Koopman-based approach to nonintrusive projection-based reduced-order modeling with black-box high-fidelity models

This paper presents a methodology that enables projection-based model reduction for black-box high-fidelity models governing nonlinear static parametric systems. The methodology specifically addresses the situation in which the high-fidelity model may be a black box, but there is knowledge of the governing equations in continuous partial differential equation (PDE) form. Drawing from the Koopman theory, it first obtains an underdetermined linear representation of the governing equations in terms of a set of observables. Then, the linear operator is extracted by a direct discretization of the linear differential terms using a suitable method, such as the finite volume method. By applying the snapshots of the observables to the discrete linear operator, a right-hand-side vector is obtained, providing the necessary system matrices for the projection step, which is done via proper orthogonal decomposition. The underdetermined linear system is closed with a set of nonlinear algebraic equations that act as constraints, whose computation is efficiently handled via the discrete empirical interpolation method. An offline database of reduced-order models (ROMs) corresponding to each parameter snapshot is generated, which are then interpolated online to predict the state for new parameter instances. The resulting ROM is posed and solved as a nonlinear constrained optimization problem. The method is tested on a canonical PDE, followed by the 2-D compressible Euler equations governing the flow past an airfoil, under both the subsonic and transonic regimes. It is demonstrated that the ROM predicts both the state and outputs within 5% of the full-order model given adequate snapshots and the computational speedup is up to two to three orders of magnitude.