@article{FackeldeyKoltaiNeviretal.2019, author = {Fackeldey, Konstantin and Koltai, Peter and Nevir, Peter and Rust, Henning and Schild, Axel and Weber, Marcus}, title = {From metastable to coherent sets - Time-discretization schemes}, volume = {29}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, doi = {10.1063/1.5058128}, pages = {012101 -- 012101}, year = {2019}, abstract = {In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.}, language = {en} } @phdthesis{Quer2018, author = {Quer, Jannes}, title = {Importance Sampling for metastable dynamical systems in molecular dynamics}, year = {2018}, language = {en} } @article{DonatiHeidaKelleretal.2018, author = {Donati, Luca and Heida, Martin and Keller, Bettina G. and Weber, Marcus}, title = {Estimation of the infinitesimal generator by square-root approximation}, volume = {30}, journal = {J. Phys.: Condens. Matter}, number = {42}, doi = {10.1088/1361-648X/aadfc8}, pages = {425201 -- 425201}, year = {2018}, abstract = {In recent years, for the analysis of molecular processes, the estimation of time-scales and transition rates has become fundamental. Estimating the transition rates between molecular conformations is—from a mathematical point of view—an invariant subspace projection problem. We present a method to project the infinitesimal generator acting on function space to a low-dimensional rate matrix. This projection can be performed in two steps. First, we discretize the conformational space in a Voronoi tessellation, then the transition rates between adjacent cells is approximated by the geometric average of the Boltzmann weights of the Voronoi cells. This method demonstrates that there is a direct relation between the potential energy surface of molecular structures and the transition rates of conformational changes. We will show also that this approximation is correct and converges to the generator of the Smoluchowski equation in the limit of infinitely small Voronoi cells. We present results for a two dimensional diffusion process and alanine dipeptide as a high-dimensional system.}, language = {en} } @article{FackeldeySikorskiWeber2018, author = {Fackeldey, Konstantin and Sikorski, Alexander and Weber, Marcus}, title = {Spectral Clustering for Non-Reversible Markov Chains}, volume = {37}, journal = {Computational and Applied Mathematics}, number = {5}, doi = {https://doi.org/10.1007/s40314-018-0697-0}, pages = {6376 -- 6391}, year = {2018}, abstract = {Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We also extend this spectral clustering method to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension non-reversible Markov chains can be analyzed. Furthermore, the chains do not need to have a positive stationary distribution. In addition to metastabilities, dominant cycles and sinks can also be identified. This novel method is called GenPCCA (i.e., generalized PCCA), since it includes the case of non-reversible processes. We also apply the method to real-world eye-tracking data.}, language = {en} } @misc{BujotzekSchuettNielsenetal.2013, author = {Bujotzek, Alexander and Sch{\"u}tt, Ole and Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {Efficient Conformational Analysis by Partition-of-Unity Coupling}, journal = {Math Chem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42570}, year = {2013}, abstract = {Obtaining a sufficient sampling of conformational space is a common problem in molecular simulation. We present the implementation of an umbrella-like adaptive sampling approach based on function-based meshless discretization of conformational space that is compatible with state of the art molecular dynamics code and that integrates an eigenvector-based clustering approach for conformational analysis and the computation of inter-conformational transition rates. The approach is applied to three example systems, namely n-pentane, alanine dipeptide, and a small synthetic host-guest system, the latter two including explicitly modeled solvent.}, language = {en} } @article{SarichBanischHartmannetal.2013, author = {Sarich, Marco and Banisch, Ralf and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Markov State Models for Rare Events in Molecular Dynamics}, volume = {16}, journal = {Entropy (Special Issue)}, number = {1}, doi = {10.3390/e16010258}, pages = {258 -- 286}, year = {2013}, language = {en} } @misc{QuerDonatiKelleretal.2017, author = {Quer, Jannes and Donati, Luca and Keller, Bettina and Weber, Marcus}, title = {An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62075}, year = {2017}, abstract = {In this article we propose an adaptive importance sampling scheme for dynamical quantities of high dimensional complex systems which are metastable. The main idea of this article is to combine a method coming from Molecular Dynamics Simulation, Metadynamics, with a theorem from stochastic analysis, Girsanov's theorem. The proposed algorithm has two advantages compared to a standard estimator of dynamic quantities: firstly, it is possible to produce estimators with a lower variance and, secondly, we can speed up the sampling. One of the main problems for building importance sampling schemes for metastable systems is to find the metastable region in order to manipulate the potential accordingly. Our method circumvents this problem by using an assimilated version of the Metadynamics algorithm and thus creates a non-equilibrium dynamics which is used to sample the equilibrium quantities.