We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a unique capability: it seamlessly integrates SciML into UQ and vice versa, which allows it to quantify the uncertainties in SciML tasks effectively and leverage SciML for improved uncertainty assessment during UQ-related tasks. The proposed surrogate model can effectively incorporate a variety of physical constraints, such as governing partial differential equations (PDEs) with associated initial and boundary conditions constraints, inequality-type constraints (e.g., monotonicity, convexity, non-negativity, among others), and additional a priori information in the training process to supplement limited data. This ensures physically realistic predictions and significantly reduces the need for expensive computational model evaluations to train the surrogate model. Furthermore, the proposed method has a built-in uncertainty quantification (UQ) feature to efficiently estimate output uncertainties. To demonstrate the effectiveness of the proposed method, we apply it to a diverse set of problems, including linear/non-linear PDEs with deterministic and stochastic parameters, data-driven surrogate modeling of a complex physical system, and UQ of a stochastic system with parameters modeled as random fields.

翻译：我们提出了一种新颖的物理约束多项式混沌展开方法，作为一种代理建模技术，能够同时执行科学机器学习（SciML）和不确定性量化（UQ）任务。该方法具有独特优势：它能无缝地将SciML融入UQ中，反之亦然，从而有效量化SciML任务中的不确定性，并利用SciML提升UQ相关任务中的不确定性评估。所提出的代理模型能够在训练过程中有效融入多种物理约束，例如控制偏微分方程（PDEs）及其初始条件与边界条件约束、不等式型约束（如单调性、凸性、非负性等），以及额外的先验信息，以补充有限数据。这确保了预测结果的物理真实性，并显著降低了训练代理模型所需的高昂计算模型评估成本。此外，该方法内置了不确定性量化功能，能够高效估计输出不确定性。为验证所提方法的有效性，我们将其应用于多种问题，包括具有确定性和随机参数的线性/非线性偏微分方程、复杂物理系统的数据驱动代理建模，以及参数建模为随机场的随机系统的UQ问题。