Machine Learning and Statistical Learning with Applications

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

Deadline for manuscript submissions: 31 May 2025 | Viewed by 717

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 (1 paper)

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Research

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 354
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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