the following lemma. Lemma D.4. BCN*
& (G) is dense in [f(p), .P2*): |/ <00 C LG), where p; Trg'.
Proof. We prove this lemma by induction. For ks = 1, the output has the following form
Σ oggg'
wro Σ aágg'
+b.
Note that Tro (Z a²g") part (in other words, forcing W.sim in the dense layer to be zero), the output is simply
(Reatp: )akpr. For any channel index i, when taking only the real
Σ ReafRep; Imaç[Ipp))
Σ
Rea Ima,Imp, = Σ
Rea Ima
(28)
25

---

Therefore, it is essentially a one- llye fully connected network on {(11.22) = R Thus the set dense.
Assume this is the case for n, and we would like to prove the lemma for n - 1. We denote 2" = N. Then the layer input has the following form:
Σ (po. PN)2)pp
(pu, ... -PN/2)0
Now the new "aa"" (denoted as b/) takes the following form:
Σ
Σ
=
p-o=č j.k
Consider the bijection F CN++ P. (C), given by F(ū) =, 4,2". Using this we define
ā t = F (F(T)
Then
6, = aaj () P)jjaj = asjk dk).
This forms a linear space Bu+1 c (I (I. 22N+1 For simplicity we henceforth omit the subscript Kn+1-
We assume (a)) could approximate any constant function of P1. PN/2- This is triiially true when n = 1, since it is function on a constant and takes arbitrary constant values.
Denoting ep (1/d), where d dim G, we have ey P 1, K. Since K is compact, there exists an open set U s.t. E OU, and b- p E (1, +0o), vb E U. E.K. K.
the other hand, easy to see that {b+b b: b (1, ..... , ) is linearly independent subset. This way we could choose 2N + elements (b*t 12N from its intersection with U, such that fba ba) C2N+1
Now given constant vector -= (bn,... b2Nn), ), there exits {ar} such that 5 a b We want to show that i can be approximated by any precision e.
Without loss of generality, assume sup = and inf = 0. Then, for all e, there exists Mo > 0 such that for all Mo/2. E (vI e, 1). This gives
d
H (p, Meo, Mea) = - v. > Mo. M2
By induction, there exists such that ||a 60 < m/2), Consider
1 at M2 H( H(p at, at) (Mo p)aa at-
Then
17
b)
(Ma p)2 1))b² b + aMMa p) b² at aa))
arelb bad 2e|66'| +
N)E.
(29)
When the coefficient functions approximate constants, the last layer is essentially one-hidden-layer fully connected network over P1.... PPN. Similar to the N = 1 case, as the width grows larger the network can approximate any function fpt..... .P2N) ). the concludes the proof of the lemma. 0
We may now complete the proof Theorem D3
Proof. (Proof of Theorem D.3
Recall that by Theorem D. the space of class functions T.2 (G) spanned by symmetric polyno- mials in the eigenvalues of group elements. Since these symmetric polynomials can be expressed
26

---

in terms of traces Tr(g), Tr(g").... ................................ Tr(g"1) it follows that any class function can be written function these traces. Now, since G is compact, we have L²(G) (G) and I|ll C'I flo.
Therefore, for all f E 7 2ass (G), and for any > 0, there exists
fn = f (PI. .. Pn) E (K)
such that |If fnll2 1/2e. By Lemma D.4 the function class BCN (G), (G') consisting of neural networks with gauge equivariant bilinear layers, can approximate any function f(p...... Pe) arbitrarily well, provided & is large enough. We deduce that there exists 9 BLN" C C such that |gg - fallæ < c/2Cc. Therefore
- - //l < (C. =
This concludes the proof of the main theorem.
Restricting our scope to unitary groups, we could further generalize to the case where the input data has more than channels. In other words, the function f991.-...999 attempt to learn has the following property,
Vg, h4 E G.
(30)
For inputs with channels, we denote the of functions that could be represented by GEBLNet as BCN & (G").
Theorem D.5. For any activation funccioion â that is bounded and non-decreasing, BLN (G") could approximate any function with the property specified in (30) in the I.2 norm
The proof requires the following theorem in [40].
Theorem D.6 (Procesi, 1976). Let be a field of character 0, and let
F[Ax (xI
(k) k S K]
polyyooomiii be the polynomial ring in KN2 variables, corresponding to scalar-valued polynomials on FNxN*K Then the subalgebra
{p E F[An] p(A Ak) = plggAg ..... ....,,AKK Vg GLN(F))
(31)
generated by { (Tr A%
Proof of Theorem D2.5 It well-known that the unitary group is not contained in the root set of any
nonzero polynomial on the set of n by n matrices (C). Therefore, for an arbitarily chosen sequence of (A....., A&) in G. if for any g G we have p( p(A,,..., Ax) = P(gAi9-1111 ..... 9 1). then this also holds for any
- Vg G.
(C). Analogously we could show that the equation in Eq. (31
equivalent Eq. (30) for polynomials. Namely, functions satisfying Eq. (30) spanned by
[Tr I nu }. and the proof, analogous to the one-dimensional case, follows.
27

