Abstract:

The effective longtime dynamics of complex multiscale systems can often be described by just a small number of essential degrees of freedom, also known as reaction coordinates (RCs). In cases where good RCs are known beforehand, the Mori Zwanzig formalism offers an elegant, self-contained scheme to compute the associated effective dynamics. However, the data-driven derivation of good RCs often proves difficult, in part due to a lack of robust mathematical criteria for judging their quality.
Recently, a new mathematical framework for the characterization and computation of RCs has been developed [1]. The derived RCs are optimal in preserving the longest implied timescales and the associated slow sub-processes of the original system. At the heart of the new theory lies the observation that state space points that are “statistically indistinguishable” under the longtime evolution of the system are “geometrically indistinguishable” after embedding into a certain function space. In this embedding space, the points thus lie on a low-dimensional manifold which can be numerically identified by established manifold learning algorithms. In particular, a newly-developed kernelized variant of the diffusion maps algorithm has proven especially well-suited for the problem [2]. The new theory has deep links to Markov state modelling and transition path theory.
This talk will present both the theoretical concepts of the new method, as well as the data-driven algorithms, and demonstrate them on realistic biochemical systems.

[1] “Transition Manifolds of Complex Metastable Systems”. Bittracher, A., Koltai, P., Klus, S., Banisch, R., Dellnitz, M., Schütte, C. J Nonlinear Sci (2018) 28: 471. https://doi.org/10.1007/s00332-017-9415-0
[2] “Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds”. Bittracher, A., Klus, S., Hamzi, B., Schütte, C. arXiv Preprint (2019). https://arxiv.org/abs/1904.08622