Search Results (102)

Atomic charges are widely used to analyze molecular electronic structure and substituent effects, yet their numerical values and interpretations are inherently dependent on the adopted density partitioning scheme. Here, we adapt the Equivariant Atomic Contribution framework to molecular systems (EAC-qm), enabling prediction of atom-resolved continuous charge densities from which atomic charges are obtained as spatial moments. The predicted densities reproduce reference density functional theory results with high accuracy and preserve global charge conservation. To assess chemical interpretability, we examine charge responses in monosubstituted aromatic systems using Hammett substituent constants as external empirical references. Atomic charges derived from EAC-qm exhibit a strong linear association with Hammett parameters, compared with values obtained from traditional density partitioning approaches applied to the same electronic structures. These correlations indicate that density-derived charges respond systematically to established substituent electronic trends. Beyond scalar charges, atom-resolved dipole moments can be evaluated as first-order moments of the same continuous density representation. Illustrative examples for formaldehyde (H₂CO) and formamide (HCONH₂) show that local dipole vectors provide directional information about intra-atomic polarization that is not captured by point-charge models. Overall, the results suggest that machine-learned continuous electron densities provide a representation-consistent basis for constructing atom-centered electronic descriptors with chemical interpretability. Full article

Quantiles are among the most fundamental constructs in probability theory and statistics, intrinsically linked to order structures, stochastic dominance, and the principles of robust statistical inference. Although the univariate theory of quantiles is by now classical and well developed, their generalization to multivariate settings remains mathematically subtle and methodologically demanding. In particular, extending the notion of “location within a distribution” beyond one dimension raises delicate questions of geometry, ordering, and equivariance. Within this landscape, the spatial—or geometric—formulation of multivariate quantiles has emerged as a rigorous and conceptually unifying framework capable of reconciling these issues. In this work we advance this paradigm by introducing a kernel-based estimation procedure for nonparametric conditional geometric quantiles of a multivariate response $Y\in {\mathbb{R}}^q$ ($q\ge 2$) given a functional covariate X that takes values in an infinite-dimensional space. The data are assumed to form a strictly stationary and ergodic process, while the responses may be subject to a missing-at-random mechanism, a feature of substantial practical relevance. Our analysis establishes strong consistency of the proposed estimator, characterizes its optimal convergence rate, and derives its asymptotic distribution. These limit theorems, in turn, provide the theoretical foundation for constructing asymptotically valid confidence regions and for performing inference in multivariate quantile regression with functional covariates. The theoretical developments rest on natural complexity conditions for the involved functional classes together with mild smoothness and regularity assumptions. This balance between generality and mathematical precision ensures that the resulting methodology is not only robust in a rigorous probabilistic sense but also widely applicable to contemporary problems in high-dimensional and functional data analysis. The proposed methodology is numerically investigated through simulations and is implemented in a real data application. Full article

This paper mainly studies the equivariant Hopf bifurcation of a delayed reaction–diffusion predator–prey model with stage structures on a two-dimensional circular domain. Firstly, we calculate the existence of steady-state solutions, and then analyze the existence of Hopf and equivariant Hopf bifurcation for the model according to bifurcation theory. Secondly, we calculate the normal form of the equivariant Hopf bifurcation. Finally, we conduct numerical simulations to verify the conclusion. And through simulation, we obtain a spatially homogeneous periodic solution, and spatially inhomogeneous periodic solution including rotating waves and standing waves on a two-dimensional circular domain, which shows rich dynamic properties on a two-dimensional space. Full article

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

We investigate how symmetries present in datasets affect the structure of the latent space learned by Variational Autoencoders (VAEs). Understanding symmetries in data is essential because symmetries determine the true degrees of freedom, constrain generalization, and provide physically interpretable coordinates. We therefore study whether a standard, non-equivariant VAE can reveal symmetry-induced dimensional reduction directly from data, without imposing the symmetry in the architecture. By training VAEs on data originating from simple mechanical systems and particle collisions, we analyze the organization of the latent space through a relevance measure that identifies the most meaningful latent directions. We show that when symmetries or approximate symmetries are present, the VAE self-organizes its latent space, effectively compressing the data along a reduced number of latent variables. This behavior captures the intrinsic dimensionality determined by the symmetry constraints and reveals hidden relations among the features. Furthermore, we provide a theoretical analysis of a simple toy model, demonstrating how, under idealized conditions, the latent space aligns with the symmetry directions of the data manifold. We illustrate these findings with examples ranging from two-dimensional datasets with $O\left(2\right)$ symmetry to realistic datasets from electron–positron and proton–proton collisions. Our results highlight the potential of unsupervised generative models to expose underlying structures in data and offer a novel approach to symmetry discovery without explicit supervision. Full article

