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Machine Learning and Statistical Learning with Applications
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Machine Learning and Statistical Learning with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 January 2026) | Viewed by 8848

Special Issue Editors

Prof. Dr. Guang Lin

E-Mail Website
Guest Editor
Departments of Mathematics/Mechanical Engineering/Statistics (Courtesy)/Earth, Atmospheric, and Planetary Sciences (Courtesy), Purdue University, West Lafayette, IN 47907, USA
Interests: machine learning; uncertainty quantification; big data analysis; scientific computing
Dr. Zecheng Zhang

E-Mail Website
Guest Editor
Department of Mathematics, Florida State University (FSU), Tallahassee, FL, USA
Interests: multiscale modeling and simulation; mathematics of machine learning; scientific machine learning
Dr. Christian Moya

E-Mail Website
Guest Editor
Departments of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Interests: machine learning; control systems

Special Issue Information

Dear Colleagues, 

With the rapid advancement of artificial intelligence, machine learning, and statistical learning, high-dimensionality, big data, data imbalance, and out-of-distribution data have posed significant challenges for academic and industrial applications. Artificial intelligence models based on machine learning (ML) and statistical learning (SL) are employed in analyzing data. ML methods play significant roles in many research directions. Various machine learning technologies have been developed in diverse application domains. Such technology has solved numerous complex engineering and science problems. Machine learning is one of the fastest-growing active research areas. The Special Issue aims to have a collection of recent advances in machine learning. This Special Issue on "Machine Learning and Statistical Learning with Applications" will focus on publishing high-quality original research studies that address challenges in machine learning and statistical learning and their applications in science and engineering. Topics include but are not limited to the following:

  • ML and SL model algorithm developments;
  • ML and SL applications for predictive science and engineering;
  • Physics-informed neural network model development and applications;
  • Operator learning model development and applications;
  • ML algorithms and approaches to handling out-of-distribution, data imbalance,  data fusion, etc.;
  • Federated learning algorithm development and applications;
  • Differential privacy-based ML algorithm development and applications;
  • Uncertainty quantification for ML and SL algorithms and applications;
  • Large-language model development and applications.

Prof. Dr. Guang Lin
Dr. Zecheng Zhang
Dr. Christian Moya
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 250 words) can be sent to the Editorial Office for assessment.

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

  • ML and SL model algorithm developments
  • ML and SL applications for predictive science and engineering
  • physics-informed neural network model development and applications
  • operator learning model development and applications
  • ML algorithms and approaches to handling out-of-distribution, data imbalance, data fusion, etc.
  • federated learning algorithm development and applications
  • differential privacy-based ML algorithm development and applications
  • uncertainty quantification for ML and SL algorithms and applications
  • large-language model development and applications

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (5 papers)

