Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

参数化微分方程的数值模拟对于研究应用科学和工程中的现实世界现象至关重要。用于此类问题的实时和多查询模拟的计算方法通常需要极高的计算成本才能获得足够准确的数值解。在过去的几十年里，模型降阶已被证明成功地提供了低复杂度高保真代理模型，该模型允许在参数变化下进行快速准确的模拟，从而能够对日益复杂的问题进行数值模拟。然而，在确保非线性时间相关问题的数值模拟所需的鲁棒性和效率方面仍然存在许多挑战。本文的目的是调查时间相关问题的简化基方法的最新技术，并总结三个主要方向的最新进展。首先，我们讨论旨在保留连续问题的关键物理特性的结构保持降阶模型。其次，我们研究了基于解空间非线性近似的局部化和自适应方法。最后，我们考虑基于非侵入式降阶模型的数据驱动技术，其中学习了参数空间和降阶基系数之间的映射近似。在每一类方法中，我们描述了不同的方法，并提供了一个比较讨论，以洞察优势、劣势和潜在的开放性问题。