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：本研究提出了一种基于经验正交的超降阶求解流程与贪心训练算法，通过有限训练数据有效降低求解平流主导问题时的计算成本。该方法通过定义由低维仿射空间与底层域双射复合构成的非线性逼近流形，规避了线性模型降阶技术处理平流主导问题时存在的$n$-宽度衰减缓慢的局限性。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。通过采用经验正交方法逼近最优性系统，使得该系统可通过网格无关操作求解，从而获得在线高效算法。采用贪心流程对参数域进行系统性采样以实现降阶模型的训练。最后通过两个以激波为主导的计算流体动力学基准测试验证了该方法的有效性。