Physics-constrained polynomial chaos expansion for scientific machine
  learning and uncertainty quantification
Himanshu Sharma Lukáš Novák Michael D. Shields
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.