---

GRAuk
Th
Tapsc: M3)
GLAd
GaCWa
Sert
Ibace
Figure 8: Architecture of GEConvNet. In this figure, the rectangles represent the spatial grid, the arrows the grid represent links, and the number of layers (Neh) represents the number of channels (y). Each circle represents a site on the grid, and quantities on different sites do not interact with each other, except for the GEConv Layers, and the last summation grids.
Table 4: Configuration of GEBL layers for the representative model. See its architecture in Figure In addition, every GEBL layer is followed by a GEAct layer and a TrNorm layer that maintains channel size.
Layer
Input Channel
Output Channel
GEBL
32
GEBL 2
32
16
GEBL
16
8
E Further Details regarding the Network
Architecture of GEConvNet Figure is is demonstration of the architecture of GEConvNet. It has GEConv layers implemented inside, therefore it takes the links U./ as inputs, beside the fluxes Wk. the kernel size is set positive.
Configuration of the representative model Table 4 lists the hyperparameters for GEBL layers in the representative model. Since the GEBL layers are consecutive with channel size-maintaining layers, GEAct and TrNorm layers in between, the former layer's output channel equals the latter's input channel.
Complexity-Accuran Comparison Figure 9 is a comparison of model complexity and accuracy across different models. All the models are trained 5 grid with 4 filled bands. Due the instability of TrMLP, there are only 3 successfull runs with this model.
General Compute Resource Requirements All experiments were conducted on computing cluster equipped with NVIDIA T4 GPUs. For the baseline task of training GEBLNet to predict Chern numbers on 5 5 grid, training typically required approximately 15 to 20 hours with I T4 GPU and 32 CPU cores. The same hardware resources are sufficient to train models for tasks on 4D systems or on larger grids; however, the computational time generally increases with the task complexity. While is possible train on personal computers, we do not provide guarantees for the feasibility such setups.
Global Loss Curve for Training on Higher Order Chern Numbers Figure TO shows the global loss curve for training on second order Chern numbers. Since we adopt the T.rmoorm ||. || here, I.g essentially measures the mean absolute error of global outputs.
28

---

100%
90%
80%
Representative Model
70%
GEBLNet DeepSpec
60%
GEConvNet, =44
GEConvNet, 5 -2
Accurscy
GEConvNet, 5, 95% Accuracy Benchmark prediction: 27.2% Accuracy
50%
40%
30%
20%
10%
104
10
10
107
Number Parametors
Figure 9: Comparison of model complexity and accuracy across different architectures. Complexity measured by the number of learnable parameters, and & denotes the kernel size for GEConv layers. Each marker represents a training run with variations in models, learnable parameters, and random seeds.
1.00
Global Loss
Global Loss
0.75
0.50
0.25
Epochs
Figure 10: Global validation loss curve of the representative model, trained on a 34 grid, with 3 filled bands to learn the second order Chern number C2.
F Learning Chern numbers using ResNets
Figure II shows an ablation over a number of different architectures considered for the task of predicting the determinant of N x N real matrices.
29

---

Architecture
3,2,,,2
3,2,2,2,2
2,2,2,2,,,2
Relative error 6
4
2,3,2,2,2
1,1
N layers
2,2
3,3
*
2-
3,2
=
1,1,1
+
1-
2,2,2
3,3,2
0
4
5
6
Matrix size
Figure 11: Architecture ablation over different layer orders (see section 3.2 and depths. Best relative error 5 for each architecture for different matrix sizes N x N.
30


