Uncertainty Quantification in High-Dimensional Engineering: Polynomial Chaos, Bayesian Inference, and Active Learning
Keywords:
Uncertainty quantification, Polynomial Chaos Expansion, Bayesian inference, Active learning, Surrogate modeling, High-dimensional engineering, Computational uncertainty, Multi-fidelity modelingAbstract
This review aims to synthesize and critically analyze the state-of-the-art methodologies in uncertainty quantification (UQ) for high-dimensional engineering systems, focusing on Polynomial Chaos Expansion, Bayesian inference, and active learning frameworks as core paradigms for scalable and interpretable uncertainty management. This qualitative review employed a systematic literature analysis approach. A total of twelve peer-reviewed journal articles published between 2010 and 2024 were purposefully selected from leading engineering and computational science databases, including IEEE Xplore, ScienceDirect, SpringerLink, and Wiley Online Library. The inclusion criteria emphasized methodological rigor, relevance to high-dimensional UQ, and the presence of at least one of the three focal paradigms. Data collection relied exclusively on a literature-based review process, followed by qualitative thematic analysis using NVivo 14 software. The coding process involved open, axial, and selective coding to identify emerging themes, ensuring theoretical saturation. The resulting conceptual framework categorized the extracted data into three major themes—spectral methods (Polynomial Chaos), probabilistic inference (Bayesian approaches), and adaptive learning (active sampling)—and their interconnections. The analysis revealed a convergent methodological evolution in UQ research. Polynomial Chaos methods demonstrated robust efficiency in surrogate modeling and spectral uncertainty propagation through sparse and adaptive expansions. Bayesian inference emerged as a statistically coherent framework for parameter calibration, model selection, and posterior uncertainty representation, supported by scalable techniques such as Hamiltonian Monte Carlo and variational inference. Active learning proved essential for adaptive data acquisition and surrogate refinement, significantly reducing computational costs through informed sampling. Collectively, the three paradigms exhibited strong complementarity, forming hybrid UQ architectures that combine interpretability, scalability, and computational sustainability. Modern high-dimensional UQ research increasingly integrates spectral, Bayesian, and adaptive learning paradigms into unified frameworks capable of handling nonlinear, data-scarce, and computationally intensive problems. This triadic convergence represents a methodological shift toward interpretable, data-efficient, and scalable uncertainty quantification suitable for next-generation engineering simulations.
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