arXiv subjects

Marcus Weber

Publications and source records attributed to Marcus Weber.

At least 19 recordsLinked to original sources

arXiv2026

Boolean Algebra -- Driven Sepsis Diagnosis

Sepsis remains a diagnostic challenge due to its heterogeneous molecular signatures and complex immune responses. In this study, we develop a logical data analysis framework based on Boolean polynomial rings. This method constructs an ideal $\mathcal{I}$ of selection criteria that isolate empty subsets of previously analyzed patient data. This approach enables the derivation of interpretable classification rules based on biomarker profiles. We demonstrate that logical data analysis identifies distinct logical patterns for positive and negative sepsis classification. For instance, elevated levels of GLP-1 and MyD88 are associated with septic states in our dataset, whereas high TRAIL and low MyD88 concentrations may suggest a non-septic condition. Importantly, a new way to integrate expert knowledge to filter out potential overfitting or dataset-specific artifacts is shown. Our findings highlight the utility of logics in generating transparent, biologically plausible rules for a data-based and expert-based understanding of sepsis. Moreover, we show how data analysis can benefit from algebraic structures.

math.RA↗

arXiv2026

Effective Dynamics and Transition Pathways from Koopman-Inspired Neural Learning of Collective Variables

The ISOKANN (Invariant Subspaces of Koopman Operators Learned by Artificial Neural Networks) framework provides a data-driven route to extract collective variables (CVs) and effective dynamics from complex molecular systems. In this work, we integrate the theoretical foundation of Koopman operators with Krylov-like subspace algorithms, and reduced dynamical modeling to build a coherent picture of how to describe metastable transitions in high-dimensional systems based on CVs. Starting from the identification of CVs based on dominant invariant subspaces, we derive the corresponding effective dynamics on the latent space and connect these to transition rates and times, committor functions, and transition pathways. The combination of Koopman-based learning and reduced-dimensional effective dynamics yields a principled framework for computing transition rates and pathways from simulation data. Numerical experiments on one-, two-, and three-dimensional benchmark potentials illustrate the ability of ISOKANN to reconstruct the coarse-grained kinetics and reproduce transition times across enthalpic and entropic barriers.

math.DS↗

arXiv2026

On-the-Fly Lifting of Coarse Reaction-Coordinate Paths to Full-Dimensional Transition Path Ensembles

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.

math.DS↗

arXiv2026

Importance sampling of unbounded random stopping times: computing committor functions and exit rates without reweighting

Rare events in molecular dynamics are often related to noise-induced transitions between different macroscopic states (e.g., in protein folding). A common feature of these rare transitions is that they happen on timescales that are on average exponentially long compared to the characteristic timescale of the system, with waiting time distributions that have (sub)exponential tails and infinite support. As a result, sampling such rare events can lead to trajectories that can be become arbitrarily long, with not too low probability, which makes the reweighting of such trajectories a real challenge. Here, we discuss rare event simulation by importance sampling from a variational perspective, with a focus on {the computation of committor functions and mean first exit times that both play a prominent role in molecular dynamics}. The idea is to design importance sampling schemes that (a) reduce the variance of a rare event estimator while controlling the average length of the trajectories and (b) that do not require the reweighting of possibly very long trajectories. In doing so, we study different stochastic control formulations for committor and mean first exit times, which we compare both from a theoretical and a computational point of view, including numerical studies of some benchmark examples.

math.PR↗

arXiv2025

Revealing the Atomistic Mechanism of Rare Events in Molecular Dynamics

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 $\chi$ representing the dominant slow process, from which transition pathways are reconstructed as minimum-energy paths aligned with the gradient of $\chi$, 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 M\"uller-Brown potential, alanine dipeptide, and the elastin-derived hexapeptide VGVAPG.