Reviews:
[{"review_id": "D5ucgkiRT4", "reviewer": "Reviewer_FXTi", "summary": "This work presents a new gauge equivariant architecture aimed at learning Chern numbers of a simple model of topological insulators, which are of interest in the condensed matter community. The paper begins by describing the model which (at a very high-level) consists of a 2-dimensional lattice, where each node carries a unitary matrix $W_{i,j} \\in U(N)$, where $N$ corresponds to the number of bands. Critically, two instances of this lattice are equivalent up to conjugation by an element $\\Omega_{i,j}$ of $U(N)$ which varies from node to node. The task involves predicting a certain formula using these $W_{i,j}$. The paper explains why this problem is hard for more familiar neural networks by exploring network performance on the task of predicting matrix determinants (which is related to the problem above). This motivates the introduction of a family of architectures GEBLNet and GEConvNet that include a novel normalization layer. The paper performs a range of experiments on synthetic data and shows that GEBLNet can successfully learn to predict Chern numbers for different numbers of $N$ and can even by trained to learn on $4$-dimensional lattices.", "questions": "- Line 154: The term ‘flux data’ appears here without a definition. It is later referenced several times. What does this mean?\n- Reviewer may have missed it, but what is the point of introducing GEConvNet it is not perform particularly well in any of the instances?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- **The paper is clearly written and should be comprehensible to a significant portion of the NeurIPS community:** The paper does an excellent job making the material intelligible to non-physics readers. Understanding the problem really only requires understanding linear algebra and some basic group theory. This is a big accomplishment as this material can be quite complex. Fortunately, the paper distills its analysis down to the aspects that will be of most interest to the NeurIPS community.\n- **The problem is a natural fit for equivariant neural networks:** The paper makes a convincing case that the problem of computing invariants of topological insulators is one where considerations of symmetry are essential to solving the learning problem. This is significant at a time where equivariance has been abandoned for several tasks in favor of just using more compute (e.g., alphafold). It seems unlikely that a similar approach would work in this setting. This problem would likely be of high interest to the geometric deep learning community.\n\nWeaknesses\n- **The architecture section could be better explained:** It is possible that it is just because this reviewer has not worked with gauge equivariant neural networks before, but he found Section 4.1 which describes different layers in GEBLNet and GEConvNet to be a little unmotivated. It would improve the paper if more time was spent describing each layer at a high-level (e.g., what does this function do, what properties does it give the network, etc.). That being said, the reviewer has seen far more impenetrable descriptions in geometric deep learning papers at past NeurIPS.\n- **Evidence why GEBLNet works but other architectures do not :** Both the determinant example in Section 3.2 and the DeepSpec example in 3.3 are a nice opportunity to explore what makes other approaches fail. While this is speculated on in these sections, it would make the paper more compelling if deeper experimental evidence was provided to support these claims. As it is, most analysis seems to be based purely on performance.\n\nNitpicks\n- Line 87: ‘…seen an explosive development…’ $\\mapsto$ ‘…seen explosive development…’\n- Figure 1: It would be interesting to see this plot but with addition curves representing different architecture variations (e.g., values of $R$).", "was_revised": true, "final_justification": "The reviewer feels that this is a valuable contribution to the machine learning community. The paper tackles an interesting and scientifically important problem that has so far only been explored to a limited extent in machine learning. This reviewer agrees with the others that the paper probably could have done more to make general readers of the work aware of the scientific and data challenges associated with the problem.", "rebuttal": "We would like to thank the reviewer for providing such a thorough and positive evaluation of our submission. In the following rebuttal, we will try to answer your questions and clear up potential misunderstandings.\n\n### Weaknesses\n\n> **The architecture section could be better explained:**  It is possible that it is just because this reviewer has not worked with gauge equivariant neural networks before, but he found Section 4.1 which describes different layers in GEBLNet and GEConvNet to be a little unmotivated. It would improve the paper if more time was spent describing each layer at a high-level (e.g., what does this function do, what properties does it give the network, etc.). That being said, the reviewer has seen far more impenetrable descriptions in geometric deep learning papers at past NeurIPS.\n> \n\nWe acknowledge the fact that the motivation for different layers were not clearly stated. We will clarify this in the revised version of the paper.\n\nIn short, there are three major layers in GEBLNet and GEConvNet: GEBL, GEConv, and GEAct. The samples can be viewed as matrix-valued multichannel “images”, and the operations of the layers are analogous to standard image processing operations, but in a gauge group setting.\n\nGEBL calculates pixel-wise matrix multiplications among channels, yielding a second order polynomial at each site (pixel).  This captures higher-order features for later processing when the matrix-valued samples are converted to scalar-valued outputs. \n\nGEConv serves a similar purpose to the traditional convolution layers where it extracts features among neighborhoods of each pixel with linear combinations. Alternatively from a physics viewpoint, the matrix channel represents an integral of a matrix valued function along a small contour, and GEConv layers combine these to yield integrals along larger curves.\n\nGEAct is a non-linearity that is invariant under gauge transformations.\n\n> • **Evidence why GEBLNet works but other architectures do not :** Both the determinant example in Section 3.2 and the DeepSpec example in 3.3 are a nice opportunity to explore what makes other approaches fail. While this is speculated on in these sections, it would make the paper more compelling if deeper experimental evidence was provided to support these claims. As it is, most analysis seems to be based purely on performance.