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).

翻译：本研究针对具有多项式结构的非线性问题，提出了一种侵入式投影基模型降阶框架，该框架在降维空间中采用牛顿求解器进行迭代求解。研究证明，对于目标类别的多项式非线性问题，投影近似残差和雅可比矩阵中出现的所有算子均可在离线阶段预计算，从而无需进行超约简。此外，投影近似残差及其雅可比矩阵的评估复杂度仅与降维空间的维度成正比，而与全阶模型的维度无关，实现了有效的离线-在线分解。所提出的免超约简框架被应用于伽辽金投影方案与最小二乘彼得罗夫-伽辽金投影方案。通过两个数值实验评估了所提HRF方案的精度与计算效率，并将其与伽辽金和LSPG方案中常用的能量守恒采样加权超约简方法进行对比。在第一个数值算例中，采用参数化伯格斯方程评估各模型降阶方法在训练快照未覆盖参数集上的预测能力。第二个算例研究了含三次反应项的参数化热方程，通过引入升维变换以呈现所需结构。本文探讨了HRF方法在准确降低升维形式系统维度方面的有效性。对于所研究问题，结果表明：HRF-G与HRF-LSPG方案在保持状态预测误差低于O(10^-2)的前提下，相较于全阶模型分别实现了两个数量级和一个数量级的加速。