Ellipsoid-Based Interval-Type Uncertainty Model Updating Based on Riemannian Manifold and Gaussian Process Model

Modern engineering systems require advanced uncertainty-aware model updating methods that address parameter correlations beyond conventional interval analysis. This paper proposes a novel framework integrating Riemannian manifold theory with Gaussian Process Regression (GPR) for systems governed by Symmetric Positive-Definite (SPD) matrix constraints. Our methodology features three key innovations: (1) A semi-definite programming-optimized Minimum Volume Ellipsoid model that explicitly quantifies parameter interdependencies while ensuring computational efficiency; (2) A manifold-embedded GPR surrogate model employing Log-Euclidean kernels to intrinsically preserve SPD constraints during uncertainty updating; (3) A Riemannian gradient optimization scheme that enables efficient parameter updates via logarithmic matrix mapping. Validated through mechanical and aerospace case studies, the framework achieves a Log-Euclidean distance of 3.12 × 10⁻⁴ in uncertainty updating (compared to a baseline of 48.48) and provides a tenfold computational acceleration over Bayesian alternatives. Robustness tests demonstrate stable performance under 5% noise perturbation, with the Log-Euclidean distance increased only marginally to 1.41 × 10⁻². By unifying differential geometry with machine learning, our approach eliminates heuristic projections required in conventional methods while advancing uncertainty quantification through structure-preserving manifold operations. This study bridges geometric consistency, computational efficiency, and physical consistency in uncertainty-aware model updating.