🤖 AI Summary
This work proposes a hyper-reduction-free projected reduced-order Newton solver framework for polynomial nonlinear dynamical systems, addressing the computational overhead and complexity typically incurred by hyper-reduction in model order reduction. The method precomputes all projected residuals and Jacobian operators during the offline phase, achieving strict decoupling between offline and online stages. It is the first to enable fully hyper-reduction-free, efficient reduced-order modeling within both Galerkin and least-squares Petrov–Galerkin (LSPG) frameworks, while handling non-polynomial nonlinearities via lifting transformations. Numerical experiments on the Burgers equation and a heat equation with cubic reaction terms demonstrate that the proposed HRF-G and HRF-LSPG methods achieve speedups of approximately 100× and 10×, respectively, with state prediction errors below 10⁻², substantially reducing online costs while preserving high accuracy.
📝 Abstract
This study proposes an intrusive projection-based model-order reduction framework for nonlinear problems with a polynomial structure, solved iteratively using a Newton solver in the reduced space. It is demonstrated that for the targeted class of polynomial nonlinearities, all operators appearing in the projected approximate residual and Jacobian can be precomputed in the offline phase, eliminating the need for hyper-reduction. Additionally, the evaluation of both the projected approximate residual and its Jacobian scales only with the dimension of the reduced space, and does not depend on the dimension of the full-order model, enabling effective offline-online decomposition. The proposed hyper-reduction-free (HRF) framework is applied to both Galerkin (HRF-G) and least-squares Petrov-Galerkin (HRF-LSPG) projection schemes. The accuracy and computational efficiency of the proposed HRF schemes are evaluated in two numerical experiments and compared with a commonly used hyper-reduction scheme, namely the energy-conserving sampling and weighting method, for both the Galerkin and LSPG schemes. In the first numerical example, a parametric Burgers' equation is used to assess the predictive capabilities of the considered model reduction approaches on parameter sets not seen in the training snapshots. In the second example, a parametric heat equation with a cubic reaction term is studied, for which a lifting transformation is employed to expose the desired structure. The efficacy of the HRF methods in accurately reducing the dimensionality of the lifted formulation is investigated. For the studied problems, the results show that HRF-G and HRF-LSPG achieve two and one order of magnitude speedup, respectively, with respect to the full-order model while resulting in state prediction errors below O(10^-2).