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.

提出了一种新颖的物理约束多项式混沌扩展方法作为代理模型方法，能够同时进行科学机器学习和不确定性量化任务，有效地量化科学机器学习任务的不确定性，并利用科学机器学习改善不确定性评估。