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Nonlinear Model Reduction by Probabilistic Manifold Decomposition.

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Title: Nonlinear Model Reduction by Probabilistic Manifold Decomposition.
Authors: Guo, Jiaming1 (AUTHOR) 2411955@tongji.edu.cn, Xiao, Dunhui2 (AUTHOR) xiaodunhui@tongji.edu.cn
Source: SIAM Journal on Scientific Computing. 2026, Vol. 48 Issue 1, pA209-A235. 27p.
Subjects: Reduced-order models, Manifolds (Mathematics), Distributed parameter systems, Prediction models, Optimization algorithms, Dimensional reduction algorithms, Flow simulations
Abstract: This paper presents a novel nonlinear model reduction method: probabilistic manifold decomposition (PMD), which provides a powerful framework for constructing nonintrusive reduced-order models by embedding a high-dimensional system into a low-dimensional probabilistic manifold and predicting the dynamics. Through explicit mappings, PMD captures both linearity and nonlinearity of the system. A key strength of PMD lies in its predictive capabilities, allowing it to generate stable dynamic states based on embedded representations. The method also offers a mathematically rigorous approach to analyze the convergence of linear feature matrices and low-dimensional probabilistic manifolds, ensuring that sample-based approximations converge to the true data distributions as sample sizes increase. These properties, combined with its computational efficiency, make PMD a versatile tool for applications requiring high accuracy and scalability, such as fluid dynamics simulations and other engineering problems. By preserving the geometric and probabilistic structures of the high-dimensional system, PMD achieves a balance between computational speed, accuracy, and predictive capabilities, positioning itself as a robust alternative to the traditional model reduction methods. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
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  Data: Nonlinear Model Reduction by Probabilistic Manifold Decomposition.
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  Data: <searchLink fieldCode="AR" term="%22Guo%2C+Jiaming%22">Guo, Jiaming</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> 2411955@tongji.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Xiao%2C+Dunhui%22">Xiao, Dunhui</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> xiaodunhui@tongji.edu.cn</i>
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  Data: <searchLink fieldCode="DE" term="%22Reduced-order+models%22">Reduced-order models</searchLink><br /><searchLink fieldCode="DE" term="%22Manifolds+%28Mathematics%29%22">Manifolds (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Distributed+parameter+systems%22">Distributed parameter systems</searchLink><br /><searchLink fieldCode="DE" term="%22Prediction+models%22">Prediction models</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Dimensional+reduction+algorithms%22">Dimensional reduction algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Flow+simulations%22">Flow simulations</searchLink>
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  Data: This paper presents a novel nonlinear model reduction method: probabilistic manifold decomposition (PMD), which provides a powerful framework for constructing nonintrusive reduced-order models by embedding a high-dimensional system into a low-dimensional probabilistic manifold and predicting the dynamics. Through explicit mappings, PMD captures both linearity and nonlinearity of the system. A key strength of PMD lies in its predictive capabilities, allowing it to generate stable dynamic states based on embedded representations. The method also offers a mathematically rigorous approach to analyze the convergence of linear feature matrices and low-dimensional probabilistic manifolds, ensuring that sample-based approximations converge to the true data distributions as sample sizes increase. These properties, combined with its computational efficiency, make PMD a versatile tool for applications requiring high accuracy and scalability, such as fluid dynamics simulations and other engineering problems. By preserving the geometric and probabilistic structures of the high-dimensional system, PMD achieves a balance between computational speed, accuracy, and predictive capabilities, positioning itself as a robust alternative to the traditional model reduction methods. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/25M1738863
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      – Code: eng
        Text: English
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        PageCount: 27
        StartPage: A209
    Subjects:
      – SubjectFull: Reduced-order models
        Type: general
      – SubjectFull: Manifolds (Mathematics)
        Type: general
      – SubjectFull: Distributed parameter systems
        Type: general
      – SubjectFull: Prediction models
        Type: general
      – SubjectFull: Optimization algorithms
        Type: general
      – SubjectFull: Dimensional reduction algorithms
        Type: general
      – SubjectFull: Flow simulations
        Type: general
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      – TitleFull: Nonlinear Model Reduction by Probabilistic Manifold Decomposition.
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            NameFull: Guo, Jiaming
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            NameFull: Xiao, Dunhui
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              M: 01
              Text: 2026
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              Y: 2026
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