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Research

17 pages, 2360 KB  
Article
Smart Meter Low Battery Voltage Status Assessment Driven by Knowledge and Data
by Wenao Liu, Xia Xiao, Zhengbo Zhang and Yihong Li
Mathematics 2026, 14(6), 1038; https://doi.org/10.3390/math14061038 - 19 Mar 2026
Abstract
As a key metering device in the smart grid, the clock battery status of smart meters directly affects the operational efficiency and economy of the grid. In response to the limitations of current evaluation methods in feature correlation analysis and model interpretability, this [...] Read more.
As a key metering device in the smart grid, the clock battery status of smart meters directly affects the operational efficiency and economy of the grid. In response to the limitations of current evaluation methods in feature correlation analysis and model interpretability, this study proposes a knowledge-and-data-driven low battery voltage status prediction method. We systematically dissected the physical mechanisms underlying battery undervoltage faults and constructed a status features knowledge graph comprising 17 state features across four dimensions. By employing Pearson correlation analysis and association rule mining techniques, we achieved a quantitative correlation analysis between multi-source heterogeneous features and battery status. Building on this foundation, we developed an interpretable model framework based on XGBoost-SHAP. Empirical studies utilized a dataset of 939,000 faulty meters recalled by a provincial power company in 2023, with 9.87% of outlier samples eliminated using the Isolation Forest algorithm during preprocessing. Results demonstrate that the proposed model achieved an R2 of 0.851 and a Mean Squared Error (MSE) of 0.0088 on the test set. The prediction performance significantly surpassed that of Random Forest (R2 = 0.692) and MLP+BP neural networks (R2 = 0.583), thereby validating the effectiveness of the approach in combining predictive accuracy with decision transparency. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)
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25 pages, 1954 KB  
Article
RMFGP: A Rotated Multi-Fidelity Gaussian Process Framework for Supervised Dimension Reduction
by Jiahao Zhang, Shiqi Zhang and Guang Lin
Mathematics 2026, 14(2), 325; https://doi.org/10.3390/math14020325 - 18 Jan 2026
Viewed by 314
Abstract
High-dimensional surrogate modeling with limited high-fidelity data poses a major challenge in uncertainty quantification. Classical supervised dimension reduction methods often fail in this setting due to insufficient accurate observations, while low-fidelity data are abundant but biased. In this work, we propose a Rotated [...] Read more.
High-dimensional surrogate modeling with limited high-fidelity data poses a major challenge in uncertainty quantification. Classical supervised dimension reduction methods often fail in this setting due to insufficient accurate observations, while low-fidelity data are abundant but biased. In this work, we propose a Rotated Multi-Fidelity Gaussian Process (RMFGP) framework that enables reliable dimension reduction and surrogate construction under severe data scarcity. The proposed method integrates nonlinear multi-fidelity Gaussian process regression with sliced average variance estimation (SAVE) to iteratively identify informative input directions. Low-fidelity data are first used to extract coarse structural information, which is exploited to rotate the input space prior to multi-fidelity model training. Predictions generated by the trained RMFGP surrogate are then used to refine the dimension reduction, allowing accurate estimation of the central sufficient dimension reduction subspace even when high-fidelity data are scarce. A Bayesian active learning strategy based on predictive uncertainty is further incorporated to adaptively select new high-fidelity samples. Numerical examples, including stochastic partial differential equations, demonstrate that RMFGP significantly improves prediction accuracy, convergence, and uncertainty propagation compared to existing Gaussian process-based dimension reduction approaches, while requiring substantially fewer high-fidelity evaluations. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)
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17 pages, 343 KB  
Article
Gaussian Process Regression with Soft Equality Constraints
by Didem Kochan and Xiu Yang
Mathematics 2025, 13(3), 353; https://doi.org/10.3390/math13030353 - 22 Jan 2025
Cited by 3 | Viewed by 3375
Abstract
This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of [...] Read more.
This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of distributions by allowing a particle’s mass matrix to vary according to a probability distribution. By integrating QHMC into the GP regression with probabilistic handling of the constraints, this approach balances the computational cost and accuracy in the resulting GP model, as the probabilistic nature of the method contributes to shorter execution times compared with existing GP-based approaches. Additionally, we introduce an adaptive learning algorithm to optimize the selection of constraint locations to further enhance the flexibility of the method. We demonstrate the effectiveness and robustness of our algorithm on synthetic examples, including 2-dimensional and 10-dimensional GP models under noisy conditions, as well as a practical application involving the reconstruction of a sparsely observed steady-state heat transport problem. The proposed approach reduces the posterior variance in the resulting model, achieving stable and accurate sampling results across all test cases while maintaining computational efficiency. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)
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28 pages, 481 KB  
Article
Convergence Analysis for an Online Data-Driven Feedback Control Algorithm
by Siming Liang, Hui Sun, Richard Archibald and Feng Bao
Mathematics 2024, 12(16), 2584; https://doi.org/10.3390/math12162584 - 21 Aug 2024
Cited by 1 | Viewed by 1633
Abstract
This paper presents convergence analysis of a novel data-driven feedback control algorithm designed for generating online controls based on partial noisy observational data. The algorithm comprises a particle filter-enabled state estimation component, estimating the controlled system’s state via indirect observations, alongside an efficient [...] Read more.
This paper presents convergence analysis of a novel data-driven feedback control algorithm designed for generating online controls based on partial noisy observational data. The algorithm comprises a particle filter-enabled state estimation component, estimating the controlled system’s state via indirect observations, alongside an efficient stochastic maximum principle-type optimal control solver. By integrating weak convergence techniques for the particle filter with convergence analysis for the stochastic maximum principle control solver, we derive a weak convergence result for the optimization procedure in search of optimal data-driven feedback control. Numerical experiments are performed to validate the theoretical findings. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)
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19 pages, 788 KB  
Article
Quadrature Based Neural Network Learning of Stochastic Hamiltonian Systems
by Xupeng Cheng, Lijin Wang and Yanzhao Cao
Mathematics 2024, 12(16), 2438; https://doi.org/10.3390/math12162438 - 6 Aug 2024
Cited by 1 | Viewed by 2109
Abstract
Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we [...] Read more.
Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error εH arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)
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