Convergence analysis of Parametric Probabilistic Manifold Decomposition

Abstract

This paper presents a convergence analysis for a newly developed nonlinear model reduction method: parametric probabilistic manifold decomposition (PPMD)~\cite{guo2026parametric}. In addition, existing analyzes of nonlinear reduced order models typically treat subspace reduction, manifold representation, regression, and nonlinear reconstruction as separate components and often remain at the level of discrete state vectors. To the best of our knowledge, no theory tracks the complete error propagation in a data-dependent model whose basis, residual geometry, spectral coordinates, parameter maps, and lifting operator are all learned from the same numerical solution data. We develop a coupled perturbation analysis for the entire PPMD procedure. A trajectory geometry induced by the spatial discretization and temporal quadrature connects discrete trajectory vectors isometrically with the corresponding PDE norm. Population spectral objects are introduced to align the empirical residual coordinates and derive a uniform coordinate error estimate, whose propagation through the Hilbert-valued kernel lifting estimator is then quantified. Combining these results with the full order discretization error, weighted low-rank approximation, parameter regression, and residual representation defect yields deterministic and high-probability trajectory error bounds and consistency in probability in the continuous PDE trajectory space. The theory identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.