Construction and applications of iterative methods for finding approximate solutions of nonlinear equations having unknown zeros of multiplicity with fractal geometry and dynamical behavior.

Quantum-enhanced SMOTE: Integration of variational quantum circuits and quantum kernel methods for optimization of perovskite oxide structure prediction

A bubble-function-based variational multiscale method with two-local Gaussian integration for stationary convection-dominated diffusion equations on surfaces

This work integrates the physics-informed neural network (PINN) approach into the neural quantum state framework to simulate open quantum system dynamics and to circumvent the computationally expensive time-dependent variational principle required in conventional variational methods. The proposed PINN-DQME method employs time-encoded neural networks within a time-domain decomposition strategy to represent the evolution governed by the dissipaton-embedded quantum master equation (DQME). We implement and validate this approach in the single-impurity Anderson model, benchmarking the PINN-DQME results against the numerically exact hierarchical equations of motion. The PINN-DQME method demonstrates high accuracy in simulating quantum dissipative dynamics at high temperatures, where non-Markovian effects are weak. However, for strongly non-Markovian dynamics at low temperatures, it encounters challenges with error accumulation during time propagation, highlighting an area for future refinement in applying PINNs to complex quantum dynamical settings.

This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the L2 Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through variational methods. Analytical solutions are derived for multidimensional domains by employing the Fourier method and spectral analysis techniques. Complementing this theoretical framework, a high-accuracy numerical approach based on the Associated Legendre Polynomials Collocation Spectral Method (ALP-CSM) with Chebyshev–Gauss–Lobatto nodes is developed. Rigorous convergence analysis confirms spectral accuracy, with numerical examples in one, two, and three dimensions demonstrating error decay from O(10−3) to machine precision O(10−15). The mathematical impact of Third-Kind boundary conditions on the diffusion rate and the steady state of the system is demonstrated. The obtained results provide a robust tool for modeling physical processes, particularly in systems involving heat exchange on the surfaces of complex-structured domains, offering both theoretical insight and computational efficiency.

In this paper, we have applied two semi-analytical techniques: the Mohand variational iteration method (MVIM) and the q-homotopy Mohand transform method (q-HMTM) to derive approximate solutions of fractional-order nonlinear partial differential equations. Particularly, time-fractional FitzHugh Nagumo equation and Fisher equation based on Caputo derivative are explored. Both of them utilize well the features of Mohand transform and fractional calculus to represent the nonlocal and memory-dependent nature of the models. The validity and reliability of the suggested methods are justified by the comparison of the obtained solutions with the exact solutions known in the integer-order limit. The graphical and tabular analysis shows how the fractional order parameter ψ has a great impact on the solution profiles, and a smoothing effect and a delaying propagation effect occur as the fractional order parameter ψ reduces. In a close comparative analysis, both q-HMTM and MVIM provide highly precise results, where MVIM in some instances has a little better accuracy. The results affirm the effectiveness of these methods in addressing new complex systems of a fractional-order that occur in biological, physical, and engineering scenarios.

Intratumor heterogeneity (ITH) impacts cancer progression, and its characterization is crucial. Clustering algorithms applied to the variant allele frequency (VAF) of mutations can facilitate the exploratory analysis of ITH. This study comparatively evaluated six clustering algorithms to characterize ITH by clustering mutations based on their VAFs. We utilized data from The Cancer Genome Atlas to analyze three cancer types by examining the distribution of clusters in the results from various methods and four internal validation metrics. The results indicated that the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) and Variational Bayesian Gaussian Mixture Model methods identified an insufficient number of clusters in most tumor samples. The Hierarchical DBSCAN (HDBSCAN) and Ordering Points to Identify the Clustering Structure (OPTICS) algorithms exhibited greater variability in the number of clusters, while Affinity Propagation (AP) showed controlled behavior, and Mean-Shift demonstrated greater consistency. The Mean-Shift and AP methods were consistently superior in the validation metrics, in contrast to HDBSCAN and OPTICS, which had inferior performance. We conclude that Mean-Shift and AP are promising and accessible alternatives for the initial exploratory analysis of ITH by VAFs. A computational pipeline is provided on the Google Colab platform to facilitate future studies.