physics.chem-ph↗

arXiv2024

Topological analysis reveals multiple pathways in molecular dynamics

Molecular Dynamics simulations are essential tools for understanding the dynamic behavior of biomolecules, yet extracting meaningful molecular pathways from these simulations remains challenging due to the vast amount of generated data. In this work, we present Molecular Kinetics via Topology (MoKiTo), a novel approach that combines the ISOKANN algorithm to determine the reaction coordinate of a molecular system with a topological analysis inspired by the Mapper algorithm. Our strategy efficiently identifies and characterizes distinct molecular pathways, enabling the detection and visualization of critical conformational transitions and rare events. This method offers deeper insights into molecular mechanisms, facilitating the design of targeted interventions in drug discovery and protein engineering.

physics.chem-ph↗

arXiv2024

Capturing the Macroscopic Behaviour of Molecular Dynamics with Membership Functions

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 from simulations based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the {\mu}-opioid receptor. With this approach we provide a new perspective on the analysis of macroscopic behaviour of Markov systems.

physics.chem-ph↗

arXiv2024

Towards a FAIR Documentation of Workflows and Models in Applied Mathematics

Modeling-Simulation-Optimization workflows play a fundamental role in applied mathematics. The Mathematical Research Data Initiative, MaRDI, responded to this by developing a FAIR and machine-interpretable template for a comprehensive documentation of such workflows. MaRDMO, a Plugin for the Research Data Management Organiser, enables scientists from diverse fields to document and publish their workflows on the MaRDI Portal seamlessly using the MaRDI template. Central to these workflows are mathematical models. MaRDI addresses them with the MathModDB ontology, offering a structured formal model description. Here, we showcase the interaction between MaRDMO and the MathModDB Knowledge Graph through an algebraic modeling workflow from the Digital Humanities. This demonstration underscores the versatility of both services beyond their original numerical domain.

cs.AI↗

arXiv2024

The Kramers turnover in terms of a macro-state projection on phase space

We have investigated how Langevin dynamics is affected by the friction coefficient using the novel algorithm ISOKANN, which combines the transfer operator approach with modern machine learning techniques. ISOKANN describes the dynamics in terms of an invariant subspace projection of the Koopman operator defined in the entire state space, avoiding approximations due to dimensionality reduction and discretization. Our results are consistent with the Kramers turnover and show that in the low and moderate friction regimes, metastable macro-states and transition rates are defined in phase space, not only in position space.

physics.chem-ph↗

arXiv2023

Tensor-SqRA: Modeling the Transition Rates of Interacting Molecular Systems in terms of Potential Energies

Estimating the rate of rare conformational changes in molecular systems is one of the goals of Molecular Dynamics simulations. In the past decades, a lot of progress has been done in data-based approaches towards 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 (PRE), 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.

physics.chem-ph↗

arXiv2023

Efficient Estimation of Transition Rates as Functions of pH

Extracting the kinetic properties of a system whose dynamics depend on the pH of the environment with which it exchanges energy and atoms requires sampling the Grand Canonical Ensemble. As an alternative, we present a novel strategy that requires simulating only the most recurrent Canonical Ensembles that compose the Grand Canonical Ensemble. The simulations are used to estimate the Gran Canonical distribution for a specific pH value by reweighting and to construct the transition rate matrix by discretizing the Fokker-Planck equation by Square Root Approximation and robust Perron Cluster Cluster Analysis. As an application, we have studied the tripeptide Ala-Asp-Ala.

physics.chem-ph↗

arXiv2022

Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN

For stochastic diffusion processes the dominant eigenfunctions of the corresponding Koopman operator contain important information about the slow-scale dynamics, that is, about the location and frequency of rare events. In this article, we reformulate the eigenproblem in terms of $\chi$-functions in the ISOKANN framework and discuss how optimal control and importance sampling allows for zero variance sampling of these functions. We provide a new formulation of the ISOKANN algorithm allowing for a proof of convergence and incorporate the optimal control result to obtain an adaptive iterative algorithm alternating between importance sampling and $\chi$-function approximation. We demonstrate the usage of our proposed method in experiments increasing the approximation accuracy by several orders of magnitude.

