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  • 1
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    Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics (2017)
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    Publication Date: 2019-04-02
    Description: 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: English
    Type: article , doc-type:article
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    OPUS
  • 2
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    Data-driven Computation of Molecular Reaction Coordinates (2017)
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    Publication Date: 2018-02-15
    Description: The identification of meaningful reaction coordinates plays a key role in the study of complex molecular systems whose essential dynamics is characterized by rare or slow transition events. In a recent publication, the authors identified a condition under which such reaction coordinates exist - the existence of a so-called transition manifold - and proposed a numerical method for their point-wise computation that relies on short bursts of MD simulations. This article represents an extension of the method towards practical applicability in computational chemistry. It describes an alternative computational scheme that instead relies on more commonly available types of simulation data, such as single long molecular trajectories, or the push-forward of arbitrary canonically-distributed point clouds. It is based on a Galerkin approximation of the transition manifold reaction coordinates, that can be tuned to individual requirements by the choice of the Galerkin ansatz functions. Moreover, we propose a ready-to-implement variant of the new scheme, that computes data-fitted, mesh-free ansatz functions directly from the available simulation data. The efficacy of the new method is demonstrated on a realistic peptide system.
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
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    A kernel-based approach to molecular conformation analysis (2018)
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    Publication Date: 2020-01-27
    Description: 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: English
    Type: article , doc-type:article
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    OPUS
  • 4
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    A probabilistic algorithm for aggregating vastly undersampled large Markov chains (2021)
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    Publication Date: 2022-11-28
    Language: English
    Type: article , doc-type:article
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  • 5
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    A weak characterization of slow variables in stochastic dynamical systems (2020)
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    Publication Date: 2022-11-28
    Language: English
    Type: incollection , doc-type:Other
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  • 6
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    Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds (2020)
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    Publication Date: 2022-11-28
    Description: 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: English
    Type: article , doc-type:article
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    OPUS
  • 7
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    Exploring the locking stage of NFGAILS amyloid fibrillation via transition manifold analysis (2021)
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    Publication Date: 2022-11-28
    Language: English
    Type: article , doc-type:article
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    OPUS
  • 8
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    A probabilistic algorithm for aggregating vastly undersampled large Markov chains (2020)
    Details
    Publication Date: 2022-11-28
    Description: Model reduction of large Markov chains is an essential step in a wide array of techniques for understanding complex systems and for efficiently learning structures from high-dimensional data. We present a novel aggregation algorithm for compressing such chains that exploits a specific low-rank structure in the transition matrix which, e.g., is present in metastable systems, among others. It enables the recovery of the aggregates from a vastly undersampled transition matrix which in practical applications may gain a speedup of several orders of mag- nitude over methods that require the full transition matrix. Moreover, we show that the new technique is robust under perturbation of the transition matrix. The practical applicability of the new method is demonstrated by identifying a reduced model for the large-scale traffic flow patterns from real-world taxi trip data.
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
  • 9
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    Optimal Reaction Coordinates: Variational Characterization and Sparse Computation (2023)
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    Publication Date: 2023-04-26
    Description: Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.
    Language: English
    Type: article , doc-type:article
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  • 10
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    Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics (2017)
    Details
    Publication Date: 2024-02-12
    Description: 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: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    OPUS
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