Predicting customer churn from transactional data is a central problem in management science, with direct implications for retention strategy, revenue forecasting, and resource allocation. This paper introduces Quantum Geometric-Entropic Optimization (Q-GEO), a framework that integrates Geometric-Entropic Optimization – combining Riemannian gradient methods with entropy-regularized optimal transport – into the training of variational quantum kernels for classification. The algorithm operates on a parameter manifold equipped with a Fisher-Wasserstein metric and incorporates Sinkhorn-type projections to enforce distributional coherence on the quantum feature space. We establish three theoretical contributions: (i) a convergence theorem for Q-GEO-trained variational quantum kernels under a combined Polyak–Łojasiewicz and Sinkhorn contraction framework, yielding linear convergence in the Riemannian condition number plus geometric contraction of the Sinkhorn residual; (ii) a margin amplification result showing that GEO-trained quantum embeddings achieve improved separation bounds over Euclidean-trained counterparts due to the spectral regularization provided by the Wasserstein component of the Fisher-Wasserstein metric; and (iii) a distributional stability result proving that Sinkhorn-projected quantum kernel matrices preserve a doubly stochastic spectral structure that mitigates kernel collapse in imbalanced settings. We validate the framework on the UCI Online Retail II dataset ( N = 5,942 customers, d=11 RFM-extended features, churn rate ≈ 37 % ), a publicly available transactional benchmark. Under nested cross-validation, Q-GEO achieves 0.8614 accuracy, 0.8103 precision, 0.7891 recall, 0.7996 F1, and 0.9047 ROC AUC, outperforming both classical baselines (including logistic regression, random forest, XGBoost, and CatBoost) and standard variational quantum kernel methods. We complement the accuracy analysis with SHAP-based explainability, computation time comparisons, and a detailed feature engineering appendix to support interpretability and reproducibility. We interpret these results as evidence that geometric optimization principles can materially enhance quantum machine learning for management science applications.

An efficient variational method is presented for estimating the diffusion coefficients and free-energy profiles along selected collective variables from projected molecular dynamics trajectories under both equilibrium and non-equilibrium conditions. This method is based on the assumption that the short-time transition probability density of the coordinate moves can be approximated by a Gaussian form. Defining a loss function as the sum of Kullback-Leibler divergences between the analytical short-time propagators of an overdamped Langevin model and those estimated directly from the projected trajectories maximizes the agreement between the two and allows for its analytic evaluation. For cases where the Gaussian approximation is insufficient, we present a robust alternative. To efficiently minimize this loss function by varying diffusion and free-energy profiles along collective variables, we use an adaptive Monte Carlo scheme. The method is applied to two model systems exhibiting diffusive dynamics, as well as to water diffusion across the interface of a biomolecular condensate, demonstrating its robustness and accuracy.

Variational Methods in Mechanics When Modeling Principles of Formation of Functional Systems for Biomechanics Objects

We prove the existence and multiplicity of solutions for a variable-exponent double-phase problem with parametric logistic reaction term. Combining variational and truncation methods with homological critical group theory, we prove the existence of at least one or two solutions with respect to a positive parameter, i.e., a bifurcation result. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/33/abstr.html