\n> \n\nWe agree that it would be interesting to further investigate the failure modes of other architectures for these examples. For the determinant task, the goal was simply to show that higher order terms (i.e. at least bilinear) are sufficient to predict the determinant of matrices with high rank, whereas standard linear MLPs do not suffice. \n\nFor the DeepSpec example, it would be very interesting to better understand why equivariance provides a better model prior even though the task output is invariant. In analogy to equivariant tasks sometimes benefitting from approximate model equivariance when traversing the loss lanscape, it might be the case that our equivariant model provides the needed flexibility in terms of training dynamics. It could also be the case that it is beneficial to perform intermediate computations on equivariant quantities rather than invariant ones. We hope that GEBLNet paves the way for future work into these questions.\n\n### Nitpicks\n\n> Line 87: ‘…seen an explosive development…’ ‘…seen explosive development…’\n> \n\nThank you for pointing out this typo.\n\n> Figure 1: It would be interesting to see this plot but with addition curves representing different architecture variations (e.g., values of $R$).\n> \n\nThank you for this suggestion. We have generated the corresponding plot and will include it in a revised version of the manuscript, however, due to the rebuttal formatting rules, we cannot include it here. The general trend we observe is that models whose total order is lower than the order of terms in the determinant fail to learn the determinant. Additionally, there seems to be an exponential increase in error with matrix size, even for sufficiently large models.\n\n### Questions\n\n> • Line 154: The term ‘flux data’ appears here without a definition. It is later referenced several times. What does this mean?\n> \n\nThank you for pointing out this oversight. With the fluxes / flux data, we just mean the Wilson loops $W_k$.\n\n> • Reviewer may have missed it, but what is the point of introducing GEConvNet it is not perform particularly well in any of the instances?\n> \n\nWhen Favoni et al. (2022) introduced their gauge equivariant models, the GEConv layers were a central building block. Therefore, it is a natural baseline to consider whether these convolutional layers would also enhance performance in our setting. A negative answer suggests a drawback when redundant neighboring information is introduced, which offers valuable insights for later studies."}, {"review_id": "xzNWxARcQt", "reviewer": "Reviewer_6HNj", "summary": "The authors are able to learn higher-band and higher-dimensional Chern numbers by adding a normalization layer to Lattice Gauge Equivariant CNNs (LGE-CNNs). They prove a universal approximation theorem for the resulting architectures.", "questions": "Are there real / existing datasets where applying your new architecture would yield scientifically interesting or instructive results? I will raise my score if you perform such an experiment. Otherwise, I may decrease the score.", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 1, "originality": 2, "strengths_and_weaknesses": "### Strengths\n\n- The authors are able to fix training stability issues in LGE-CNNs by introducing a new normalization layer.\n- Using the introduced architectures, the authors are able to learn Chern numbers with more than 3 bands for the first time. \n- Similarly, higher-dimensional Chern numbers are learned.\n- The paper is clear, easy to understand. The plots are readable, and the text synthesizes the technical results and experiments quite well.\n\n### Weaknesses\n\n- Relatively little methodological and technical novelty. \n    - The main technical contribution is the introduction of a normalization layer. The authors nicely motivate that this is useful to stabilize training. However, because it is a straightforward / incremental patch, it probably shouldn’t merit being listed as a contribution.\n    - The main methodological contributions are applications in synthetic datasets. The scientific or technical significance of these constructed tasks is not motivated in the main body.\n- The universal approximation theorem is stated as a main theoretical result. However, it uses standard analysis and comparable results already exist for similar equivariant architectures.\n- The evaluations are largely toy. These motivate the need for gauge equivariant architectures to learn Chern numbers, but e.g. do not perform new science or have a clear path to doing so.", "was_revised": true, "final_justification": "Am still not convinced that the current work has sufficient (a) methodological novelty (a normalization layer that is, in the reviewer's opinion, straightforward) , (b) technical novelty, or (c) experimental novelty to merit publication at this time. The authors are encouraged to apply the architecture to more difficult problems to better motivate the methodological usefulness.", "rebuttal": "We would like to thank the reviewer for providing such a thorough and detailed evaluation of our submission. In the following rebuttal, we will try to answer your questions and clear up potential misunderstandings.\n\n### Weaknesses and Questions\n\n> • The main technical contribution is the introduction of a normalization layer. The authors nicely motivate that this is useful to stabilize training. However, because it is a straightforward / incremental patch, it probably shouldn’t merit being listed as a contribution.\n> \n\nWe respectfully disagree with the assessment that the normalization layer is merely an incremental patch and not a significant contribution. While normalization is a common technique, numerical instability was a well-known but unsolved issue in the context of gauge equivariant networks, and to the best of our knowledge, no prior work has systematically addressed it.\n\nIn particular, our work proposes a tailored normalization mechanism that specifically handles complex-valued representations in this setting. This is not a trivial adaptation of existing techniques:\n\nInstead of standard forms like $\\frac{x - \\mu}{\\sigma}$, which proved ineffective in our setup, we instead divide by the mean. Furthermore, we normalize by the norm of the traces rather than their raw values, respecting the complex valued nature and the gauge symmetry. Other gauge equivariant normalization layers using e.g. the determinant are conceivable but we decided for the current version based on a statistical analysis of the gauge invariants of the preactivations.\n\nWe therefore believe this component is both methodological novel, and practically valuable for stabilizing training in this class of networks.\n\n> • The universal approximation theorem is stated as a main theoretical result. However, it uses standard analysis and comparable results already exist for similar equivariant architectures.\n> \n\nUniversal approximation has indeed been proven for various globally group equivariant architectures. However, as far as we know, prior to our work no such results existed for *gauge* equivariant architectures (i.e. where the symmetry is *local)*. \n\n> • The main methodological contributions are applications in synthetic datasets. The scientific or technical significance of these constructed tasks is not motivated in the main body.