@article{DonatiHartmannKeller, author = {Donati, Luca and Hartmann, Carsten and Keller, Bettina G.}, title = {Girsanov reweighting for path ensembles and Markov state models}, series = {The Journal of Chemical Physics}, volume = {146}, journal = {The Journal of Chemical Physics}, number = {24}, doi = {10.1063/1.4989474}, pages = {244112}, abstract = {The sensitivity of molecular dynamics on changes in the potential energy function plays an important role in understanding the dynamics and function of complex molecules. We present a method to obtain path ensemble averages of a perturbed dynamics from a set of paths generated by a reference dynamics. It is based on the concept of path probability measure and the Girsanov theorem, a result from stochastic analysis to estimate a change of measure of a path ensemble. Since Markov state models (MSMs) of the molecular dynamics can be formulated as a combined phase-space and path ensemble average, the method can be extended to reweight MSMs by combining it with a reweighting of the Boltzmann distribution. We demonstrate how to efficiently implement the Girsanov reweighting in a molecular dynamics simulation program by calculating parts of the reweighting factor "on the fly" during the simulation, and we benchmark the method on test systems ranging from a two-dimensional diffusion process and an artificial many-body system to alanine dipeptide and valine dipeptide in implicit and explicit water. The method can be used to study the sensitivity of molecular dynamics on external perturbations as well as to reweight trajectories generated by enhanced sampling schemes to the original dynamics.}, language = {en} } @misc{HartmannBanischSarichetal., author = {Hartmann, Carsten and Banisch, Ralf and Sarich, Marco and Badowski, Thomas and Sch{\"u}tte, Christof}, title = {Characterization of Rare Events in Molecular Dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42410}, abstract = {A good deal of molecular dynamics simulations aims at predicting and quantifying rare events, such as the folding of a protein or a phase transition. Simulating rare events is often prohibitive, especially if the equations of motion are high-dimensional, as is the case in molecular dynamics. Various algorithms have been proposed for efficiently computing mean first passage times, transition rates or reaction pathways. This article surveys and discusses recent developments in the field of rare event simulation and outlines a new approach that combines ideas from optimal control and statistical mechanics. The optimal control approach described in detail resembles the use of Jarzynski's equality for free energy calculations, but with an optimized protocol that speeds up the sampling, while (theoretically) giving variance-free estimators of the rare events statistics. We illustrate the new approach with two numerical examples and discuss its relation to existing methods.}, language = {en} } @article{HartmannBanischSarichetal., author = {Hartmann, Carsten and Banisch, Ralf and Sarich, Marco and Badowski, Thomas and Sch{\"u}tte, Christof}, title = {Characterization of Rare Events in Molecular Dynamics}, series = {Entropy (Special Issue)}, volume = {16}, journal = {Entropy (Special Issue)}, number = {1}, doi = {10.3390/e16010350}, pages = {350 -- 376}, language = {en} } @incollection{HartmannRichter, author = {Hartmann, Carsten and Richter, Lorenz}, title = {Transgressing the Boundaries: Towards a Rigorous Understanding of Deep Learning and Its (Non )Robustness}, series = {AI - Limits and Prospects of Artificial Intelligence}, volume = {4}, booktitle = {AI - Limits and Prospects of Artificial Intelligence}, editor = {Klimczak, Peter and Petersen, Christer}, publisher = {transcript Verlag}, doi = {10.1515/9783839457320}, pages = {43 -- 82}, abstract = {The recent advances in machine learning in various fields of applications can be largely attributed to the rise of deep learning (DL) methods and architectures. Despite being a key technology behind autonomous cars, image processing, speech recognition, etc., a notorious problem remains the lack of theoretical understanding of DL and related interpretability and (adversarial) robustness issues. Understanding the specifics of DL, as compared to, say, other forms of nonlinear regression methods or statistical learning, is interesting from a mathematical perspective, but at the same time it is of crucial importance in practice: treating neural networks as mere black boxes might be sufficient in certain cases, but many applications require waterproof performance guarantees and a deeper understanding of what could go wrong and why it could go wrong. It is probably fair to say that, despite being mathematically well founded as a method to approximate complicated functions, DL is mostly still more like modern alchemy that is firmly in the hands of engineers and computer scientists. Nevertheless, it is evident that certain specifics of DL that could explain its success in applications demands systematic mathematical approaches. In this work, we review robustness issues of DL and particularly bridge concerns and attempts from approximation theory to statistical learning theory. Further, we review Bayesian Deep Learning as a means for uncertainty quantification and rigorous explainability.}, language = {en} } @article{HartmannRichter, author = {Hartmann, Carsten and Richter, Lorenz}, title = {Nonasymptotic bounds for suboptimal importance sampling}, language = {en} } @misc{HartmannRichterSchuetteetal., author = {Hartmann, Carsten and Richter, Lorenz and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Variational characterization of free energy: Theory and algorithms}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-65045}, abstract = {The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated.}, language = {en} } @article{HartmannRichterSchuetteetal., author = {Hartmann, Carsten and Richter, Lorenz and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Variational characterization of free energy: theory and algorithms}, series = {Entropy}, volume = {19}, journal = {Entropy}, number = {11}, doi = {10.3390/e19110626}, pages = {626}, language = {en} } @article{HartmannSchuette2012, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Efficient rare event simulation by optimal nonequilibrium forcing}, series = {J. Stat. Mech. Theor. Exp.}, volume = {2012}, journal = {J. Stat. Mech. Theor. Exp.}, doi = {10.1088/1742-5468/2012/11/P11004}, pages = {P11004}, year = {2012}, language = {en} } @article{HartmannSchuette2005, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {A Geometric Approach to Constrained Molecular Dynamics and Free Energy}, series = {Comm. Math. Sci.}, volume = {3}, journal = {Comm. Math. Sci.}, number = {1}, pages = {1 -- 20}, year = {2005}, language = {en} } @article{HartmannSchuette2008, author = {Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Balancing of partially-observed stochastic differential equations}, series = {47th IEEE Conference on Decision and Control}, journal = {47th IEEE Conference on Decision and Control}, doi = {10.1109/CDC.2008.4739161}, pages = {4867 -- 4872}, year = {2008}, language = {en} }