}, language = {en} } @misc{WeberFackeldeySchuette2017, author = {Weber, Marcus and Fackeldey, Konstantin and Sch{\"u}tte, Christof}, title = {Set-free Markov State Building}, issn = {1438-0064}, doi = {10.1063/1.4978501}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62167}, year = {2017}, abstract = {Molecular dynamics (MD) simulations face challenging problems since the timescales of interest often are much longer than what is possible to simulate and even if sufficiently long simulation are possible the complex nature of the resulting simulation data makes interpretation difficult. Markov State Models (MSMs) help to overcome these problems by making experimentally relevant timescales accessible via coarse grained representations that also allows for convenient interpretation. However, standard set-based MSMs exhibit some caveats limiting their approximation quality and statistical significance. One of the main caveats results from the fact that typical MD trajectories repeatedly re-cross the boundary between the sets used to build the MSM which causes statistical bias in estimating the transition probabilities between these sets. In this article, we present a set-free approach to MSM building utilizing smooth overlapping ansatz functions instead of sets and an adaptive refinement approach. This kind of meshless discretization helps to overcome the recrossing problem and yields an adaptive refinement procedure that allows to improve the quality of the model while exploring state space and inserting new ansatz functions into the MSM.}, language = {en} } @misc{Weber2017, author = {Weber, Marcus}, title = {Eigenvalues of non-reversible Markov chains - A case study}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62191}, year = {2017}, abstract = {Finite reversible Markov chains are characterized by a transition matrix P that has real eigenvalues and pi-orthogonal eigenvectors, where pi is the stationary distribution of P. This means, that a transition matrix with complex eigenvalues corresponds to a non-reversible Markov chain. This observation leads to the question, whether the imaginary part of that eigendecomposition corresponds to or indicates the "pattern" of the nonreversibility. This article shows that the direct relation between imaginary parts of eigendecompositions and the non-reversibility of a transition matrix is not given. It is proposed to apply the Schur decomposition of P instead of the eigendecomposition in order to characterize its nonreversibility.}, language = {en} } @article{HartmannSchuetteWeberetal.2017, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Weber, Marcus and Zhang, Wei}, title = {Importance sampling in path space for diffusion processes with slow-fast variables}, journal = {Probability Theory and Related Fields}, doi = {10.1007/s00440-017-0755-3}, pages = {1 -- 52}, year = {2017}, abstract = {Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.}, language = {en} } @article{SarichDjurdjevacConradBruckneretal.2014, author = {Sarich, Marco and Djurdjevac Conrad, Natasa and Bruckner, Sharon and Conrad, Tim and Sch{\"u}tte, Christof}, title = {Modularity revisited: A novel dynamics-based concept for decomposing complex networks}, volume = {1}, journal = {Journal of Computational Dynamics}, number = {1}, doi = {10.3934/jcd.2014.1.191}, pages = {191 -- 212}, year = {2014}, language = {en} } @misc{Zwingel2015, type = {Master Thesis}, author = {Zwingel, Oliver}, title = {Die Cross-Entropy-Methode f{\"u}r stochastische Optimierungsprobleme}, year = {2015}, language = {de} } @misc{Lange2015, type = {Master Thesis}, author = {Lange, Christoph}, title = {Approximating and Clustering Markov Chains via Kullback-Leibler-Divergence}, year = {2015}, language = {en} } @misc{Nielsen2016, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59933}, year = {2016}, abstract = {In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the \$L^2\$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains.}, language = {en} } @article{Nielsen2016, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, volume = {118}, journal = {Statistics \& Probability Letters}, publisher = {Elsevier}, doi = {10.1016/j.spl.2016.06.011}, pages = {163 -- 170}, year = {2016}, abstract = {In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the \$L^2\$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains.}, language = {en} } @article{HartmannRichterSchuetteetal.2017, author = {Hartmann, Carsten and Richter, Lorenz and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Variational characterization of free energy: theory and algorithms}, volume = {19}, journal = {Entropy}, number = {11}, doi = {10.3390/e19110626}, pages = {626}, year = {2017}, language = {en} } @article{QuerDonatiKelleretal.2018, author = {Quer, Jannes and Donati, Luca and Keller, Bettina and Weber, Marcus}, title = {An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates}, volume = {40}, journal = {SIAM Journal on Scientific Computing}, number = {2}, doi = {10.1137/17m1124772}, pages = {A653 -- A670}, year = {2018}, abstract = {In this article we propose an adaptive importance sampling scheme for dynamical quantities of high dimensional complex systems which are metastable. The main idea of this article is to combine a method coming from Molecular Dynamics Simulation, Metadynamics, with a theorem from stochastic analysis, Girsanov's theorem. The proposed algorithm has two advantages compared to a standard estimator of dynamic quantities: firstly, it is possible to produce estimators with a lower variance and, secondly, we can speed up the sampling. One of the main problems for building importance sampling schemes for metastable systems is to find the metastable region in order to manipulate the potential accordingly. Our method circumvents this problem by using an assimilated version of the Metadynamics algorithm and thus creates a non-equilibrium dynamics which is used to sample the equilibrium quantities.}, language = {en} } @misc{Weber2018, author = {Weber, Marcus}, title = {Supplementary: Implications of PCCA+ in Molecular Simulation}, year = {2018}, abstract = {Matlab-software and data sets to recapitulate the presented results in M. Weber: Implications of PCCA+ in Molecular Simulation. Computation, 6(1):20, 2018.}, language = {en} } @article{KoltaiWuNoeetal.2018, author = {Koltai, Peter and Wu, Hao and No{\´e}, Frank and Sch{\"u}tte, Christof}, title = {Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics}, volume = {6}, journal = {Computation}, number = {1}, publisher = {MDPI}, address = {Basel, Switzerland}, doi = {10.3390/computation6010022}, pages = {22}, year = {2018}, language = {en} } @article{Weber2018, author = {Weber, Marcus}, title = {Implications of PCCA+ in Molecular Simulation}, volume = {6}, journal = {Computation}, number = {1}, doi = {10.3390/computation6010020}, pages = {20}, year = {2018}, abstract = {Upon ligand binding or during chemical reactions the state of a molecular system changes in time. Usually we consider a finite set of (macro-) states of the system (e.g., 'bound' vs. 'unbound'), although the process itself takes place in a continuous space. In this context, the formula chi=XA connects the micro-dynamics of the molecular system to its macro-dynamics. Chi can be understood as a clustering of micro-states of a molecular system into a few macro-states. X is a basis of an invariant subspace of a transfer operator describing the micro-dynamics of the system. The formula claims that there is an unknown linear relation A between these two objects. With the aid of this formula we can understand rebinding effects, the electron flux in pericyclic reactions, and systematic changes of binding rates in kinetic ITC experiments. We can also analyze sequential spectroscopy experiments and rare event systems more easily. This article provides an explanation of the formula and an overview of some of its consequences.}, language = {en} } @article{HartmannJoesterSchuetteetal.2026, author = {Hartmann, Carsten and J{\"o}ster, Annika and Sch{\"u}tte, Christof and Sikorski, Alexander and Weber, Marcus}, title = {Importance sampling of unbounded random stopping times: computing committor functions and exit rates without reweighting}, arxiv = {http://arxiv.org/abs/2601.01489}, year = {2026}, abstract = {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 applications in molecular dynamics, in particular the computation of committor functions. 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.