This study proposes a new formalized approach to the stabilization of linear transformations in artificial neural networks, based on discrete algebraic properties. In contrast to existing stability methods that rely on spectral norms, regularization techniques, or empirical heuristics, this work introduces the concept of algebraic stabilization—stability that arises from the structural properties of the matrices defining linear operators. The central object of investigation is the class of integer-valued matrices for which exponentiation to a form of the type ${\mathrm{W}}^{\mathrm{k}}=\mathrm{I}+\mathsf{\mu }\mathrm{D}$ is possible, where $\mathrm{D}\in {\mathbb{Z}}^{\mathrm{n}\times \mathrm{n}},\mathsf{\mu }\in {\mathbb{Z}}_{>1}$. A well-known problem in group algebra is considered that guarantees the existence of such an exponent under the condition that $\mathsf{\mu }$ is coprime with the determinant of $\mathrm{W}$. Within this framework, modular arithmetic, reduction modulo $\mathsf{\mu }$, and the group structure of ${\mathrm{G}\mathrm{L}}_{\mathrm{n}}\left({\mathbb{Z}}_{\mathsf{\mu }}\right)$ are employed, thereby linking the proposed method to the theory of finite groups and linear automata. The advantages of the approach are discussed, including formal control over the iterative behavior of transformations, compatibility with quantized and finitely representable networks, the possibility of embedding stabilizing conditions directly into the network architecture, and the potential to improve model interpretability and reliability. At the same time, limitations are identified, particularly those related to constructive implementation, the selection of suitable hyperparameters, and generalization to broader classes of transformations. Full article

Object recognition in remote sensing images is essential for applications such as land resource monitoring, maritime vessel detection, and emergency disaster assessment. However, detection accuracy is often limited by complex backgrounds, densely distributed targets, and multiscale variations. To address these challenges, this study aims to improve the detection of small-scale and densely distributed objects in complex remote sensing scenes. An improved object detection network is proposed, called omnidirectional and adaptive YOLOv8 (OA-YOLOv8), based on the YOLOv8 architecture. Two targeted enhancements are introduced. First, an omnidirectional perception refinement (OPR) network is embedded into the backbone to strengthen multiscale feature representation through the incorporation of receptive-field convolution with a triplet attention mechanism. Second, an adaptive channel dynamic upsampling (ACDU) module is designed by combining DySample, the Haar wavelet transform, and a self-supervised equivariant attention mechanism (SEAM) to dynamically optimize channel information and preserve fine-grained features during upsampling. Experiments on the satellite imagery multi-vehicle dataset (SIMD) demonstrate that OA-YOLOv8 outperforms the original YOLOv8 by 4.6%, 6.7%, and 4.1% in terms of mAP@0.5, precision, and recall, respectively. Visualization results further confirm its superior performance in detecting small and dense targets, indicating strong potential for practical remote sensing applications. Full article

The nonholonomic constraints of fixed-wing UAVs, characterized by coupled heading-curvature feasibility and asymmetric costs, fundamentally deviate from classical Euclidean routing assumptions. While standard neural combinatorial optimization (NCO) architectures could theoretically incorporate Dubins costs via reward signals, such naive adaptations lack the capacity to explicitly model the intrinsic SE(2) geometric invariance and directional asymmetry of fixed-wing motion, leading to suboptimal generalization. To bridge this gap, we propose a Dubins-Aware NCO framework. We design a dual-channel embedding to decouple asymmetric physical distances from rotation-stable geometric features. Furthermore, we introduce a Rotary Phase Encoding (RoPhE) mechanism that theoretically guarantees strict SO(2) equivariance within the attention layer. Extensive sensitivity, ablation, and cross-distribution generalization experiments are conducted on tasks spanning varying turning radii and problem variants with instance scales of 10, 20, 36, and 52 nodes. The results consistently validate the superior optimality and stability of our approach compared with state-of-the-art DRL and NCO baselines, while maintaining significant computational efficiency advantages over classical heuristics. Our results highlight the importance of explicitly embedding geometry-physics consistency, rather than relying on scalar reward signals, for real-world fixed-wing autonomous scheduling. Full article

