@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{SchulzePeppertSchuetteetal.2025, author = {Schulze, Kenrick and Peppert, Felix and Sch{\"u}tte, Christof and Sunkara, Vikram}, title = {Chimeric U-Net - Modifying the standard U-Net towards Explainability}, volume = {338}, journal = {Artificial Intelligence}, doi = {10.1016/j.artint.2024.104240}, year = {2025}, abstract = {Healthcare guided by semantic segmentation has the potential to improve our quality of life through early and accurate disease detection. Convolutional Neural Networks, especially the U-Net-based architectures, are currently the state-of-the-art learning-based segmentation methods and have given unprecedented performances. However, their decision-making processes are still an active field of research. In order to reliably utilize such methods in healthcare, explainability of how the segmentation was performed is mandated. To date, explainability is studied and applied heavily in classification tasks. In this work, we propose the Chimeric U-Net, a U-Net architecture with an invertible decoder unit, that inherently brings explainability into semantic segmentation tasks. We find that having the restriction of an invertible decoder does not hinder the performance of the segmentation task. However, the invertible decoder helps to disentangle the class information in the latent space embedding and to construct meaningful saliency maps. Furthermore, we found that with a simple k-Nearest-Neighbours classifier, we could predict the Intersection over Union scores of unseen data, demonstrating that the latent space, constructed by the Chimeric U-Net , encodes an interpretable representation of the segmentation quality. Explainability is an emerging field, and in this work, we propose an alternative approach, that is, rather than building tools for explaining a generic architecture, we propose constraints on the architecture which induce explainability. With this approach, we could peer into the architecture to reveal its class correlations and local contextual dependencies, taking an insightful step towards trustworthy and reliable AI. Code to build and utilize the Chimeric U-Net is made available under: https://github.com/kenrickschulze/Chimeric-UNet---Half-invertible-UNet-in-Pytorch}, language = {en} } @article{SherrattSrivastavaAinslieetal.2024, author = {Sherratt, Katharine and Srivastava, Ajitesh and Ainslie, Kylie and Singh, David E. and Cublier, Aymar and Marinescu, Maria Cristina and Carretero, Jesus and Garcia, Alberto Cascajo and Franco, Nicolas and Willem, Lander and Abrams, Steven and Faes, Christel and Beutels, Philippe and Hens, Niel and M{\"u}ller, Sebastian and Charlton, Billy and Ewert, Ricardo and Paltra, Sydney and Rakow, Christian and Rehmann, Jakob and Conrad, Tim and Sch{\"u}tte, Christof and Nagel, Kai and Abbott, Sam and Grah, Rok and Niehus, Rene and Prasse, Bastian and Sandmann, Frank and Funk, Sebastian}, title = {Characterising information gains and losses when collecting multiple epidemic model outputs}, volume = {47}, journal = {Epidemics}, publisher = {Elsevier BV}, issn = {1755-4365}, doi = {10.1016/j.epidem.2024.100765}, year = {2024}, abstract = {Collaborative comparisons and combinations of epidemic models are used as policy-relevant evidence during epidemic outbreaks. In the process of collecting multiple model projections, such collaborations may gain or lose relevant information. Typically, modellers contribute a probabilistic summary at each time-step. We compared this to directly collecting simulated trajectories. We aimed to explore information on key epidemic quantities; ensemble uncertainty; and performance against data, investigating potential to continuously gain information from a single cross-sectional collection of model results. Methods We compared July 2022 projections from the European COVID-19 Scenario Modelling Hub. Five modelling teams projected incidence in Belgium, the Netherlands, and Spain. We compared projections by incidence, peaks, and cumulative totals. We created a probabilistic ensemble drawn from all trajectories, and compared to ensembles from a median across each model's quantiles, or a linear opinion pool. We measured the predictive accuracy of individual trajectories against observations, using this in a weighted ensemble. We repeated this sequentially against increasing weeks of observed data. We evaluated these ensembles to reflect performance with varying observed data. Results. By collecting modelled trajectories, we showed policy-relevant epidemic characteristics. Trajectories contained a right-skewed distribution well represented by an ensemble of trajectories or a linear opinion pool, but not models' quantile intervals. Ensembles weighted by performance typically retained the range of plausible incidence over time, and in some cases narrowed this by excluding some epidemic shapes. Conclusions. We observed several information gains from collecting modelled trajectories rather than quantile distributions, including potential for continuously updated information from a single model collection. The value of information gains and losses may vary with each collaborative effort's aims, depending on the needs of projection users. Understanding