}, language = {en} } @article{ZhangWangHartmannetal.2014, author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, volume = {36}, journal = {Siam Journal on Scientific Computing}, number = {6}, doi = {10.1137/14096493X}, pages = {A2654 -- A2672}, year = {2014}, language = {en} } @misc{SarichSchuette2014, author = {Sarich, Marco and Sch{\"u}tte, Christof}, title = {Markov Model Theory}, volume = {797}, journal = {An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation}, editor = {Bowman, Gregory R. and Pande, Vijay S. and No{\´e}, Frank}, publisher = {Springer}, doi = {10.1007/978-94-007-7606-7}, pages = {23 -- 44}, year = {2014}, abstract = {This section reviews the relation between the continuous dynamics of a molecular system in thermal equilibrium and the kinetics given by a Markov State Model (MSM). We will introduce the dynamical propagator, an error-less, alternative description of the continuous dynamics, and show how MSMs result from its discretization. This allows for an precise understanding of the approximation quality of MSMs in comparison to the continuous dynamics. The results on the approximation quality are key for the design of good MSMs. While this section is important for understanding the theory of discretization and related systematic errors, practitioners wishing only to learn how to construct MSMs may skip directly to the discussion of Markov model estimation.}, language = {en} } @article{SchuetteNielsenWeber2015, author = {Sch{\"u}tte, Christof and Nielsen, Adam and Weber, Marcus}, title = {Markov State Models and Molecular Alchemy}, volume = {113}, journal = {Molecular Physics}, number = {1}, doi = {10.1080/00268976.2014.944597}, pages = {69 -- 78}, year = {2015}, abstract = {In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.}, language = {en} } @misc{Lie2012, type = {Master Thesis}, author = {Lie, Han Cheng}, title = {A Meshless Method for Conformational Analysis}, year = {2012}, language = {en} } @misc{WangSchuette2014, author = {Wang, Han and Sch{\"u}tte, Christof}, title = {Building Markov State Models for Periodically Driven Non-Equilibrium Systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53167}, year = {2014}, abstract = {Recent years have seen an increased interest in non-equilibrium molecular dynamics (NEMD) simulations, especially for molecular systems with periodic forcing by external fields, e.g., in the context of studying effects of electromagnetic radiation on the human body tissue. Lately, an NEMD methods with local thermostating has been proposed that allows for studying non-equilibrium processes in a statistically reliable and thermodynamically consistent way. In this article, we demonstrate how to construct Markov State Models (MSMs) for such NEMD simulations. MSM building has been well-established for systems in equilibrium where MSMs with just a few (macro-)states allow for accurate reproduction of the essential kinetics of the molecular system under consideration. Non-equilibrium MSMs have been lacking so far. The article presents how to construct such MSMs and illustrates their validity and usefulness for the case of conformation dynamics of alanine dipeptide in an external electric field.}, language = {en} } @article{NielsenWeber2015, author = {Nielsen, Adam and Weber, Marcus}, title = {Computing the nearest reversible Markov chain}, volume = {22}, journal = {Numerical Linear Algebra with Applications}, number = {3}, doi = {10.1002/nla.1967}, pages = {483 -- 499}, year = {2015}, abstract = {Reversible Markov chains are the basis of many applications. However, computing transition probabilities by a finite sampling of a Markov chain can lead to truncation errors. Even if the original Markov chain is reversible, the approximated Markov chain might be non-reversible and will lose important properties, like the real valued spectrum. In this paper, we show how to find the closest reversible Markov chain to a given transition matrix. It turns out that this matrix can be computed by solving a convex minimization problem.}, language = {en} } @book{OPUS4-5339, title = {Molecular Dynamics Simulation}, journal = {Entropy (special issue)}, editor = {Ciccotti, Giovanni and Ferrario, Mauro and Sch{\"u}tte, Christof}, publisher = {Multidisciplinary Digital Publishing Institute (MDPI)}, year = {2014}, language = {en} } @article{HartmannSchuetteZhang2016, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Model reduction algorithms for optimal control and importance sampling of diffusions}, volume = {29}, journal = {Nonlinearity}, number = {8}, doi = {10.1088/0951-7715/29/8/2298}, pages = {2298 -- 2326}, year = {2016}, abstract = {We propose numerical algorithms for solving optimal control and importance sampling problems based on simplified models. The algorithms combine model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only a few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either a reaction coordinate onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behavior in the small noise regime. No explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples.}, language = {en} } @article{KlusKoltaiSchuette2016, author = {Klus, Stefan and Koltai, Peter and Sch{\"u}tte, Christof}, title = {On the numerical approximation of the Perron-Frobenius and Koopman operator}, volume = {3}, journal = {Journal of Computational Dynamics}, number = {1}, doi = {10.3934/jcd.2016003}, pages = {51 -- 77}, year = {2016}, abstract = {Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di� fferent methods that have been developed over the last decades to compute � infinite-dimensional approximations of these in� finite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di� fferences between these approaches. The results will be illustrated using simple stochastic di� fferential equations and molecular dynamics examples.}, language = {en} } @article{KlusNueskeKoltaietal.2018, author = {Klus, Stefan and N{\"u}ske, Feliks and Koltai, Peter and Wu, Hao and Kevrekidis, Ioannis and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {Data-driven model reduction and transfer operator approximation}, volume = {28}, journal = {Journal of Nonlinear Science}, number = {3}, doi = {10.1007/s00332-017-9437-7}, pages = {985 -- 1010}, year = {2018}, language = {en} } @article{RuedrichSarichSchuette2017, author = {R{\"u}drich, S. and Sarich, Marco and Sch{\"u}tte, Christof}, title = {Utilizing hitting times for finding metastable sets in non-reversible Markov chains}, journal = {Journal of Comp. Dynamics}, year = {2017}, language = {en} } @article{WeberFackeldeySchuette2017, author = {Weber, Marcus and Fackeldey, Konstantin and Sch{\"u}tte, Christof}, title = {Set-Free Markov State Model Building}, volume = {146}, journal = {Journal of Chemical Physics}, number = {12}, doi = {10.1063/1.4978501}, year = {2017}, language = {en} } @article{QuerLie2017, author = {Quer, Jannes and Lie, Han Cheng}, title = {Some connections between importance sampling and enhanced sampling methods in molecular dynamics}, journal = {Journal of Chemical Physics}, year = {2017}, abstract = {Enhanced sampling methods play an important role in molecular dynamics, because they enable the collection of better statistics of rare events that are important in many physical phenomena. We show that many enhanced sampling methods can be viewed as methods for performing importance sampling, by identifying important correspondences between the language of molecular dynamics and the language of probability theory. We illustrate these connections by highlighting the similarities between the rare event simulation method of Hartmann and Sch{\"u}tte (J. Stat. Mech. Theor. Exp., 2012), and the enhanced sampling method of Valsson and Parrinello (Phys. Rev. Lett. 113, 090601). We show that the idea of changing a probability measure is fundamental to both enhanced sampling and importance sampling.}, language = {en} } @misc{QuerLie2017, author = {Quer, Jannes and Lie, Han Cheng}, title = {Some connections between importance sampling and enhanced sampling methods in molecular dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64289}, year = {2017}, abstract = {Enhanced sampling methods play an important role in molecular dynamics, because they enable the collection of better statistics of rare events that are important in many physical phenomena. We show that many enhanced sampling methods can be viewed as methods for performing importance sampling, by identifying important correspondences between the language of molecular dynamics and the language of probability theory. We illustrate these connections by highlighting the similarities between the rare event simulation method of Hartmann and Sch{\"u}tte (J. Stat. Mech. Theor. Exp., 2012), and the enhanced sampling method of Valsson and Parrinello (Phys. Rev. Lett. 113, 090601). We show that the idea of changing a probability measure is fundamental to both enhanced sampling and importance sampling.}, language = {en} } @article{FackeldeyNiknejadWeber2017, author = {Fackeldey, Konstantin and Niknejad, Amir and Weber, Marcus}, title = {Finding Metastabilities in Reversible Markov Chains based on Incomplete Sampling: Case of Molecular Simulation}, journal = {Special Matrices}, number = {5/1}, doi = {10.1515/spma-2017-0006}, pages = {73 -- 81}, year = {2017}, language = {en} } @article{FackeldeyWeber2017, author = {Fackeldey, Konstantin and Weber, Marcus}, title = {GenPCCA -- Markov State Models for Non-Equilibrium Steady States}, journal = {Big data clustering: Data preprocessing, variable selection, and dimension reduction. WIAS Report No. 29}, doi = {10.20347/WIAS.REPORT.29}, pages = {70 -- 80}, year = {2017}, language = {en} } @misc{ZhangWangHartmannetal.2014, author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, issn = {1438-0064}, doi = {10.1137/14096493X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-49720}, year = {2014}, abstract = {We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.