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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]
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Database: Engineering Source
Description
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]
ISSN:10648275
DOI:10.1137/25M1738863