the differing information potential of methods to collect model projections can support the accuracy, sustainability, and communication of collaborative infectious disease modelling efforts. Data availability All code and data available on Github: https://github.com/covid19-forecast-hub-europe/aggregation-info-loss}, language = {en} } @article{GaskinConradPavliotisetal.2024, author = {Gaskin, Thomas and Conrad, Tim and Pavliotis, Grigorios A. and Sch{\"u}tte, Christof}, title = {Neural parameter calibration and uncertainty quantification for epidemic forecasting}, volume = {19}, journal = {PLOS ONE}, number = {10}, arxiv = {http://arxiv.org/abs/2312.03147}, doi = {10.1371/journal.pone.0306704}, year = {2024}, abstract = {The recent COVID-19 pandemic has thrown the importance of accurately forecasting contagion dynamics and learning infection parameters into sharp focus. At the same time, effective policy-making requires knowledge of the uncertainty on such predictions, in order, for instance, to be able to ready hospitals and intensive care units for a worst-case scenario without needlessly wasting resources. In this work, we apply a novel and powerful computational method to the problem of learning probability densities on contagion parameters and providing uncertainty quantification for pandemic projections. Using a neural network, we calibrate an ODE model to data of the spread of COVID-19 in Berlin in 2020, achieving both a significantly more accurate calibration and prediction than Markov-Chain Monte Carlo (MCMC)-based sampling schemes. The uncertainties on our predictions provide meaningful confidence intervals e.g. on infection figures and hospitalisation rates, while training and running the neural scheme takes minutes where MCMC takes hours. We show convergence of our method to the true posterior on a simplified SIR model of epidemics, and also demonstrate our method's learning capabilities on a reduced dataset, where a complex model is learned from a small number of compartments for which data is available.}, language = {en} } @article{KostreSunkaraSchuetteetal.2022, author = {Kostr{\´e}, Margarita and Sunkara, Vikram and Sch{\"u}tte, Christof and Djurdjevac Conrad, Natasa}, title = {Understanding the Romanization Spreading on Historical Interregional Networks in Northern Tunisia}, volume = {7}, journal = {Applied Network Science}, publisher = {Springer Nature}, doi = {10.1007/s41109-022-00492-w}, pages = {18}, year = {2022}, abstract = {Spreading processes are important drivers of change in social systems. To understand the mechanisms of spreading it is fundamental to have information about the underlying contact network and the dynamical parameters of the process. However, in many real-wold examples, this information is not known and needs to be inferred from data. State-of-the-art spreading inference methods have mostly been applied to modern social systems, as they rely on availability of very detailed data. In this paper we study the inference challenges for historical spreading processes, for which only very fragmented information is available. To cope with this problem, we extend existing network models by formulating a model on a mesoscale with temporal spreading rate. Furthermore, we formulate the respective parameter inference problem for the extended model. We apply our approach to the romanization process of Northern Tunisia, a scarce dataset, and study properties of the inferred time-evolving interregional networks. As a result, we show that (1) optimal solutions consist of very different network structures and spreading rate functions; and that (2) these diverse solutions produce very similar spreading patterns. Finally, we discuss how inferred dominant interregional connections are related to available archaeological traces. Historical networks resulting from our approach can help understanding complex processes of cultural change in ancient times.}, language = {en} } @misc{KostreSunkaraSchuetteetal.2022, author = {Kostr{\´e}, Margarita and Sunkara, Vikram and Sch{\"u}tte, Christof and Djurdjevac Conrad, Nataša}, title = {Understanding the Romanization Spreading on Historical Interregional Networks in Northern Tunisia}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-86764}, year = {2022}, abstract = {Spreading processes are important drivers of change in social systems. To understand the mechanisms of spreading it is fundamental to have information about the underlying contact network and the dynamical parameters of the process. However, in many real-wold examples, this information is not known and needs to be inferred from data. State-of-the-art spreading inference methods have mostly been applied to modern social systems, as they rely on availability of very detailed data. In this paper we study the inference challenges for historical spreading processes, for which only very fragmented information is available. To cope with this problem, we extend existing network models by formulating a model on a mesoscale with temporal spreading rate. Furthermore, we formulate the respective parameter inference problem for the extended model. We apply our approach to the romanization process of Northern Tunisia, a scarce dataset, and study properties of the inferred time-evolving interregional networks. As a result, we show that (1) optimal solutions consist of very different network structures and spreading rate functions; and that (2) these diverse solutions produce very similar spreading patterns. Finally, we discuss how inferred dominant interregional connections are related to available archaeological traces. Historical networks resulting from our approach can help understanding complex processes of cultural change in ancient times.