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Representation and regression problems in neural networks: Relaxation, generalization, and numerics

MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 2025年第6期35卷 1471-1521页

作者： Liu, Kang Zuazua, Enrique Univ Bourgogne Europe Inst Math Bourgogne CNRS F-21000 Dijon France Friedrich Alexander Univ Erlangen Nurnberg Dept Math D-91058 Erlangen Germany Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain Fdn Deusto Bilbao Bilbao 48007 Basque Country Spain

In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets, analyzing the impact of key hyperparameters on these bounds, and proposing optimal choices. On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problems.

关键词： Neural network mean-field relaxation representer theorem generalization numerical algorithm

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Efficient simulation of mixed boundary value problems and conformal mappings

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APPLIED MATHEMATICS AND COMPUTATION 2025年 488卷

作者： Han, Qiansheng Rasila, Antti Sottinen, Tommi Guangdong Technion Israel Inst Technol Dept Math Comp Sci Shantou 515063 Guangdong Peoples R China Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel Univ Vaasa Sch Technol & Innovat POB 700 FIN-65101 Vaasa Finland

We present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on- Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an Lshaped insulated polyhedron.

关键词： Brownian motion Monte Carlo methods Walk on spheres Harmonic measure Conformal mappings Conformal modulus numerical algorithm Heat flow

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Description of the Lubricant Behavior Based on the Theory of Elasto-Viscoplastic

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MATERIALS 2025年第6期18卷 1360-1360页

作者： Nosov, Yuriy O. Kamenskikh, Anna A. Bogdanova, Anastasia P. Perm Natl Res Polytech Univ Dept Computat Math Mech & Biomech Perm 614990 Russia

The aim of the work is to provide a mathematical description of the lubricant's behavior model used in sliding bearings of bridge structures. It was previously established that the Maxwell model does not correctly describe the lubricant's behavior in a wide range of temperatures and deformation rates. The lubricant model should take into account not only viscosity but also plasticity. The Anand model, which was adapted by introducing temperature dependencies for a number of material parameters, was chosen to describe the lubricant behavior. The functionality of the previously created procedure for identifying material properties was also expanded on the modified Anand model. This made it possible to obtain a lubricant mathematical model with an error of less than 5% in the operating temperature range from -40 to +80 degrees C. The study included a description of the behavior model for two lubricants: CIATIM-221 and CIATIM-221F. CIATIM-221F differs from CIATIM-221 by including superfine particles of polytetrafluoroethylene (PTFE) to improve properties. The study confirmed that the modified Anand model allows describing the material behavior more accurately than the Maxwell model. It was found that the samples behave as a solid over the entire temperature range (from -40 to +80 degrees C). A comparative analysis of the thermal behavior of CIATIM-221 and CIATIM-221F was performed.

关键词： numerical algorithm mathematical model Anand model lubricant thermophysical properties computer-aided design systems

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Cyclic cohesive zone modeling for fatigue prediction of concrete-concrete interface in CRTS III slab ballastless track: Theory, algorithm, validation, and application

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ENGINEERING STRUCTURES 2025年 333卷

作者： Yan, Daobin Shi, Zixuan Xu, Yude Tongji Univ Key Lab Rd & Traff Engn Minist Educ Shanghai 201804 Peoples R China Tongji Univ Shanghai Key Lab Rail Infrastruct Durabil & Syst S Shanghai 201804 Peoples R China

This paper presents a cyclic cohesive zone model (CCZM) for predicting the interfacial fatigue of concrete- concrete composites in the CRTS III slab ballastless track subjected to high-cycle fatigue loading. The mathematical formulation of the CCZM is detailed, in which the static and the fatigue damage variables representing the interfacial fatigue evolution are derived based on a mixed-mode bilinear cohesive zone model. The numerical algorithm of the CCZM, combined with a valid cycle jump strategy, is programmed in Fortran as a user-defined material subroutine in Abaqus. The predictive capacity of the CCZM under Mode I, Mode II, and Mode I/II loading conditions is validated against interfacial fatigue benchmarks, including the double-cantilever beam (DCB), end-notched flexure (ENF), and single-leg bending (SLB) tests. Furthermore, the CCZM is applied in a finite element (FE) analysis of the interfacial fatigue at the concrete-concrete interface between the track-slab and self-compacting concrete layer in the CRTS III slab ballastless track under cyclic temperature gradient loading, and the results show that the interfacial fatigue may reach level-I and level-II damage within 11 and 15 years, respectively, indicating a potential need for debonding maintenance.

