Entropic transfer operators

Abstract

We propose a new concept for the regularization and discretization of transfer 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 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.

Gromov-Wasserstein Transfer Operators

“…In both references [14,15], the assumption on the underlying dynamics is that they are compliant with an optimal transport. For certain situations this might not be the case.…”

“…This makes precise knowledge of the formation of coherent structures very appealing, since it may lead to a deeper understanding of the dynamics or computational advancements. In [14], the authors assumed instead that the exact transfer operator is known on a finite subset of the full state space. Then, using regularized OT, a finite-dimensional approximation is constructed which limit is a regularized version of the ground truth and exhibits desirable properties, such as retention of the spectral information.…”