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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.
The dynamical behavior of social systems can be described by agent-based models. Although single agents follow easily explainable rules, complex time-evolving patterns emerge due to their interaction. The simulation and analysis of such agent-based models, however, is often prohibitively time-consuming if the number of agents is large. In this paper, we show how Koopman operator theory can be used to derive reduced models of agent-based systems using only simulation or real-world data. Our goal is to learn coarse-grained models and to represent the reduced dynamics by ordinary or stochastic differential equations. The new variables are, for instance, aggregated state variables of the agent-based model, modeling the collective behavior of larger groups or the entire population. Using benchmark problems with known coarse-grained models, we demonstrate that the obtained reduced systems are in good agreement with the analytical results, provided that the numbers of agents is sufficiently large.
Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds
(2021)
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.
The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept–representing highly nonlinear dynamical systems by infinite-dimensional linear operators–has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.
Transfer operators such as the Perron-Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. In fact, most of the existing methods can be derived from our framework, providing a unifying view on the approximation of transfer operators. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.
More and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, or some cancer types. Thanks to modern high-throughput omics technologies, it becomes possible to directly analyze human microbiome and its influence on the health status. Microbial communities are monitored over long periods of time and the associations between their members are explored. These relationships can be described by a time-evolving graph. In order to understand responses of the microbial community members to a distinct range of perturbations such as antibiotics exposure or diseases and general dynamical properties, the time-evolving graph of the human microbial communities has to be analyzed. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to solving this problem is the representation of the time-evolving graphs as fixed-length feature vectors preserving the original dynamics. We propose a method for learning the embedding of the time-evolving graph that is based on the spectral analysis of transfer operators and graph kernels. We demonstrate that our method can capture temporary changes in the time-evolving graph on both synthetic data and real-world data. Our experiments demonstrate the efficacy of the method. Furthermore, we show that our method can be applied to human microbiome data to study dynamic processes.
We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.