Surface crack detection and temporal evolution analysis are fundamental tasks in remote sensing and photogrammetry, providing critical information for slope stability assessment, infrastructure safety inspection, and long-term geohazard monitoring. However, current unmanned aerial vehicle (UAV)-based crack detection pipelines typically treat spatial detection and temporal change analysis as separate processes, leading to weak geometric consistency across time and limiting the interpretability of crack evolution patterns. To overcome these limitations, we propose the Longitudinal Crack Fitting Network (LCFNet), a unified and physically interpretable framework that achieves, for the first time, integrated time-series crack detection and evolution analysis from UAV remote sensing imagery. At its core, the Longitudinal Crack Fitting Convolution (LCFConv) integrates Fourier-series decomposition with affine Lie group convolution, enabling anisotropic feature representation that preserves equivariance to translation, rotation, and scale. This design effectively captures the elongated and oscillatory morphology of surface cracks while suppressing background interference under complex aerial viewpoints. Beyond detection, a Lie-group-based Temporal Crack Change Detection (LTCCD) module is introduced to perform geometrically consistent matching between bi-temporal UAV images, guided by a partial differential equation (PDE) formulation that models the continuous propagation of surface fractures, providing a bridge between discrete perception and physical dynamics. Extensive experiments on the constructed UAV-Filiform Crack Dataset (10,588 remote sensing images) demonstrate that LCFNet surpasses advanced detection frameworks such as You only look once v12 (YOLOv12), RT-DETR, and RS-Mamba, achieving superior performance (mAP_50:95 = 75.3%, F1 = 85.5%, and CDR = 85.6%) while maintaining real-time inference speed (88.9 FPS). Field deployment on a UAV–IoT monitoring platform further confirms the robustness of LCFNet in multi-temporal remote sensing applications, accurately identifying newly formed and extended cracks under varying illumination and terrain conditions. This work establishes the first end-to-end paradigm that unifies spatial crack detection and temporal evolution modeling in UAV remote sensing, bridging discrete deep learning inference with continuous physical dynamics. The proposed LCFNet provides both algorithmic robustness and physical interpretability, offering a new foundation for intelligent remote sensing-based structural health assessment and high-precision photogrammetric monitoring. Full article

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra $GL\left(V\right)$ built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$, with $\psi ,\varphi \in GL\left(V\right)$ and integers $i,j$. We prove that $(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\tilde{L}={\psi }^{-1}L\psi$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. Full article

Graph neural networks (GNNs) have rapidly matured into a unifying, end-to-end framework for energy-materials discovery. By operating directly on atomistic graphs, modern angle-aware and equivariant architectures achieve formation-energy errors near 10 meV atom⁻¹, sub-0.1 V voltage predictions, and quantum-level force fidelity—enabling nanosecond molecular dynamics at classical cost. In this review, we provide an overview of the basic principles of GNNs, widely used datasets, and state-of-the-art architectures, including multi-GPU training, calibrated ensembles, and multimodal fusion with large language models, followed by a discussion of a wide range of recent applications of GNNs in the rapid screening of battery electrodes, solid electrolytes, perovskites, thermoelectrics, and heterogeneous catalysts. Full article