\n> \n\n> • The evaluations are largely toy. These motivate the need for gauge equivariant architectures to learn Chern numbers, but e.g. do not perform new science or have a clear path to doing so.\n> \n\n> Are there real / existing datasets where applying your new architecture would yield scientifically interesting or instructive results? I will raise my score if you perform such an experiment. Otherwise, I may decrease the score.\n> \n\nWe appreciate the need for additional motivation for the chosen dataset. As is standard in the literature in this domain (see e.g. [13], [34], [35] as well as [I] and [II]), we train on synthetically generated data. These datasets are based on Hamiltonians which model physical topological materials. However, to date, there is no large-scale dataset of band structures obtained by measuring angular-resolved photoemission spectra which would allow one to train a machine-learning model in this way.\n\nConsidering higher-band topological insulators, or generally more complex gauge groups, is an important problem as it opens up for the application of machine learning to related domains such as emergent gauge fields in strongly correlated / topologically ordered phases or artificial non‑Abelian gauge fields in cold atoms and photonics as well as condensed matter physics. Before this work, no machine learning system existed which could predict Chern numbers of higher-band topological materials. Our model significantly extends the SOTA by being able to predict Chern numbers of materials with as much as 7 filled bands.\n\nIn particular, we train on uniformly distributed link variables (Appendix C.1) and on link variables whose distribution was adjusted for a specific distribution of Chern numbers (Appendix C.2). These allow us to test our model on input configurations spanning a wide range of Chern numbers with many filled bands, going beyond the limited Hamiltonian models common in the literature. To verify that our model is indeed able to learn also the simpler data from those Hamiltonians, we trained our model on the Bloch Hamiltonian and obtained an accuracy of 95.5%.\n\nReferences:\nArabic references correspond to our manuscript.\n\n[I] Che, Y., Gneiting, C., Liu, T., & Nori, F. (2020). *Topological quantum phase transitions retrieved through unsupervised machine learning*. Phys. Rev. B, **102**(13), 134213.\n\n[II] Scheurer, M. S., & Slager, R.-J. (2020). *Unsupervised Machine Learning and Band Topology*. Phys. Rev. Lett., **124**(22), 226401."}, {"review_id": "GK1OBoeP49", "reviewer": "Reviewer_cwof", "summary": "The paper introduces a gauge equivariant neural network that allows to predict Chern numbers associated with multiband topological insulators. The model respects local gauge symmetry through the novel gauge equivariant normalization layer as well as some of the established equivariant layers, contributing to the drastic reduction in the size of training dataset. The model is evaluated from multiple aspects and is shown to produce promising results.", "questions": "- How is the model for 4D grid implemented?", "limitations": "- Scalability. The input size is N^2 * Grid size.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strength\n- The motivation of incorporation equivariant model is clear\n- Benchmarking standard methods, whose results clearly point out downsides or weakness of the relevant methods.\n- Solid proof of the universality of the model\n- Exhaustive experiments. Some of the ideas to overcome difficulties are based on the authors' deep understanding on the topic, which makes the paper strong. For example, using the fact that each orbit has a diagonal matrix for the dataset generation is clever.   \n\nWeakness\n- The importance of the Chern number in the context of topological insulators is not very clearly stated.", "was_revised": true, "final_justification": "Topological insulator is actively studied in mathematical physics community. Chern number (or character) is a common mathematical concept, but in general highly abstract notion and not computable at all. I support this paper based on the following two reasons: The authors cleverly use an alternative expression of Chern characters in the context of topological insulator, and the idea to overcome the difficulty in training their proposed models is also clever and mathematically-grounded. The major drawback is, as the authors also agreed (partially), its potential associated computational cost and scalability. However, even when taking the drawback into account, the significance of contribution of the paper outweighs the drawback, and the paper could plays a leading and influential role in the community.", "rebuttal": "We would like to thank the reviewer for providing such a thorough and positive evaluation of our submission. In the following rebuttal, we will try to answer your questions and clear up potential misunderstandings.\n\n### Weakness\n\n> • The importance of the Chern number in the context of topological insulators is not very clearly stated.\n> \n\n### Question\n\nWe agree that the physics background was very brief in the manuscript. This was intentional, in order to focus on the mathematical background and the machine learning aspects of the problem. In the following, we will provide some further background about the importance of the Chern number.\n\nTopological insulators are materials that are insulators in the bulk but have conducting states on the surface (edge in the case of a 2D insulator). Topological insulators characterized by a Chern number are the prime example of such materials. The fact that a material with finite Chern number cannot be continuously deformed into a material with Chern number 0 without closing the energy gap (which make the materials insulating) implies that there has to be a zero energy (conducting) state at the boundary. The quantized conductance of the quantum Hall effect is one celebrated physical system which is related to such a conducting boundary mode. \n\nWe will add further details to the revised version of the manuscript.\n\n> • How is the model for 4D grid implemented?\n> \n\nThe 4D model is implemented analogously to the 2D case. In 2D, the input consists of a single Wilson loop matrix at each lattice site. In 4D, on the other hand, there are $\\binom{4}{2} = 6$ independent Wilson loops, each defined on a different 2D subplane. These six Wilson loop matrices serve as the input channels of the model, with gauge transformations acting simultaneously on each of them. Structurally, the 4D model therefore follows the same architecture as the 2D model, but with six input channels instead of one.\n\n### Limitation\n\n> • Scalability. The input size is N^2 * Grid size.\n> \n\nYou are certainly correct in pointing out that scaling of the input size is a potential problem. However, this is precisely a key strength of our gauge equivariant model: since it considers *local* symmetries the scaling problem in grid size is not an issue."}]

