On the Equivariant Learning of the $Q$-tensor Order Parameter

• URL: http://arxiv.org/abs/2605.27679v1
• Date: Tue, 26 May 2026 20:54:38 GMT
• Title: On the Equivariant Learning of the $Q$-tensor Order Parameter
• Abstract summary: We construct and evaluate group-equivariant neural networks for the prediction of the two-dimensional $Q$-tensor order parameter of nematic liquid crystals.<n>We find that the equivariant models consistently achieve lower errors and generalize more robustly to unseen defect configurations.
• Score: 0.7734726150561086
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We construct and evaluate group-equivariant neural networks for the prediction of the two-dimensional $Q$-tensor order parameter of nematic liquid crystals from synthetically generated microscopic textures. Seven architectures, equivariant to cyclic groups $C_k$ of order $k$ for $k=4,\,8,\,16,\,32,\,64,\,128,\, 256$, are built using a combination of weight-sharing constraints, equivariant activations and regularization techniques. To do this, we construct rotation-like permutation matrix groups with elements $\varrho_{C_k}(g)$ that act on row-wise vectorized images, thereby approximating a $\frac{2π}{k}$ rotation of the circular subdomain on square images. We show that all seven equivariant models satisfy the $Q$-tensor equivariance constraint to within single-precision floating point accuracy. Comparing against approximate parameter-matched non-equivariant benchmarks, with and without data augmentation, we find that the equivariant models consistently achieve lower errors and generalize more robustly to unseen defect configurations. Performance increases with group order, suggesting that the incorporation of finer rotational symmetry leads to lower errors.

Related papers

• A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of $\mathrm{O}(2)$ [8.549379020585462]
  We introduce a family of vision transformers equivariant to arbitrary discrete subgroups of $mathrmO(2)$.<n>Our construction yields equivariant analogues of the core transformer components, together with expressivity guarantees for the resulting layers.
  arXiv  Detail & Related papers  (2026-06-26T09:05:47Z)
• Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery [17.340037184664208]
  We introduce the $star_G$ tensor algebra, in which any finite group $G$ defines the algebraic multiplication rule.<n>The framework provides capabilities that redesign neural networks (ENNs) structurally cannot: a closed-form per-irreducible equi- decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset.
  arXiv  Detail & Related papers  (2026-05-19T19:47:40Z)
• Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration [71.51826063983434]
  Image restoration is an inherently ill posed inverse problem.<n>We use non strict symmetry as a constraint to formulate the restoration inverse problem.<n>We show that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size.
  arXiv  Detail & Related papers  (2026-05-13T16:22:19Z)
• High-dimensional Adaptive MCMC with Reduced Computational Complexity [0.8594140167290097]
  We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation.<n>We show that it outperforms diagonal preconditioning in terms of absolute performance.
  arXiv  Detail & Related papers  (2026-04-10T12:58:00Z)
• Any-Subgroup Equivariant Networks via Symmetry Breaking [47.72921214239821]
  equivariant architectures are usually highly constrained, designed for symmetries chosen a priori.<n>This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly.<n>We build a single model that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature.
  arXiv  Detail & Related papers  (2026-03-19T21:30:41Z)
• Parameter-free Algorithms for the Stochastically Extended Adversarial Model [59.81852138768642]
  Existing approaches for the Extended Adversarial (SEA) model require prior knowledge of problem-specific parameters, such as the diameter of the domain $D$ and the Lipschitz constant of the loss functions $G$.<n>We develop parameter-free methods by leveraging the Optimistic Online Newton Step (OONS) algorithm to eliminate the need for these parameters.
  arXiv  Detail & Related papers  (2025-10-06T10:53:37Z)
• Statistical Learning Guarantees for Group-Invariant Barron Functions [8.94770625611836]
  Group-invariant structures introduce a group-dependent factor $delta_G,Gamma,sigma le 1$ into the approximation rate.<n>Our results show that encoding group-invariant structures in neural networks leads to clear statistical advantages for symmetric target functions.
  arXiv  Detail & Related papers  (2025-09-27T19:52:14Z)
• Relaxed Rotational Equivariance via $G$-Biases in Vision [19.814324876189772]
  Group Equivariant Convolution (GConv) can capture rotational equivariance from original data.<n>However, the presentation or distribution of real-world data rarely conforms to strict rotational equivariance.<n>We propose a simple but highly effective method to address this problem, which utilizes a set of learnable biases called $G$-Biases.<n> Experiments demonstrate that the proposed RREConv-based methods achieve excellent performance compared to existing GConv-based methods in both classification and 2D object detection tasks.
  arXiv  Detail & Related papers  (2024-08-22T14:52:53Z)
• Transformers as Support Vector Machines [54.642793677472724]
  We establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem. We characterize the implicit bias of 1-layer transformers optimized with gradient descent. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.
  arXiv  Detail & Related papers  (2023-08-31T17:57:50Z)
• Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of $S_n$ [15.711517003382484]
  Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of equivariant dynamics related to neural nets. It is shown that spurious minima do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic $S_n$-equivariant bifurcation. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterback &
  arXiv  Detail & Related papers  (2021-07-06T06:43:06Z)
• 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)
• Robustly Learning any Clusterable Mixture of Gaussians [55.41573600814391]
  We study the efficient learnability of high-dimensional Gaussian mixtures in the adversarial-robust setting. We provide an algorithm that learns the components of an $epsilon$-corrupted $k$-mixture within information theoretically near-optimal error proofs of $tildeO(epsilon)$. Our main technical contribution is a new robust identifiability proof clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system.
  arXiv  Detail & Related papers  (2020-05-13T16:44:12Z)

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.