This work introduces an empirical quadrature-based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection-dominated problems with limited training. The proposed approach circumvents the slowly decaying $n$-width limitation of linear model reduction techniques applied to convection-dominated problems by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with bijections of the underlying domain. The reduced-order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online-efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh-independent operations. The proposed reduced-order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock-dominated computational fluid dynamics benchmarks.

翻译：本文提出了一种基于经验求积的超降阶过程与贪婪训练算法，旨在通过有限训练有效降低对流主导问题的计算成本。该方法通过将低维仿射空间与底层域的双射进行复合，系统性地构建非线性近似流形，从而规避了线性模型降阶技术在对流主导问题中缓慢衰减的$n$宽度限制。降阶模型被定义为在该非线性流形上求解残差最小化问题的解。通过采用经验求积逼近最优性系统，使得该问题可通过与网格无关的操作进行求解，从而实现了在线高效计算方法。所提出的降阶模型采用贪婪过程进行训练，以系统性地对参数域进行采样。该方法的有效性通过两个激波主导的计算流体动力学基准测试得到了验证。