关键词： CRTS III slab ballastless track Concrete-concrete composites Interfacial bonding High-cycle fatigue Cyclic cohesive zone model numerical algorithm

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numerical algorithmS FOR MOORE-PENROSE INVERSE OF A MATRIX - ITERATIVE METHODS

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1972年第3期24卷 621-629页

作者： SHINOZAK.N SIBUYA, M TANABE, K KEIO UNIV FAC ENGNTOKYOJAPAN INST STATISTICAL MATH MINATOTOKYO 106JAPAN IBM CORP SCI CTRTOKYOJAPAN

The iterative methods by Ben-Israel and others for computing the Moore-Penrose inverse of a matrix are examined. Ill conditioned test matrices are inverted by the methods and some difficulties are found out. The itera... 详细信息

关键词： Iterative Method numerical algorithm Generalize Inverse Zero Element Dual algorithm

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The Split Quaternion Least Squares Problem With Equality Constraint and Its Application

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年

作者： Zhou, Zhi-Han Meng, Zi-Xiang Li, Ying Zhang, Feng-Xia Liaocheng Univ Coll Math & Sci Liaocheng Peoples R China

By applying the real representation matrix and the Moore-Penrose inverse of the split quaternion matrix, we first study the split quaternion least squares problem with equality constraint (SQLSE) and obtain its general solution and minimum norm solution. Then, we obtain the Karush-Kuhn-Tucker equation, which is an equivalent problem of the SQLSE problem. Immediately after, we propose a numerical algorithm on the real number field for the minimum norm solution of the SQLSE problem and enumerate three examples to display its validity. Eventually, we provide an application of the SQLSE problem in the color image restoration.

关键词： color image restoration equality constraint numerical algorithm real representation matrix split quaternion least squares problem

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Dynamics of solar radiation absorption for Darcy-Forchheimer Carreau nanofluid considering variable conductivity with chemical reaction: A solar thermal energy application

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RESULTS IN ENGINEERING 2025年 26卷

作者： Rashid, Madiha Irfan, Muhammad Muhammad, Taseer Abu-Jrai, A. Anwar, Muhammad Shoaib Skyline Univ Coll Dept Gen Educ Sharjah U Arab Emirates PMAS Arid Agr Univ Dept Math Rawalpindi 46300 Pakistan King Khalid Univ Coll Sci Dept Math Abha Saudi Arabia Appl Sci Univ ASU Coll Engn Al Eker Bahrain Univ Jhang Dept Math Gojra Rd Jhang 35200 Pakistan

Background: Recently the crucial challenge in developing the uses of nanofluids exists in confirming their constancy over comprehensive phases. Furthermore, the appropriateness of nanofluid is subject to comprehensively on particular situations and desires and enactment of nanofluid is pointedly reliant on functioning factors. Fluctuations in functioning situations can harmfully influence the efficiency of nanofluids, with thermophoresis affectation a noteworthy task. The nanofluids mainly used for thermal targets, understanding quick constancy decrease because of thermophoresis. It is essential to wisely measure temperature gradients and pressure falls to raise thermal enactment. Also, scrupulous fusion of nanofluids is significant to check contagion and confirm actual thermal enactment. The modern growth in the use of nano-particles and their nanofluids in surplus heat regaining, fuel chambers, astrophysical plates, storm, hydro-power turbines and geo-thermal dynamism. Objective: The present effort elaborates the Darcy-Forchheimer concept in magneto hydrodynamics Carreau nanofluid considering the phenomena of variable conductivity and radiative heat flux. The Buongiorno theory has been working to scrutinize Brownian diffusion and thermophoresis properties. The chemical reaction in 3D Carreau porous surface is also presented. Methodology: The bvp4c algorithm has been exploited after altering the nNar structure of ordinary differential equations into nonlinear partial differential equations. The bvp4c approach is a finite difference code that adopted the Lobatto IIIa (three-stage) procedure. For the attaining solution the algorithm needs to describe the differential equations and boundary conditions along with initial estimate. Furthermore, bvp4c provides trustworthy solutions even for inflexible difficulties and multifaceted boundary conditions. Computational results: The temperature field exaggerates for higher values of porosity and Forchheimer factors;however, d

关键词： Darcy Forchheimer Carreau Nanofluid Thermal conductivity Radiative heat flux numerical algorithm

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Radiative Darcy-Forchheimer hybrid nanofluid flow with thermal and bioconvection impacts over a stretching/shrinking sheet

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JOURNAL OF RADIATION RESEARCH AND APPLIED SCIENCES 2025年第1期18卷

作者： Eladeb, Aboulbaba Sambas, Aceng Bilal, Muhammad Alshammari, Badr M. Kolsi, Lioua Northern Border Univ Coll Engn Dept Chem & Mat Engn POB 1321 Ar Ar Saudi Arabia Univ Sultan Zainal Abidin Fac Informat & Comp Besut Campus Besut 22200 Terengganu Malaysia Univ Muhammadiyah Tasikmalaya Dept Mech Engn Tasikmalaya 46196 Indonesia Univ Sultan Zainal Abidin Artificial Intelligence Sutainabil & Islamic Res C Gongbadak 21300 Terengganu Malaysia Univ Peshawar Dept Math Sheikh Taimur Acad Block II Khyber Pakhtunkhwa 25120 Pakistan Univ Hail Coll Engn Dept Elect Engn Hail City 81451 Saudi Arabia Univ Hail Coll Engn Dept Mech Engn Hail City 81451 Saudi Arabia

