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.

We explore the problem of sampling graph signals in scenarios where the graph structure is not predefined and must be inferred from data. In this scenario, existing approaches rely on a two-step process, where a graph is learned first, followed by sampling. More generally, graph learning and graph signal sampling have been studied as two independent problems in the literature. This work provides a foundational step towards jointly optimizing the graph structure and sampling set. Our main contribution, Vertex Importance Sampling (VIS), is to show that the sampling set can be effectively determined from the vertex importance (node weights) obtained from graph learning. We further propose Vertex Importance Sampling with Repulsion (VISR), a greedy algorithm where spatially -separated "important" nodes are selected to ensure better reconstruction. Empirical results on simulated data show that sampling using VIS and VISR leads to competitive reconstruction performance and lower complexity than the conventional two-step approach of graph learning followed by graph sampling.