A finite-orthogonal-basis reduced-order model for solving partial differential equations

Projection-based reduced order model (PROM) and the hyper projection-based reduced order model (HPROM) require users to embed the projection and approximation processes within numerical calculation framework. These requirements are exceedingly challenging for most users of commercial software. Accordingly, we propose an innovative reduced-order framework comprising finite-orthogonal-basis reduced-order model (FB-ROM) and its hyper-reduced form, namely, finite-orthogonal-basis hyper-reduced-order model (FB-HROM). FB-ROM and FB-HROM directly discretize PDEs via simple matrix addition and pointwise multiplication between reduced order bases Phi and its corresponding differential operator. L(Phi). rather than via element-wise traversal used in PROMs and HPROMs. This approach gives the FB-ROM and FB-HROM a non-intrusive characteristic in terms of model discretization during the online phase, allowing users of commercial software to discretize the PDEs without relying on complex numerical computation models that have a high barrier and a non-intrusive characteristic in terms of data acquisition during the offline phase, allowing users to obtain all necessary snapshots through commercial software. Specially, FB-HROM organically omits the approximation processes of nonlinear operators, relying solely on the projection process inherent in the FB-ROM to achieve hyper-reduction. Diverse case studies, namely, on the heat conduction equation, reaction-diffusion equation, Burgers' equation, and the Navier-Stokes equations, reveal that FB-ROM and FB-HROM exhibit remarkable performance in terms of acceleration with acceleration factors of 158.2x, 180.1x, 202.5x, and 393.4x, respectively, stability, and generalizability and high applicability.