math.DS↗

arXiv2022

Research-Data Management Planning in the German Mathematical Community

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.

math.HO↗

arXiv2022

Qualitative Euclidean embedding of Disjoint Sets of Points

We consider two disjoint sets of points. If at least one of the sets can be embedded into an Euclidean space, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean embedding, the distances between the points are generated by a specific relation-preserving function. Consequently, the mutual distances between two points of the same set are specific qualitative transformations of their mutual distances in their original space; the pairwise distances between the points of different sets can be constructed from an arbitrary proximity function.

math.GM↗

arXiv2022

A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models

Dynamical reweighting methods permit to estimate kinetic observables of a stochastic process governed by a target potential $\tilde{V}(x)$ from trajectories that have been generated at a different potential $V(x)$. In this article, we present Girsanov reweighting and Square Root Approximation (SqRA): the first method reweights path probabilities exploiting the Girsanov theorem and can be applied to Markov State Models (MSMs) to reweight transition probabilities; the second method was originally developed to discretize the Fokker-Planck operator into a transition rate matrix, but here we implement it into a reweighting scheme for transition rates. We begin by reviewing the theoretical background of the methods, then present two applications relevant to Molecular Dynamics (MD), highlighting their strengths and weaknesses.

physics.chem-ph↗

arXiv2022

Assessing transition rates as functions of environmental variables

We present a method to estimate the transition rates of molecular systems under different environmental conditions which cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios", governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then we use the Square Root Approximation (SqRA) method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis (PCCA+) method to coarse-grain the kinetic model. This permits to efficiently estimate the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work we formalize the theoretical framework of the procedure and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies which take into account how the cellular environment influences biochemical processes.

physics.chem-ph↗

arXiv2022

Coding Reliability with Aclus -- Did I correctly characterize my observations?

Describing observations or objects in non-mathematical disciplines can often be accomplished by answering a list of questions. These questions can be formulated in such a way that the only possible answers always are ``yes'' or ``no''. This article is about automatically checking such given binary data sets for inconsistencies and about finding possible logical rules valid for the analyzed objects.

math.AC↗

arXiv2022

Deposition of nanosized amino acid functionalized bismuth oxido clusters on gold surfaces

Bismuth compounds are of growing interest with regard to potential applications in catalysis, medicine and electronics, for which their environmentally benign nature is one of the key factors. The most common starting material is bismuth nitrate, which easily hydrolyses to give a large number of condensation products. The so-called bismuth subnitrates are composed of bismuth oxido clusters of varying composition and nuclearity. One reason that hampers the further development of bismuth oxido-based materials, however is the low solubility of the subnitrates, which makes targeted immobilisation on substrates challenging. We present an approach towards solubilisation of bismuth oxido clusters by introducing an amino carboxylate as functional group and a study of the growth mode of these atom-precise nanoclusters on gold surfaces. For this purpose the bismuth oxido cluster [Bi38O45(NO3)20(dmso)28](NO3)4*4dmso (dmso=dimethyl sulfoxide) was reacted with the sodium salt of tert-butyloxycabonyl(Boc)-protected phenylalanine (Phe) to give the soluble and chiral nanocluster [Bi38O45(Boc-Phe)24(dmso)9]. The hydrodynamic diameter of the cluster was estimated with (1.4-1.6) nm (in CH3CN) and (2.2 nm-2.9) nm (in Ethanol) based on dynamic light scattering (DLS). The full exchange of the nitrates by the amino carboxylates was proven by NMR and FTIR as well as elemental analysis (EA) and XPS. The solubility of the bismuth oxido cluster in a protic as well as an aprotic polar organic solvent and the growth mode of the clusters on Au upon spin-, dip-, and drop-coating on gold surfaces were studied. Successful deposition of bismuth oxido cluster was proven by powder XRD, FTIR, and XPS while the microstructure of the resulting films was investigated as a function of the deposition method and the solvent used by SEM, AFM, and optical microscopy.

cond-mat.mtrl-sci↗