Physics-Informed Polynomial Chaos Expansions
Lukáš Novák Himanshu Sharma Michael D. Shields
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.