Two-stage ROM: a hybrid reduced order model for offline and online efficiency

Physics-based reduced-order models are one of the key enablers for improving the predictive capabilities of digital twins, offering a balance between computational efficiency and physical fidelity. However, their efficiency for nonlinear problems remains a challenge because evaluating nonlinear operators in reduced dimensions can be very costly, and offline construction often requires many high-fidelity simulations. In this paper, we address this issue with a reduced-order modeling strategy for parametrized nonlinear partial differential equations that is efficient both online and offline. Offline efficiency is achieved using a greedy algorithm for reduced basis construction combined with a quasi-optimal hyper-reduction procedure called Reduced Basis Empirical Quadrature Procedure (RBEQP) [1]. Then, Gaussian process regression is integrated into the reduced-order model to learn the reduced coefficients and enhance online efficiency. Additionally, we evaluate the effectiveness of our proposed method on two model problems: a steady-state heat conduction problem with a nonlinear, temperature-dependent conductivity, and a neo-Hookean hyperelastic beam under self-weight load.