}, language = {en} } @misc{WeberQuer2015, author = {Weber, Marcus and Quer, Jannes}, title = {Estimating exit rates in rare event dynamical systems via extrapolation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-56622}, year = {2015}, abstract = {In this article we present a new idea for approximating exit rates for diffusion processes living in a craggy landscape. We are especially interested in the exit rates of a process living in a metastable regions. Due to the fact that Monte Carlo simulations perform quite poor and are very computational expensive in this setting we create several similar situations with a smoothed potential. For this we introduce a new parameter \$\lambda \in [0,1]\$ (\$\lambda = 1\$ very smoothed potential, \$\lambda=0\$ original potential) into the potential which controls the influence the smoothing. We then sample the exit rate for different parameters \$\lambda\$ the exit rate from a given region. Due to the fact that \$\lambda\$ is connected to the exit rate we can use this dependency to approximate the real exit rate. The method can be seen as something between hyperdynamics and temperature accelerated MC.}, language = {en} } @article{BujotzekSchuettNielsenetal.2014, author = {Bujotzek, Alexander and Sch{\"u}tt, Ole and Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {ZIBgridfree: Efficient Conformational Analysis by Partition-of-Unity Coupling}, volume = {52}, journal = {Journal of Mathematical Chemistry}, number = {3}, doi = {10.1007/s10910-013-0265-1}, pages = {781 -- 804}, year = {2014}, language = {de} } @misc{FackeldeyKoltaiNeviretal.2017, author = {Fackeldey, Konstantin and Koltai, P{\´e}ter and N{\´e}vir, Peter and Rust, Henning and Schild, Axel and Weber, Marcus}, title = {From Metastable to Coherent Sets - time-discretization schemes}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66074}, year = {2017}, abstract = {Given a time-dependent stochastic process with trajectories x(t) in a space \$\Omega\$, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space \$M\subset\Omega\$, coherent sets \$M(t)\subset\Omega\$ are defined in space and time. Hence, if we extend the space by the time-variable t, coherent sets are metastable sets in \$\Omega\times[0,\infty]\$. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.}, language = {en} } @article{LorenzWolfBraunetal.2017, author = {Lorenz, Philipp and Wolf, Frederik and Braun, Jonas and Djurdjevac Conrad, Natasa and H{\"o}vel, Philipp}, title = {Capturing the Dynamics of Hashtag-Communities}, journal = {International Workshop on Complex Networks and their Applications}, doi = {10.1007/978-3-319-72150-7_33}, pages = {401 -- 413}, year = {2017}, language = {en} } @article{HeidaSikorskiWeber2022, author = {Heida, Martin and Sikorski, Alexander and Weber, Marcus}, title = {Consistency and order 1 convergence of cell-centered finite volume discretizations of degenerate elliptic problems in any space dimension}, journal = {SIAM Journal on Numerical Analysis}, doi = {10.20347/WIAS.PREPRINT.2913}, year = {2022}, abstract = {We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension \$d \geq 2\$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete \$L^2\$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has \$H^2\$ regularity.}, language = {en} } @article{SechiWeberHeyne2021, author = {Sechi, Renata and Weber, Marcus and Heyne, Karsten}, title = {MSM building and projection for the analysis of time-resolved spectra}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {21}, doi = {https://doi.org/10.1002/pamm.202100102}, url = {http://nbn-resolving.de/https://onlinelibrary.wiley.com/doi/10.1002/pamm.202100102}, pages = {e202100102}, year = {2021}, abstract = {Understanding the kinetics between the components of time-resolved spectra is a crucial step in the study of photo-activatedprocesses. However, modeling the kinetics requires usually some a priori knowledge about the system. In our approach, webuild a Markov State Model (MSM) from the spectral data, and obtain a Koopman transition matrix K(t). With genPCCA,an invariant subspace projection, we project the process into its metastable components. The result of the application of gen-PCCA is a transition matrix Kc(t), from which we can read the transition probability between the metastable components of the reaction. We discuss the application of this analysis method to the transient absorption spectrum of brominated Al-corrole}, language = {en} } @misc{Sikorski2024, author = {Sikorski, Alexander}, title = {ISOKANN.jl}, doi = {10.5281/zenodo.11519359}, year = {2024}, abstract = {Julia pacarkge implementing the ISOKANN algorithm for the computation of invariant subspaces of Koopman operators}, language = {en} } @misc{RiberaBorrellQuerRichteretal.2021, author = {Ribera Borrell, Enric and Quer, Jannes and Richter, Lorenz and Sch{\"u}tte, Christof}, title = {Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics}, issn = {1438-0064}, year = {2021}, abstract = {Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings.}, language = {en} } @article{SchuetteKlusHartmann2023, author = {Sch{\"u}tte, Christof and Klus, Stefan and Hartmann, Carsten}, title = {Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning}, volume = {32}, journal = {Acta Numerica}, doi = {10.1017/S0962492923000016}, pages = {517 -- 673}, year = {2023}, abstract = {One of the main challenges in molecular dynamics is overcoming the 'timescale barrier': in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.}, language = {en} } @misc{Sikorski2023, author = {Sikorski, Alexander}, title = {VoronoiGraph.jl}, doi = {10.5281/zenodo.6417636}, year = {2023}, abstract = {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.}, language = {en} } @article{DonatiWeberKeller2021, author = {Donati, Luca and Weber, Marcus and Keller, Bettina G.}, title = {Markov models from the square root approximation of the Fokker-Planck equation: calculating the grid-dependent flux}, volume = {33}, journal = {Journal of Physics: Condensed Matter}, number = {11}, doi = {10.1088/1361-648X/abd5f7}, pages = {115902}, year = {2021}, abstract = {Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to \$\mathcal{O}\left(1{0}^{6}\right)\$ states can be obtained within seconds on a single high-performance compute server if regular grids are used.}, language = {en} } @article{CoomberChewleSeckeretal.2025, author = {Coomber, Celvic and Chewle, Surahit and Secker, Christopher and Fackeldey, Konstantin and Weber, Marcus and Winkelmann, Stefanie and Sch{\"u}tte, Christof and Sunkara, Vikram}, title = {Investigating Endogenous Opioids Unravels the Mechanisms Behind Opioid-Induced Constipation, a Mathematical Modeling Approach}, volume = {26}, journal = {International Journal of Molecular Sciences}, number = {13}, doi = {10.3390/ijms26136207}, year = {2025}, abstract = {Endogenous opioids, such as Endomorphin-2, are not typically associated with severe constipation, unlike pharmaceutical opioids, which induce opioid-induced constipation (OIC) by activating μ-opioid receptors in the gastrointestinal tract. In this study, we present a mathematical model, which integrates the serotonergic and opioid pathways, simulating the interaction between serotonin and opioid signaling within the enteric nervous system (ENS). The model explores the mechanisms underlying OIC, with a focus on the change in adenylyl cyclase (AC) activity, cAMP accumulation, and the distinct functionalities of Endomorphin-2 compared to commonly used pharmaceutical opioids. We study the effects of Morphine, Fentanyl, and Methadone and contrast them with Endomorphin-2. Our findings reveal that opioids do not perturb the signaling of serotonin, but only the activity of AC, suggesting that serotonin levels have no influence on improving opioid-induced constipation. Furthermore, this study reveals that the primary difference between endogenous and pharmaceutical opioids is their degradation rates. This finding shows that modulating opioid degradation rates significantly improves cAMP recovery. In conclusion, our insights steer towards exploring opioid degrading enzymes, localized to the gut, as a strategy for mitigating OIC.}, language = {en} } @article{RiberaBorrellQuerRichteretal.2023, author = {Ribera Borrell, Enric and Quer, Jannes and Richter, Lorenz and Sch{\"u}tte, Christof}, title = {Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics}, volume = {89}, journal = {SIAM Journal on Scientific Computing (SISC)}, number = {1}, doi = {10.1137/22M1503464}, year = {2023}, abstract = {Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings.}, language = {en} } @article{SikorskiRiberaBorrellWeber2024, author = {Sikorski, Alexander and Ribera Borrell, Enric and Weber, Marcus}, title = {Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN}, volume = {65}, journal = {Journal of Mathematical Physics}, arxiv = {http://arxiv.org/abs/2301.00065}, doi = {10.1063/5.0140764}, pages = {013502}, year = {2024}, abstract = {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.}, language = {en} } @article{SikorskiWeberSchuette2021, author = {Sikorski, Alexander and Weber, Marcus and Sch{\"u}tte, Christof}, title = {The Augmented Jump Chain}, volume = {4}, journal = {Advanced Theory and Simulations}, number = {4}, publisher = {Wiley-VCH}, arxiv = {http://arxiv.org/abs/2008.04624}, doi = {10.1002/adts.202000274}, year = {2021}, abstract = {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.}, language = {en} } @article{SechiSikorskiWeber2021, author = {Sechi, Renata and Sikorski, Alexander and Weber, Marcus}, title = {Estimation of the Koopman Generator by Newton's Extrapolation}, volume = {19}, journal = {Multiscale Modeling and Simulation}, number = {2}, publisher = {SIAM}, doi = {10.1137/20M1333006}, pages = {758 -- 774}, year = {2021}, abstract = {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.