In this work, a time-fractional form of Richards' equation is considered to study the infiltration phenomenon in unsaturated porous media. The memory effects in the flow process are described using the Caputo-Fabrizio fractional operator. To obtain an approximate analytical solution of the governing nonlinear fractional partial differential equation, a hybrid analytical technique combining the Natural transform method and the variational iteration method is employed. The proposed Natural transform variational iteration method (NVIM) provides a rapidly convergent series solution and avoids complicated discretization or linearization procedures. A rigorous convergence and uniqueness analysis is carried out, which confirms the validity and accuracy of the proposed approach. The effectiveness and reliability of the method are demonstrated through the solution of the considered problem. The obtained solutions are illustrated through graphical representations using MATHEMATICA, where a comparison between the approximate analytical solution and the exact solution is carried out to validate the accuracy and effectiveness of the proposed method. In addition, the influence of the fractional parameter and other model parameters on the infiltration process is analyzed through graphical illustrations. The results demonstrate that the proposed approach provides accurate and efficient solutions and can serve as a useful analytical tool for solving nonlinear fractional differential equations arising in groundwater hydrology and related physical processes.

We study the following type of magnetic system \begin{equation} -(\nabla +iA(x))^2 u + V(x) u + \lambda\phi_{|u|} (x) u =|u|^{p-1}u,\ x\in \mathbb{R}^3, \tag*{(0.1)} \end{equation} where $p\in (1,2)$, $\lambda$ is a parameter, and $\phi_{|u|}(x)$ is a nonlocal convolution potential. Under suitable assumptions on the potentials $A(x)$ and $V(x)$, we show that problem (0.1) has a ground state by using variational methods. Moreover, the asymptotical behavior of ground states as $\lambda\to 0$ has also been discussed.

Broad learning system (BLS), as an innovative type of neural network, has demonstrated exceptional performance in regression tasks. Nonetheless, the majority of BLS methods, which rely on the least squares criterion, are highly sensitive to outliers and noisy data, resulting in reduced prediction accuracy. To improve the robustness of broad networks, a sparse Bayesian BLS via adaptive Lasso priors (AL-SBBLS) is proposed in this article to handle regression tasks with data contaminated by outliers and noise. Specifically, adaptive Lasso constraints are first applied to enhance the adaptive sparsity of output weights, which facilitates the automatic selection of highly correlated features. Subsequently, a multilayer Bayesian framework is constructed to provide an adaptive Lasso prior to the output weights, allowing the model for the adaptive learning of regularization factors and the estimation of probability distributions for output values, while further sparsifying the network. By selecting highly correlated features and estimating the probability distributions of output values, the impact of outliers and noise can be effectively mitigated. To effectively train the networks, corresponding optimization algorithms are designed for AL-SBLS and AL-SBBLS using the alternating direction method of multipliers (ADMMs) and variational Bayesian inference methods, respectively. The effectiveness and robustness of the proposed methods are validated through robust regression experiments on 14 real-world datasets and complex nonlinear data. Quantitative results demonstrate that the proposed AL-SBBLS achieves the best performance on most datasets, attaining the lowest average ranking of 1.44 in Friedman tests compared with 11 state-of-the-art BLS variants, which confirms its superior predictive accuracy and robustness. The resource code of AL-SBBLS proposed in this article is available at: https://github.com/taocheny/AL-SBBLS.

• Introduces VSPYCT, a variational Bayesian oblique decision tree for struc- tured output prediction. • Captures parameter uncertainty through variational inference in each tree split. • Matches or outperforms SPYCT ensembles on classification and multi-target regression tasks. • Provides feature importance scores and visual interpretability within a single-tree framework. • Demonstrates robustness to spurious features and performs well across varied dataset properties. Oblique predictive clustering trees (SPYCTs) are semi-supervised multi-target prediction models mainly used for structured output prediction (SOP) problems. They are computationally efficient and when combined in ensembles they achieve state-of-the-art results. However, one major issue is that it is challenging to interpret an ensemble of SPYCTs without the use of a model-agnostic method. We propose variational oblique predictive clustering trees, which address this challenge. The parameters of each split node are treated as random variables, described with a probability distribution, and they are learned through the Variational Bayes method. We evaluate the model on several benchmark datasets of different sizes. The experimental analyses show that a single variational oblique predictive clustering tree (VSPYCT) achieves competitive, and sometimes better predictive performance than the ensemble of standard SPYCTs. We also present a method for extracting feature importance scores from the model. Finally, we present a method to visually interpret the model’s decision making process through analysis of the relative feature importance in each split node.