}, language = {en} } @book{OPUS4-5024, title = {MATHEON - Mathematics for Key Technologies}, volume = {1}, editor = {Deuflhard, Peter and Gr{\"o}tschel, Martin and H{\"o}mberg, Dietmar and Horst, Ulrich and Kramer, J{\"u}rg and Mehrmann, Volker and Polthier, Konrad and Schmidt, Frank and Sch{\"u}tte, Christof and Skutella, Martin and Sprekels, J{\"u}rgen}, publisher = {European Mathematical Society}, isbn = {978-3-03719-137-8}, doi = {10.4171/137}, year = {2014}, language = {en} } @article{CardereraPokuttaSchuetteetal.2025, author = {Carderera, Alejandro and Pokutta, Sebastian and Sch{\"u}tte, Christof and Weiser, Martin}, title = {An efficient first-order conditional gradient algorithm in data-driven sparse identification of nonlinear dynamics to solve sparse recovery problems under noise}, volume = {470}, journal = {Journal of Computational and Applied Mathematics}, doi = {10.1016/j.cam.2025.116675}, year = {2025}, abstract = {Governing equations are essential to the study of nonlinear dynamics, often enabling the prediction of previously unseen behaviors as well as the inclusion into control strategies. The discovery of governing equations from data thus has the potential to transform data-rich fields where well-established dynamical models remain unknown. This work contributes to the recent trend in data-driven sparse identification of nonlinear dynamics of finding the best sparse fit to observational data in a large library of potential nonlinear models. We propose an efficient first-order Conditional Gradient algorithm for solving the underlying optimization problem. In comparison to the most prominent alternative framework, the new framework shows significantly improved performance on several essential issues like sparsity-induction, structure-preservation, noise robustness, and sample efficiency. We demonstrate these advantages on several dynamics from the field of synchronization, particle dynamics, and enzyme chemistry.}, language = {en} } @misc{ZhangSchuette2017, author = {Zhang, Wei and Sch{\"u}tte, Christof}, title = {Reliable approximation of long relaxation timescales in molecular dynamics}, issn = {1438-0064}, doi = {10.3390/e19070367}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63718}, year = {2017}, abstract = {Many interesting rare events in molecular systems like ligand association, protein folding or con- formational changes happen on timescales that often are not accessible by direct numerical simulation. Therefore rare event approximation approaches like interface sampling, Markov state model building or advanced reaction coordinate based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches: How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so they also allow for understanding deep connections between the different approaches.}, 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{ZhangLiSchuette2022, author = {Zhang, Wei and Li, Tiejun and Sch{\"u}tte, Christof}, title = {Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks}, volume = {465}, journal = {Journal of Computational Physics}, arxiv = {http://arxiv.org/abs/2110.14523}, doi = {10.1016/j.jcp.2022.111377}, year = {2022}, abstract = {In this paper, we consider the eigenvalue PDE problem of the infinitesimal generators of metastable diffusion processes. We propose a numerical algorithm based on training artificial neural networks for solving the leading eigenvalues and eigenfunctions of such high-dimensional eigenvalue problem. The algorithm is useful in understanding the dynamical behaviors of metastable processes on large timescales. We demonstrate the capability of our algorithm on a high-dimensional model problem, and on the simple molecular system alanine dipeptide.}, language = {en} } @article{ZhangKlusConradetal.2019, author = {Zhang, Wei and Klus, Stefan and Conrad, Tim and Sch{\"u}tte, Christof}, title = {Learning chemical reaction networks from trajectory data}, volume = {18}, journal = {SIAM Journal on Applied Dynamical Systems (SIADS)}, number = {4}, arxiv = {http://arxiv.org/abs/1902.04920}, doi = {10.1137/19M1265880}, pages = {2000 -- 2046}, year = {2019}, abstract = {We develop a data-driven method to learn chemical reaction networks from trajectory data. Modeling the reaction system as a continuous-time Markov chain and assuming the system is fully observed,our method learns the propensity functions of the system with predetermined basis functions by maximizing the likelihood function of the trajectory data under l^1 sparse regularization. We demonstrate our method with numerical examples using synthetic data and carry out an asymptotic analysis of the proposed learning procedure in the infinite-data limit.}, 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