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参数化偏微分方程模型降阶的配准方法，适用于具有主导对流效应和移动特征的问题。配准是指利用依赖于参数的映射，使该类方程的解集更适用于经典降阶基方法的逼近。所提出的方法利用最优传输理论中的概念，通过蒙日嵌入以纯数据驱动方式构建这些映射，该方法仅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关联。数值结果验证了配准的效果，包括一个解本身为概率密度函数的模型问题，以及一个解非概率密度的模型问题。