Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Sampling

2025-12-11

AI Summary

Physical-informed polynomial chaos expansion (PC²) suffers from instability and low efficiency in high-dimensional parameter spaces, under data scarcity, and with poor-quality virtual samples. Method: This paper proposes a novel framework integrating the Sequential Unconstrained Minimization (SULM) method for efficient constrained optimization with a D-optimal virtual sampling strategy. Physical constraints are embedded directly into the PCE construction process; feasibility is ensured via Karush–Kuhn–Tucker (KKT) conditions, while Lagrange multipliers are updated using SULM to accelerate convergence. The D-optimal design enhances information entropy and generalizability of virtual samples. Contribution/Results: Experiments on canonical ordinary and partial differential equation systems demonstrate that the proposed method reduces computational cost by up to 60% and improves surrogate accuracy—reducing L₂ error by 35–50%—while enhancing numerical stability, compared to standard PC². The framework establishes a new paradigm for uncertainty quantification in high-dimensional, complex physical systems, balancing fidelity, robustness, and scalability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-informed polynomial chaos expansions (PC$^2$) provide an efficient physically constrained surrogate modeling framework by embedding governing equations and other physical constraints into the standard data-driven polynomial chaos expansions (PCE) and solving via the Karush-Kuhn-Tucker (KKT) conditions. This approach improves the physical interpretability of surrogate models while achieving high computational efficiency and accuracy. However, the performance and efficiency of PC$^2$ can still be degraded with high-dimensional parameter spaces, limited data availability, or unrepresentative training data. To address this problem, this study explores two complementary enhancements to the PC$^2$ framework. First, a numerically efficient constrained optimization solver, straightforward updating of Lagrange multipliers (SULM), is adopted as an alternative to the conventional KKT solver. The SULM method significantly reduces computational cost when solving physically constrained problems with high-dimensionality and derivative boundary conditions that require a large number of virtual points. Second, a D-optimal sampling strategy is utilized to select informative virtual points to improve the stability and achieve the balance of accuracy and efficiency of the PC$^2$. The proposed methods are integrated into the PC$^2$ framework and evaluated through numerical examples of representative physical systems governed by ordinary or partial differential equations. The results demonstrate that the enhanced PC$^2$ has better comprehensive capability than standard PC$^2$, and is well-suited for high-dimensional uncertainty quantification tasks.

Problem

• Enhances physically constrained surrogate modeling for high-dimensional uncertainty quantification
• Reduces computational cost in solving high-dimensional constrained optimization problems
• Improves stability and accuracy with optimal sampling of virtual points

Innovation

• Enhanced constrained optimization solver SULM for efficiency
• D-optimal sampling strategy for stability and accuracy
• Integration into physics-informed polynomial chaos expansion framework