We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold; reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual; and the dual-weighted residual (DWR) method, which provides an error representation formula for the quantity of interest. To find the RQ rules, we develop an empirical quadrature procedure (EQP) for time-dependent problems; we analyze the output error due to hyperreduction using a space–time DWR framework and identify appropriate constraints so that the output error due to hyperreduction is controlled. We in addition equip our reduced model with an online-efficient DWR a posteriori error estimate for the output; we again analyze the error in the hyperreduced dual problem and DWR expression to find appropriate constraints for the EQP so that the error in the error estimate is controlled. In the offline stage, the RBs and RQs, as well as the finite element mesh, are simultaneously constructed using a POD-greedy algorithm that leverages the online-efficient output error estimate. We demonstrate the framework for parametrized unsteady flows in a lid-driven cavity and over a NACA0012 airfoil. Reduced models achieve over two orders of magnitude reduction in the number of degrees of freedom, number of quadrature points, and wall-clock computational time, while achieving less than 0.5% output error and providing efficient error estimates in predictive settings.

我们为参数化的时间相关非线性偏微分方程 (PDE) 提出了一种基于投影的模型简化公式。我们的方法建立在以下成分之上：减少的基数 (RB)，它提供参数-时间解流形的快速收敛近似值；减少正交 (RQ) 规则，提供非线性残差的超减少；和双加权残差（DWR）方法，它为感兴趣的数量提供了一个误差表示公式。为了找到 RQ 规则，我们为瞬态问题开发了经验求积程序 (EQP)；我们使用时空 DWR 框架分析由超还原引起的输出误差，并确定适当的约束，以便控制由超还原引起的输出误差。输出的后验误差估计；我们再次分析 hyperreduced 对偶问题和 DWR 表达式中的误差，以找到适当的 EQP 约束，从而控制误差估计中的误差。在离线阶段，RBs 和 RQs 以及有限元网格是使用 POD-greedy 算法同时构建的，该算法利用了在线有效的输出误差估计。我们展示了盖子驱动腔中和 NACA0012 翼型上的参数化非定常流动的框架。简化模型在自由度数、正交点数和挂钟计算时间方面实现了两个数量级的减少，同时实现了小于 0.5% 的输出误差并在预测设置中提供了有效的误差估计。