We present a registration method for model reduction of parametric partial differential equations with dominating advection effects and moving features. Registration refers to the use of a parameter-dependent mapping to make the set of solutions to these equations more amicable for approximation using classical reduced basis methods. The proposed approach utilizes concepts from optimal transport theory, as we utilize Monge embeddings to construct these mappings in a purely data-driven way. The method relies on one interpretable hyper-parameter. We discuss how our approach relates to existing works that combine model order reduction and optimal transport theory. Numerical results are provided to demonstrate the effect of the registration. This includes a model problem where the solution is itself a probability density and one where it is not.

翻译：我们提出了一种针对具有主导对流效应和移动特征的参数化偏微分方程模型降阶的配准方法。配准是指利用参数依赖的映射，使这些方程的解集更易于通过经典降阶基方法进行近似。所提出的方法利用最优传输理论中的概念，采用蒙日嵌入以纯数据驱动的方式构建这些映射。该方法依赖于一个可解释的超参数。我们讨论了本方法与现有结合模型降阶与最优传输理论的工作之间的联系。提供了数值结果以展示配准的效果，包括解本身为概率密度的模型问题以及非概率密度的情况。