@article{BittracherKoltaiKlusetal., author = {Bittracher, Andreas and Koltai, P{\´e}ter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Sch{\"u}tte, Christof}, title = {Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics}, series = {Jounal of Nonlinear Science}, volume = {28}, journal = {Jounal of Nonlinear Science}, number = {2}, doi = {10.1007/s00332-017-9415-0}, pages = {471 -- 512}, abstract = {We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.}, language = {en} } @article{KlusBittracherSchusteretal., author = {Klus, Stefan and Bittracher, Andreas and Schuster, Ingmar and Sch{\"u}tte, Christof}, title = {A kernel-based approach to molecular conformation analysis}, series = {Journal of Chemical Physics}, volume = {149}, journal = {Journal of Chemical Physics}, number = {24}, doi = {10.1063/1.5063533}, abstract = {We present a novel machine learning approach to understanding conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov State Models, EDMD, and TICA can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular the alanine dipeptide and the protein NTL9.}, language = {en} } @article{BittracherKlusHamzietal., author = {Bittracher, Andreas and Klus, Stefan and Hamzi, Boumediene and Sch{\"u}tte, Christof}, title = {Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds}, series = {Journal of Nonlinear Science}, volume = {31}, journal = {Journal of Nonlinear Science}, doi = {10.1007/s00332-020-09668-z}, abstract = {We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches.}, language = {en} } @misc{BittracherKoltaiKlusetal., author = {Bittracher, Andreas and Koltai, P{\´e}ter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Sch{\"u}tte, Christof}, title = {Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63822}, abstract = {We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.}, language = {en} }