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: 31 May 2025 | Viewed by 2998

Special Issue Editors


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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

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Guest Editor
Department of Mathematics, Florida State University (FSU), Tallahassee, FL, USA
Interests: multiscale modeling and simulation; mathematics of machine learning; scientific machine learning

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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

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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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Published Papers (3 papers)

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Research

17 pages, 343 KiB  
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
Viewed by 469
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 KiB  
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
Viewed by 774
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 KiB  
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
Viewed by 907
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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