Nonlinear Model Reduction by Probabilistic Manifold Decomposition

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.