The magnetohydrodynamics hybrid nanofluid (HNF) flow across a stretching & shrinking sheet is numerically studied. The HNF is synthesized by the scattering of SiO2 and MoS2 nanoparticles (NPs) in the ethylene glycol (EG). The impact of thermal radiation, surface permeability, heat absorption/generation, aligned magnetic field, and convective condition are also considered. The present research endeavors to discourse the research gap and offer remarkable perceptions of the above particular physical factors. The governing equations of HNF are renovated into the nonlinear dimensionless differential equations, which are numerically cracked through the well-known parametric continuation approach (PCA). The range of physical flow parameters on the flow rate, thermal field, and mass distributions have been considered according to the better convergence of the solution. From tabular and graphical results, it can be determined that the flow rate of HNF declines with the effect of the magnetic field and the rising number of SiO2 and MoS2 NPs in the EG. Furthermore, the outcome of bioconvective Schmidt number and Peclet number declines the density of microorganism distribution. The fluid velocity boosts with the effect of rotation parameter, while declines with consequences of the aligned magnetic field.

关键词： 3D spinning flow Hybrid nanofluid Forchheimer medium Aligned magnetic field numerical algorithm Stretching sheet

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Physics-based numerical implementation framework towards multi-scale contact problem

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TRIBOLOGY INTERNATIONAL 2025年 202卷

作者： Yang, Tao Tang, Xiongfeng Yan, Zhixue Wang, Guoqing Zhao, Gai Peng, Hanmin Nanjing Univ Aeronaut & Astronaut State Key Lab Mech & Control Aerosp Struct Nanjing 210016 Peoples R China

This paper establishes a general computational framework to solve the muti-scale contact problem by integrating the statistical contact model with the finite element format. Compared to existing models, the proposed method is applicable to most geometric configurations and can effectively evaluate the pressure distribution. In this work, a modified Karush-Kuhn-Tucker (KKT) condition is proposed by the assumption that asperity height obeys the Gaussian distribution. Therefore, in the variational formula, the contact contribution is decomposed into body contribution and asperity contribution, corresponding to the nominal smooth surface and roughness, respectively. Then the linearization and constraint enforcement of these two components are derived, followed by a nonlinear Newton-Raphson-based iterative algorithm. The contact patch test and Hertz contact test are conducted, and the predicted results are consistent with the theoretical and experimental values, confirming the effectiveness and accuracy of the proposed approach. It is worth noting that in the Hertz contact test, the contact pressure distribution varies progressively with the roughness level and external force, tending to the Hertz limit or Gaussian limit. This means that the proposed method can be applied to any roughness and load. Finally, the contact behaviors of the transmission interface of a piezoelectric actuator, i.e., a typical multi-scale contact problem, are studied as an engineering application case.

关键词： Roughness Asperity contact numerical algorithm Multi-scale modelling

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Comparative Analysis of Caputo and Variable-Order Fractional Derivative algorithms Across Various Applications

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International Journal of Applied and Computational Mathematics 2025年第3期11卷 1-40页

作者： Amilo, David Sadri, Khadijeh Hincal, Evren Mathematics Research Center Near East University Nicosia 99138 Turkey Department of Mathematics Near East University Nicosia 99138 Turkey Faculty of Art and Science University of Kyrenia Kyrenia Turkey

Fractional derivative models have emerged as powerful tools for describing complex dynamics in various fields. However, selecting the appropriate fractional derivative method for specific applications remains a challenge due to the diverse nature of real-world systems. This paper presents a comprehensive comparison between the Caputo fractional derivative and the variable-order fractional derivative methods for multiple applications. By exploring diverse examples, we illustrate the strengths and limitations of each approach in modeling real-world phenomena. numerical simulations are conducted using the Grunwald–Letnikov method to approximate the solutions. In addition, we propose a new algorithm for comparing and implementing fixed and variable-order fractional derivatives efficiently. The motivation for this study stems from the growing demand for accurate, adaptable, and computationally efficient tools to model systems with memory and dynamic changes. The results reveal distinct differences in performance and accuracy, highlighting the scenarios in which each method excels. This comparative study aims to provide information for researchers and practitioners in selecting the appropriate fractional derivative model and algorithm for their specific applications. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2025.

关键词： Caputo derivative operator Mathematical model numerical algorithm Variable-order derivative operator

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