On the 1D homogeneous and localized solutions of variational phase field method for pressurized fracture

Abstract In this paper, we investigate the stability of a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. We employ variational methods to prove the stability in the functional inequality setting. Moreover, overcoming the absence of an explicit extremal function, we employ the asymptotic behavior of extremal functions to calculate some crucial estimates. By utilizing the finite‐dimensional reduction method, we establish a sharp stability result in the critical points setting.

Quantifying shear lag effects (SLE) in tapered box girders with steel trapezoidal corrugated webs (BGSTCWs) is crucial for improving design accuracy and structural safety, especially as these girders are increasingly utilized in long-span bridges. However, existing analytical methods exhibit significant limitations in predicting SLE in tapered BGSTCWs. The energy variational method (EVM), although theoretically rigorous, faces practical challenges of incorporating warping displacement functions in girders with complex geometries and material variations. Similarly, the analogy bar method (ABM), though effective for prismatic BGSTCWs, fails to accurately capture shear flow in tapered counterparts due to significant differences in shear transfer mechanisms governed by the Resal effect. These limitations are addressed by proposing a modified analogy bar method (MABM) that explicitly incorporates Resal effect into SLE analysis for tapered BGSTCWs. The MABM derives the equivalent stiffening bar area and shear flow distribution considering cross-sectional variability, given the subsequent establishment and analytical solution of the governing differential equations under prescribed boundary conditions. The feasibility of the proposed MABM is verified through finite element (FE) simulations and experimental tests conducted on a tapered cantilever BGSTCWs given concentrated tip load. At the fixed end of tapered cantilever BGSTCWs, where the Resal effect is significantly intensified given maximum hogging moments, the MABM improves SLE prediction accuracy by 21.36% and 15.80% in the top and bottom slabs, respectively, compared to the conventional ABM. Additionally, this study systematically examines how geometric tapered configurations (regular versus reverse tapered) govern the development of either positive or negative Resal effects, which in turn directly determine the corresponding positive or negative SLE in the structural response. The degree of the SLE in tapered BGSTCWs is found to be directly influenced by the intensity of the Resal effect.

This paper aims to integrate the Laplace transformation method with the variational iteration method to deliver an analytical approximate solution for fractional-order integro-differential equations, where the fractional-order derivative and integration are defined in the conformable sense. The iterative solution sequence is obtained using the Laplace variational iteration method, and the convergence of this sequence of approximate solutions to the exact solution is established and demonstrated. First, we shall study the approximate solution of a linear fractional integro-differential equation, and secondly, solve the nonlinear fractional integro-differential equations modeled using conformable differointegration. Some illustrative examples are considered to verify the validity and accuracy of the proposed technique, in which approximate solutions are compared with the exact solutions if they exist. Through the comparison, we conclude that the present hybrid approach is very effective for solving this type of problem.

Abstract A primary challenge in multi-radar three-dimensional (3D) wind field retrieval is the significant error in the vertical wind component. To address this issue, this study derived theoretical models and simulated wind fields to analyze the causes of errors during the retrieval process, revealing the primary sources of retrieval errors and their correlation with retrieval height. Consequently, a height-dependent filtering strategy is proposed to optimize radar observational data for error suppression. Validation experiments utilizing a network of six phased array radars in Guangzhou, China, indicate that the proposed method reduces the Root Mean Square Error (RMSE) of direct retrieval by 52.84%–69.11% within the effective coverage area (0–10 km). Furthermore, accuracy improvements of 15.41%–30.94% are observed in the boundary layer, even when applied to physically constrained variational assimilation methods. These findings demonstrate that the proposed strategy is a critical preprocessing step, independent of the specific retrieval method, for obtaining accurate 3D wind fields, thereby improving meteorological analysis and forecasting capabilities.