}, language = {en} } @misc{SchuetteKlusHartmann2022, author = {Sch{\"u}tte, Christof and Klus, Stefan and Hartmann, Carsten}, title = {Overcoming the Timescale Barrier in Molecular Dynamics: Transfer Operators, Variational Principles, and Machine Learning}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88637}, year = {2022}, abstract = {One of the main challenges in molecular dynamics is overcoming the "timescale barrier", a phrase used to describe that in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, not even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics, and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behavior on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory as well as the algorithmic development from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in MD. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.}, language = {en} } @article{QuerRiberaBorrell2024, author = {Quer, Jannes and Ribera Borrell, Enric}, title = {Connecting Stochastic Optimal Control and Reinforcement Learning}, volume = {65}, journal = {Journal of Mathematical Physics}, number = {8}, doi = {10.1063/5.0140665}, year = {2024}, abstract = {In this paper the connection between stochastic optimal control and reinforcement learning is investigated. Our main motivation is to apply importance sampling to sampling rare events which can be reformulated as an optimal control problem. By using a parameterised approach the optimal control problem becomes a stochastic optimization problem which still raises some open questions regarding how to tackle the scalability to high-dimensional problems and how to deal with the intrinsic metastability of the system. To explore new methods we link the optimal control problem to reinforcement learning since both share the same underlying framework, namely a Markov Decision Process (MDP). For the optimal control problem we show how the MDP can be formulated. In addition we discuss how the stochastic optimal control problem can be interpreted in the framework of reinforcement learning. At the end of the article we present the application of two different reinforcement learning algorithms to the optimal control problem and a comparison of the advantages and disadvantages of the two algorithms.}, language = {en} } @article{FrankSikorskiRoeblitz2024, author = {Frank, Anna-Simone and Sikorski, Alexander and R{\"o}blitz, Susanna}, title = {Spectral clustering of Markov chain transition matrices with complex eigenvalues}, volume = {444}, journal = {Journal of Computational and Applied Mathematics}, arxiv = {http://arxiv.org/abs/2206.14537}, doi = {10.1016/j.cam.2024.115791}, pages = {115791}, year = {2024}, abstract = {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.}, language = {de} } @article{LelievreZhang2019, author = {Leli{\`e}vre, Tony and Zhang, Wei}, title = {Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates}, journal = {Multiscale Modeling and Simulation}, number = {17}, arxiv = {http://arxiv.org/abs/1805.01928}, doi = {10.1137/18M1186034}, pages = {1019 -- 1051}, year = {2019}, abstract = {Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Leli{\`e}vre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Leli{\`e}vre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.}, language = {en} } @article{HartmannSchuetteZhang2019, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms}, volume = {175}, journal = {Journal of Statistical Physics}, number = {6}, arxiv = {http://arxiv.org/abs/1803.09347}, doi = {10.1007/s10955-019-02286-4}, pages = {1214 -- 1261}, year = {2019}, abstract = {In this paper, we study Jarzynski's equality and fluctuation theorems for diffusion processes. While some of the results considered in the current work are known in the (mainly physics) literature, we review and generalize these nonequilibrium theorems using mathematical arguments, therefore enabling further investigations in the mathematical community. On the numerical side, variance reduction approaches such as importance sampling method are studied in order to compute free energy differences based on Jarzynski's equality.}, language = {en} } @article{AkhyarZhangStoltzetal.2025, author = {Akhyar, Fatima-Zahrae and Zhang, Wei and Stoltz, Gabriel and Sch{\"u}tte, Christof}, title = {Generative modeling of conditional probability distributions on the level-sets of collective variables}, arxiv = {http://arxiv.org/abs/2512.17374}, year = {2025}, abstract = {Given a probability distribution \$\mu\$ in \$\mathbb{R}^d\$ represented by data, we study in this paper the generative modeling of its conditional probability distributions on the level-sets of a collective variable \$\xi: \mathbb{R}^d \rightarrow \mathbb{R}^k\$, where \$1 \le k