TY - JOUR A1 - Sikorski, Alexander A1 - Heida, Martin T1 - Voronoi Graph - Improved raycasting and integration schemes for high dimensional Voronoi diagrams N2 - The computation of Voronoi Diagrams, or their dual Delauney triangulations is difficult in high dimensions. In a recent publication Polianskii and Pokorny propose an iterative randomized algorithm facilitating the approximation of Voronoi tesselations in high dimensions. In this paper, we provide an improved vertex search method that is not only exact but even faster than the bisection method that was previously recommended. Building on this we also provide a depth-first graph-traversal algorithm which allows us to compute the entire Voronoi diagram. This enables us to compare the outcomes with those of classical algorithms like qHull, which we either match or marginally beat in terms of computation time. We furthermore show how the raycasting algorithm naturally lends to a Monte Carlo approximation for the volume and boundary integrals of the Voronoi cells, both of which are of importance for finite Volume methods. We compare the Monte-Carlo methods to the exact polygonal integration, as well as a hybrid approximation scheme. Y1 - 2024 ER - TY - GEN A1 - Sikorski, Alexander T1 - ISOKANN.jl N2 - Julia pacarkge implementing the ISOKANN algorithm for the computation of invariant subspaces of Koopman operators Y1 - 2024 U6 - https://doi.org/10.5281/zenodo.11519359 ER - TY - GEN A1 - Sikorski, Alexander T1 - VoronoiGraph.jl N2 - This Package implements a variation of the Voronoi Graph Traversal algorithm by Polianskii and Pokorny [1]. It constructs a Voronoi Diagram from a set of points by performing a random walk on the graph of the vertices of the diagram. Unlike many other Voronoi implementations this algorithm is not limited to 2 or 3 dimensions and promises good performance even in higher dimensions. Y1 - 2023 U6 - https://doi.org/10.5281/zenodo.6417636 ER - TY - GEN A1 - Sikorski, Alexander A1 - Sechi, Renata A1 - Helfmann, Luzie T1 - cmdtools N2 - Python implementation of severals tools (PCCA, AJC, SQRA, P/Q estimation) for the analysis of dynamical systems from the transfer operator perspective. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.5281/zenodo.4749331 ER - TY - CHAP A1 - Yousefian, Maryam A1 - Donati, Luca A1 - Sikorski, Alexander A1 - Weber, Marcus A1 - Röblitz, Susanna T1 - Exploring Metastable Dynamics of Gene Regulatory Networks with ISOKANN T2 - Computational Methods in Systems Biology. CMSB 2025 N2 - Stochastic dynamical systems like gene regulatory networks (GRNs) often exhibit behavior characterized by metastable sets (representing cellular phenotypes), in which trajectories remain for long times, whereas switches between these sets in the phase space are rare events. One way to capture these rare events is to infer the system’s long-term behavior from the spectral characteristics (eigenvalues and eigenvectors) of its Koopman operator. For GRNs, the Koopman operator is based on the chemical master equation (CME), which provides a precise mathematical modeling framework for stochastic GRNs. Since the CME is typically analytically intractable, methods based on discretizing the CME operator have been developed. However, determining the number and location of metastable sets in the phase space as well as the transition rates between them remains computationally challenging, especially for large GRNs with many genes and interactions. A promising alternative method, called ISOKANN (invariant subspaces of Koopman operators with artificial neural networks) has been developed in the context of molecular dynamics. ISOKANN uses a combination of the power iteration and neural networks to learn the basis functions of an invariant subspace of the Koopman operator. In this paper, we extend the application of ISOKANN to the CME operator and apply it to two small GRNs: a genetic toggle switch model and a model for macrophage polarization. Our work opens a new field of application for the ISOKANN algorithm and demonstrates the potential of this algorithm for studying large GRNs. Y1 - 2026 U6 - https://doi.org/10.1007/978-3-032-01436-8_8 VL - 15959 SP - 126 EP - 149 ER - TY - JOUR A1 - Sikorski, Alexander A1 - Ribera Borrell, Enric A1 - Weber, Marcus T1 - Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN JF - Journal of Mathematical Physics N2 - The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude. Y1 - 2024 U6 - https://doi.org/10.1063/5.0140764 VL - 65 SP - 013502 ER - TY - JOUR A1 - Sikorski, Alexander A1 - Weber, Marcus A1 - Schütte, Christof T1 - The Augmented Jump Chain JF - Advanced Theory and Simulations N2 - Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. A representation of non-autonomous Markov jump processes is presented as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so-called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated. Y1 - 2021 U6 - https://doi.org/10.1002/adts.202000274 VL - 4 IS - 4 PB - Wiley-VCH ER - TY - JOUR A1 - Sechi, Renata A1 - Sikorski, Alexander A1 - Weber, Marcus T1 - Estimation of the Koopman Generator by Newton's Extrapolation JF - Multiscale Modeling and Simulation N2 - This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multi-scale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods. Y1 - 2020 U6 - https://doi.org/10.1137/20M1333006 VL - 19 IS - 2 SP - 758 EP - 774 PB - SIAM ER - TY - JOUR A1 - Sikorski, Alexander A1 - Niknejad, Amir A1 - Weber, Marcus A1 - Donati, Luca T1 - Tensor-SqRA: Modeling the transition rates of interacting molecular systems in terms of potential energies JF - Journal of Chemical Physics N2 - Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA. Y1 - 2024 U6 - https://doi.org/10.1063/5.0187792 VL - 160 SP - 104112 ER - TY - CHAP A1 - Sikorski, Alexander A1 - Rabben, Robert Julian A1 - Chewle, Surahit A1 - Weber, Marcus ED - Fackeldey, K. T1 - Capturing the Macroscopic Behaviour of Molecular Dynamics with Membership Functions T2 - Mathematical Optimization for Machine Learning: Proceedings of the MATH+ Thematic Einstein Semester 2023 N2 - Markov processes serve as foundational models in many scientific disciplines, such as molecular dynamics, and their simulation forms a common basis for analysis. While simulations produce useful trajectories, obtaining macroscopic information directly from microstate data presents significant challenges. This paper addresses this gap by introducing the concept of membership functions being the macrostates themselves. We derive equations for the holding times of these macrostates and demonstrate their consistency with the classical definition. Furthermore, we discuss the application of the ISOKANN method for learning these quantities from simulation data. In addition, we present a novel method for extracting transition paths based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the 𝜇-opioid receptor. With this approach we provide a new perspective on analyzing the macroscopic behaviour of Markov systems. Y1 - 2024 U6 - https://doi.org/10.1515/9783111376776-004 SP - 41 EP - 58 PB - De Gruyter ER - TY - JOUR A1 - Frank, Anna-Simone A1 - Sikorski, Alexander A1 - Röblitz, Susanna T1 - Spectral clustering of Markov chain transition matrices with complex eigenvalues JF - Journal of Computational and Applied Mathematics N2 - The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided. Y1 - 2024 U6 - https://doi.org/10.1016/j.cam.2024.115791 VL - 444 SP - 115791 ER - TY - JOUR A1 - Kresse, Jakob A1 - Sikorski, Alexander A1 - Chewle, Surahit A1 - Sunkara, Vikram A1 - Weber, Marcus T1 - Revealing the Atomistic Mechanism of Rare Events in Molecular Dynamics JF - Journal of Chemical Theory and Computation N2 - Interpretable reaction coordinates are essential for understanding rare conformational transitions in molecular dynamics. The Atomistic Mechanism Of Rare Events in Molecular Dynamics (AMORE-MD) framework enhances interpretability of deep-learned reaction coordinates by connecting them to atomistic mechanisms, without requiring any a priori knowledge of collective variables, pathways, or endpoints. Here, AMORE-MD employs the ISOKANN algorithm to learn a neural membership function χ representing the dominant slow process, from which transition pathways are reconstructed as minimum-energy paths aligned with the gradient of χ, and atomic contributions are quantified through gradient-based sensitivity analysis. Iterative enhanced sampling further enriches transition regions and improves coverage of rare events enabling recovery of known mechanisms and chemically interpretable structural rearrangements at atomic resolution for the Müller-Brown potential, alanine dipeptide, and the elastin-derived hexapeptide VGVAPG. Y1 - 2026 U6 - https://doi.org/10.1021/acs.jctc.5c01906 VL - 22 IS - 5 SP - 2380 EP - 2389 ER - TY - JOUR A1 - Schütte, Christof A1 - Sikorski, Alexander A1 - Kresse, Jakob A1 - Weber, Marcus T1 - On-the-Fly Lifting of Coarse Reaction-Coordinate Paths to Full-Dimensional Transition Path Ensembles N2 - Effective dynamics on a low-dimensional collective-variable (CV) or latent space can be simulated far more cheaply than the underlying high-dimensional stochastic system, but exploiting such coarse predictions requires lifting: turning a coarse CV trajectory into dynamically consistent full-dimensional states and path ensembles, without relying on global sampling of invariant or conditional fiber measures. We present a local, on-the-fly lifting strategy based on guided full-system trajectories. First an effective model in CV space is used to obtain a coarse reference trajectory. Then, an ensemble of full-dimensional trajectories is generated from a guided version of the original dynamics, where the guidance steers the trajectory to track the CV reference path. Because guidance biases the path distribution, we correct it via pathwise Girsanov reweighting, yielding a correct-by-construction importance-sampling approximation of the conditional law of the uncontrolled dynamics. We further connect the approach to stochastic optimal control, clarifying how coarse models can inform variance-reducing guidance for rare-event quantities. Numerical experiments demonstrate that inexpensive coarse transition paths can be converted into realistic full-system transition pathways (including barrier crossings and detours) and can accelerate estimation of transition pathways and statistics while providing minimal bias through weighted ensembles. Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2603.23266 ER -