We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analysed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the $n$-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

翻译：我们提出了一种用于动力系统中传递算子与Koopman算子正则化及离散化的新概念。该方法基于两个概率测度之间的熵正则化最优传输。具体而言，我们利用最优传输计划构造可进行计算分析的传递算子或Koopman算子的有限维近似。我们证明了离散化算子的谱收敛于正则化原始算子的谱，详细分析了$n$维环面上旋转映射的离散化算子与原始外围谱之间的关系，并提供了三个数值实验的代码，其中包含基于小生物分子原始轨迹数据恢复其主导构象的实验。