Physics-Informed Polynomial Chaos Expansions

TL;DR

This paper introduces a new physics-informed polynomial chaos expansion method that integrates physical constraints from differential equations into surrogate modeling, improving accuracy without significant computational costs.

Contribution

The paper presents a novel methodology for constructing physics-informed PCEs that incorporate differential equations and boundary conditions, enhancing surrogate model accuracy.

Findings

Physically constrained PCEs outperform standard sparse PCE in accuracy.

The proposed algorithms are computationally efficient.

Physically constrained PCEs can be built without model evaluations.

Abstract

Surrogate modeling of costly mathematical models representing physical systems is challenging since it is typically not possible to create a large experimental design. Thus, it is beneficial to constrain the approximation to adhere to the known physics of the model. This paper presents a novel methodology for the construction of physics-informed polynomial chaos expansions (PCE) that combines the conventional experimental design with additional constraints from the physics of the model. Physical constraints investigated in this paper are represented by a set of differential equations and specified boundary conditions. A computationally efficient means for construction of physically constrained PCE is proposed and compared to standard sparse PCE. It is shown that the proposed algorithms lead to superior accuracy of the approximation and does not add significant computational burden. Although the main purpose of the proposed method lies in combining data and physical constraints, we show that physically constrained PCEs can be constructed from differential equations and boundary conditions alone without requiring evaluations of the original model. We further show that the constrained PCEs can be easily applied for uncertainty quantification through analytical post-processing of a reduced PCE filtering out the influence of all deterministic space-time variables. Several deterministic examples of increasing complexity are provided and the proposed method is applied for uncertainty quantification.