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Mathematics
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Modeling and Numerical Analysis with Neural Networks
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Modeling and Numerical Analysis with Neural Networks

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 2391

Special Issue Editors

Prof. Dr. Roberto Alejo Eleuterio

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Guest Editor
Division of Postgraduate Studies and Research, National Technological of Mexico, Campus Toluca, Metepec 52149, Mexico
Interests: big data; deep learning; sampling methods; neural networks
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Vicente García

E-Mail Website
Guest Editor
División Multidisciplinaria en Ciudad Universitaria, Universidad Autónoma de Ciudad Juárez, Av. José de Jesús Delgado 18100, Ciudad Juárez 32310, Chihuahua, Mexico
Interests: big data classification; meta-learning; class imbalance; time series; ensembles, neural networks
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Rosa María Valdovinos-Rosas

E-Mail Website
Guest Editor
Faculty of Engineering, Universidad Autónoma del Estado de México, Cerro de Coatepec s/n, Ciudad Universitaria, Toluca 50100, Mexico
Interests: big data classification; machine learning; neural Networks

Special Issue Information

Dear Colleagues,

We invite researchers and scholars to contribute to this Special Issue of Mathematics on "Modeling and Numerical Analysis with Neural Networks". This Issue aims to promote the application of contemporary neural networks to solving real-world challenges across various scientific disciplines, including engineering, social sciences, chemistry, medicine, business, life sciences, and others. Traditionally, problems in these fields have been approached through techniques such as function approximation, simulation, estimation, iterative methods, and differential equations. However, the emergence of artificial neural networks has provided an alternative mathematical and computational framework for numerical analysis, and modeling in the field of mathematics. We encourage researchers to submit studies that propose innovative methods and applications of neural networks in numerical analysis and modeling. We are particularly interested in publishing manuscripts that address complex real-world problems in science and engineering; moreover, whether you have developed new neural network architectures or have conducted comprehensive surveys in this field, we welcome your contributions. Join us in exploring the potential of neural networks as a powerful tool for solving numerical problems and advancing scientific research. 

Prof. Dr. Roberto Alejo Eleuterio
Prof. Dr. Vicente García
Prof. Dr. Rosa María Valdovinos-Rosas
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • neural networks
  • prediction
  • optimization
  • engineering applications
  • scientific areas

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Published Papers (2 papers)

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Research

21 pages, 1337 KiB  
Article
Automatic Differentiation-Based Multi-Start for Gradient-Based Optimization Methods
by Francesco Della Santa
Mathematics 2024, 12(8), 1201; https://doi.org/10.3390/math12081201 - 17 Apr 2024
Viewed by 789
Abstract
In global optimization problems, diversification approaches are often necessary to overcome the convergence toward local optima. One approach is the multi-start method, where a set of different starting configurations are taken into account to designate the best local minimum returned by the multiple [...] Read more.
In global optimization problems, diversification approaches are often necessary to overcome the convergence toward local optima. One approach is the multi-start method, where a set of different starting configurations are taken into account to designate the best local minimum returned by the multiple optimization procedures as the (possible) global optimum. Therefore, parallelization is crucial for multi-start. In this work, we present a new multi-start approach for gradient-based optimization methods that exploits the reverse Automatic Differentiation to perform efficiently. In particular, for each step, this Automatic Differentiation-based method is able to compute the N gradients of N optimization procedures extremely quickly, exploiting the implicit parallelization guaranteed by the computational graph representation of the multi-start problem. The practical advantages of the proposed method are illustrated by analyzing the time complexity from a theoretical point of view and showing numerical examples where the speed-up is between ×40 and ×100, with respect to classic parallelization methods. Moreover, we show that our AD-based multi-start approach can be implemented by using tailored shallow Neural Networks, taking advantage of the built-in optimization procedures of the Deep Learning frameworks. Full article
(This article belongs to the Special Issue Modeling and Numerical Analysis with Neural Networks)
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16 pages, 1160 KiB  
Article
Solving the Electronic Schrödinger Equation by Pairing Tensor-Network State with Neural Network Quantum State
by Bowen Kan, Yingqi Tian, Daiyou Xie, Yangjun Wu, Yi Fan and Honghui Shang
Mathematics 2024, 12(3), 433; https://doi.org/10.3390/math12030433 - 29 Jan 2024
Viewed by 1061
Abstract
Neural network methods have shown promise for solving complex quantum many-body systems. In this study, we develop a novel approach through incorporating the density-matrix renormalization group (DMRG) method with the neural network quantum state method. The results demonstrate that, when tensor-network pre-training is [...] Read more.
Neural network methods have shown promise for solving complex quantum many-body systems. In this study, we develop a novel approach through incorporating the density-matrix renormalization group (DMRG) method with the neural network quantum state method. The results demonstrate that, when tensor-network pre-training is introduced into the neural network, a high efficiency can be achieved for quantum many-body systems with strong correlations. Full article
(This article belongs to the Special Issue Modeling and Numerical Analysis with Neural Networks)
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