In this paper, we establish the existence and uniqueness of the minimum intrinsic risk equivariant (MIRE) estimator for univariate linear normal models. The estimator is derived under the action of the subgroup of the affine group that preserves the column space of the design matrix, within the framework of intrinsic statistical analysis based on the squared Rao distance as the loss function. This approach provides a parametrization-free assessment of risk and bias, differing substantially from the classical quadratic loss, particularly in small-sample settings. The MIRE is compared with the maximum likelihood estimator (MLE) in terms of intrinsic risk and bias, and a simple approximate version (a-MIRE) is also proposed. Numerical evaluations show that the a-MIRE performs closely to the MIRE while significantly reducing the intrinsic bias and risk of the MLE for small samples. The proposed intrinsic methods could extend to other invariant frameworks and connect with recent developments in robust estimation procedures. Full article

We recast cooperative localization and scheduling in heterogeneous multi-agent systems through the lens of symmetry and symmetry breaking. On the geometric side, the Fisher Information Matrix (FIM) objective is invariant to rigid Euclidean transformations of the global frame, while its maximization admits symmetric optimal sensor formations; on the algorithmic side, heterogeneity and task constraints break permutation symmetry across agents, requiring policies that are sensitive to role asymmetries. We model communication as a random graph and quantify structural symmetry via topology metrics (average path length, clustering, betweenness) and graph automorphism-related indices, connecting these to estimation uncertainty. We then design a hybrid reward for reinforcement learning (RL) that is equivariant to agent relabeling within roles yet intentionally introduces asymmetry through distance/FIM terms to avoid degenerate symmetric configurations with poor observability. Simulations show that (i) symmetry-aware, FIM-optimized path planning reduces localization error versus symmetric but non-informative placements; and (ii) controlled symmetry breaking in policy learning improves robustness and data rate–reward trade-offs over baselines. Our results position symmetry/asymmetry as first-class design principles that unify estimation-theoretic invariances with learning-based coordination in complex heterogeneous networks. Under DDPG training, the total data rate (SDR) reaches $6.63\pm 0.97$ and the average reward per step (ARPS) is $-80.70\pm 6.94$, representing improvements of approximately $11.8\%$ over the baseline $(5.93\pm 3.51)$ and $11.1\%$ over SAC $(5.97\pm 2.66)$, respectively. The network’s mean shortest-path length is $L=1.721$, and the average betweenness centrality of the coordination nodes is ≈0.098. Moreover, the FIM-optimized path-planning strategy achieves the lowest localization error among all evaluated policies. Full article

The spread of computer viruses poses a critical threat to networked systems and requires accurate modeling tools. Classical integer-order approaches had failed to capture memory effects inherent in real digital environments. To address this, we developed a four-compartment fractional-order model using the Atangana–Baleanu–Caputo (ABC) derivative with Mittag-Leffler kernels. We established fundamental properties such as positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Analytical solutions were derived via Laplace transform and homotopy series, while the Variation-of-Parameters Method and a dedicated numerical scheme provided approximations. Simulation results showed that the fractional order strongly influenced infection dynamics: smaller orders delayed peaks, prolonged latency, and slowed recovery. Compared to classical models, the ABC framework captured realistic memory-dependent behavior, offering valuable insights for designing timely and effective cybersecurity interventions. The model exhibits structural symmetries: the infection flux depends on the symmetric combination $L+I$ and the feasible region (probability simplex) is invariant under the flow. Under the parameter constraint $\delta =\theta$ (and equal linear loss terms), the system is equivariant under the involution $(L,I)\mapsto (I,L)$, which is reflected in identical Hyers–Ulam stability bounds for the latent and infectious components. Full article

This paper investigates the role of symmetry and asymmetry in the learning process of modern machine learning models, with a specific focus on feature representation and optimization. We introduce a novel symmetry-aware learning framework that identifies and preserves symmetric properties within high-dimensional datasets, while allowing model asymmetries to capture essential discriminative cues. Through analytical modeling and empirical evaluations on benchmark datasets, we demonstrate how symmetrical transformations of features (e.g., rotation, mirroring, permutation invariance) impact learning efficiency, interpretability, and generalization. Furthermore, we explore asymmetric regularization techniques that prioritize informative deviations from symmetry in model parameters, thereby improving classification and clustering performance. The proposed approach is validated using a variety of classifiers including neural networks and tested across domains such as image recognition, biomedical data, and social networks. Our findings highlight the critical importance of leveraging domain-specific symmetries to enhance both the performance and explainability of machine learning systems. Full article