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

Abstract

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.

Keywords

ellipsoidal convex model / Gaussian process regression / model updating / Riemannian manifold / uncertainty quantification

References

[1] J. E. Mottershead and M. I. Friswell, “Model Updating in Structural Dynamics: A Survey,” Journal of Sound and Vibration 167 (1993): 347–375.

[2] J. E. Mottershead, M. Link, and M. I. Friswell, “The Sensitivity Method in Finite Element Model Updating: A Tutorial,” Mechanical Systems and Signal Processing 25 (2011): 2275–2296.

[3] F. M. Hemez and S. W. Doebling, “Review and Assessment of Model Updating for Non-Linear, Transient Dynamics,” Mechanical Systems and Signal Processing 15 (2001): 45–74.

[4] J. J. Rubio, “Stability Analysis of the Modified Levenberg-Marquardt Algorithm for the Artificial Neural Network Training,” IEEE Transactions on Neural Networks and Learning Systems 32 (2021): 3510–3524.

[5] D. Sedlar, Z. Lozina, and D. Vucina, “Comparison of Genetic and Bees Algorithms in the Finite Element Model Update,” Transactions of FAMENA 35 (2011): 1–12.

[6] H. Tran-Ngoc, S. Khatir, G. De Roeck, T. Bui-Tien, L. Nguyen-Ngoc, and M. Abdel Wahab, “Model Updating for Nam O Bridge Using Particle Swarm Optimization Algorithm and Genetic Algorithm,” Sensors 18 (2018): 4131.

[7] C. Berzuini, N. G. Best, W. R. Gilks, and C. Larizza, “Dynamic Conditional Independence Models and Markov Chain Monte Carlo Methods,” Journal of the American Statistical Association 92 (1997): 1403–1412.

[8] C. Robert and G. Casella, “Convergence Monitoring and Adaptation for MCMC Algorithms,” Springer NewYork 8 (2010): 237–268.

[9] J. Ching and Y. C. Chen, “Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging,” Journal of Engineering Mechanics 133 (2007): 816–832.

[10] D. Straub and I. Papaioannou, “Bayesian Updating With Structural Reliability Methods,” Journal of Engineering Mechanics 141 (2015): 1–13.

[11] D. Wu, W. Gao, F. Tin-Loi, and Y. L. Pi, “Probabilistic Interval Limit Analysis for Structures With Hybrid Uncertainty,” Engineering Structures 114 (2016): 195–208.

[12] B. R. Mathon, M. M. Ozbek, and G. F. Pinder, “Dempster–Shafer Theory Applied to Uncertainty Surrounding Permeability,” Mathematical Geosciences 42 (2010): 293–307.

[13] S. Ferson, V. Kreinovich, L. Ginzburg, D. S. Myers, and K. Sentz, Constructing Probability Boxes and Dempster-Shafer Structures. Sandia Report. 2003; SAND2002-4015.

[14] S. Bai, D. Li, and Z. Kang, “Construction of Ellipsoid Convex Model of Bounded Uncertainties With Outlier Detection for Application in Non-Probabilistic Topology Optimization,” Computers & Structures 296 (2024): 107322.

[15] S. Jayasumana, R. Hartley, M. Salzmann, H. Li, and M. Harandi, “Kernel Methods on Riemannian Manifolds With Gaussian RBF Kernels,” IEEE Transactions on Pattern Analysis and Machine Intelligence 37 (2015): 2464–2477.

[16] Y. Zhao, L. Chen, Q. Zhou, J. Zuo, H. Wang, and M. Ren, “A Registration Method of Overlap Aware Point Clouds Based on Transformer-To-Transformer Regression,” Remote Sensing 16 (2024): 1898.

[17] B. Shen, L. Yao, Z. Yang, and Z. Ge, “Mode Information Separated β-VAE Regression for Multimode Industrial Process Soft Sensing,” IEEE Sensors Journal 23 (2023): 10231–10240.

[18] Z. Zhang, Z. Zou, E. Kuhl, and G. E. Karniadakis, Brown Univ., Providence, RI (United States), “Discovering a Reaction–Diffusion Model for Alzheimer's Disease by Combining Pinns With Symbolic Regression,” Computer Methods in Applied Mechanics and Engineering 419 (2024): 116647.

[19] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (MIT Press, 2005).

[20] D. Kim and S. H. Rhee, “Data-Driven Modeling and Regression Analysis on Ship Resistance of In-Service Performance,” International Journal of Naval Architecture and Ocean Engineering 16 (2024): 100623.

[21] R. Badenbroek and J. Dahl, “An Algorithm for Nonsymmetric Conic Optimization Inspired by MOSEK,” Optimization Methods & Software 37 (2022): 1027–1064.

[22] J. R. Cardoso and F. Silva Leite, “Theoretical and Numerical Considerations About Padé Approximants for the Matrix Logarithm,” Linear Algebra and Its Applications 330 (2001): 31–42.

[23] W. L. Oberkampf and C. J. Roy, Verification and Validation in Scientific Computing, 1st ed. (Cambridge University Press, 2010).

[24] B. Liao, R. Zhao, K. Yu, and C. Liu, “A Novel Interval Model Updating Framework Based on Correlation Propagation and Matrix-Similarity Method,” Mechanical Systems and Signal Processing 162 (2022): 108039.

[25] T. T. Ding, S. S. Liu, Z. J. Wang, P. Huang, M. T. Tao, and Z. W. Gu, “A Novel Mixture Sampling Strategy Combining Latin Hypercube Sampling With Optimized One Factor at a Time Method: A Case Study on Mixtures of Antibiotics and Pesticides,” Journal of Hazardous Materials 461 (2024): 132568-132568.

[26] Y. Xu, X. Kong, and Z. Cai, “Cross-Validation Strategy for Performance Evaluation of Machine Learning Algorithms in Underwater Acoustic Target Recognition,” Ocean Engineering 299 (2024): 117236.

[27] Y. Yang, C. Barnes, A. Adams, and A. Finkelstein, “Aδ: Autodiff for Discontinuous Programs - Applied to Shaders,” ACM Transactions on Graphics 41 (2022): 1–24.

[28] L. Devroye, P. Epstein, J. R. Sack, and J. R. Sack, “On Generating Random Intervals and Hyperrectangles,” Journal of Computational and Graphical Statistics 2 (1993): 291–307.

[29] Y. Song, J. Mi, Y. Cheng, L. Bai, and K. Chen, “A Dependency Bounds Analysis Method for Reliability Assessment of Complex System With Hybrid Uncertainty,” Reliability Engineering & System Safety 204 (2020): 107119.

[30] A. Gray, M. de Angelis, E. Patelli, and S. Ferson, “Bivariate Dependency Tracking in Interval Arithmetic,” Mechanical Systems and Signal Processing 186 (2023): 109771.

[31] H. Haario, M. Laine, A. Mira, and E. Saksman, “DRAM: Efficient Adaptive MCMC,” Statistics and Computing 16 (2006): 339–354.

[32] H. Haario, E. Saksman, and J. Tamminen, “An Adaptive Metropolis Algorithm,” Bernoulli 7 (2001): 223–242.

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2025 The Author(s). International Journal of Mechanical System Dynamics published by John Wiley & Sons Australia, Ltd on behalf of Nanjing University of Science and Technology.