Author rebuttal:

Meta-review: This paper introduces GEBLNet, a gauge-equivariant neural network architecture for learning in the presence of local gauge symmetries. Contributions include a gauge-equivariant normalization layer that stabilizes training, a universal approximation theorem showing the model can approximate any continuous gauge-invariant function, and extensive empirical validation. The model is applied to the prediction of Chern numbers from synthetic lattice gauge configurations, and a notable achievement is that it successfully learns non-trivial and higher Chern numbers for the first time, even when trained only on trivial cases. The authors also extend experiments to 4D Chern numbers and show generalization across grid sizes, with only a moderate linear decrease in accuracy attributed to error accumulation.

Reviewers praised the clarity and accessibility of the paper. The architectural advances are nontrivial, and the demonstration that a gauge-equivariant network can capture global, integer-valued topological invariants represents an important proof-of-concept. Limitations are that experiments are restricted to synthetic, clean data, without tests on DFT/Wannier-derived Hamiltonians, and that the task itself is solvable by analytic methods. A natural next step would be to validate on realistic Hamiltonians, which would strengthen the link to practical physics workflows. However, the main point of this work is to demonstrate that such networks can learn these types of topological quantities at all, rather than to propose a practical replacement for established computational methods.

Overall: While the immediate practical impact is limited, this work makes a meaningful methodological contribution to equivariant ML. Showing that gauge-equivariant networks can stably learn higher Chern numbers is a first, and the techniques developed here should be broadly applicable beyond this specific application.
Decision: Accept (poster)