@article{MaoutsaReichOpper2020, author = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred}, title = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {8}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22080802}, pages = {35}, year = {2020}, abstract = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.}, language = {en} } @misc{Reich1990, author = {Reich, Sebastian}, title = {On a geometrical interpretation of differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46683}, year = {1990}, abstract = {The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.}, language = {en} } @misc{Reich1994, author = {Reich, Sebastian}, title = {Momentum conserving symplectic integrators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16824}, year = {1994}, abstract = {In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).}, language = {en} } @misc{Reich1995, author = {Reich, Sebastian}, title = {On the local qualitative behavior of differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46739}, year = {1995}, abstract = {A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.}, language = {en} } @misc{Reich1991, author = {Reich, Sebastian}, title = {On an existence and uniqueness theory for nonlinear differential-algebraic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-46706}, year = {1991}, abstract = {An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0066080}, pages = {8}, year = {2021}, abstract = {We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.}, language = {en} } @article{GottwaldMitchellReich2011, author = {Gottwald, Georg A. and Mitchell, Lewis and Reich, Sebastian}, title = {Controlling overestimation of error covariance in ensemble kalman filters with sparse observations a variance-limiting kalman filter}, series = {Monthly weather review}, volume = {139}, journal = {Monthly weather review}, number = {8}, publisher = {American Meteorological Soc.}, address = {Boston}, issn = {0027-0644}, doi = {10.1175/2011MWR3557.1}, pages = {2650 -- 2667}, year = {2011}, abstract = {The problem of an ensemble Kalman filter when only partial observations are available is considered. In particular, the situation is investigated where the observational space consists of variables that are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. To limit the variance of the latter poorly resolved variables a variance-limiting Kalman filter (VLKF) is derived in a variational setting. The VLKF for a simple linear toy model is analyzed and its range of optimal performance is determined. The VLKF is explored in an ensemble transform setting for the Lorenz-96 system, and it is shown that incorporating the information of the variance of some unobservable variables can improve the skill and also increase the stability of the data assimilation procedure.}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Supervised learning from noisy observations}, series = {Physica : D, Nonlinear phenomena}, volume = {423}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2021.132911}, pages = {15}, year = {2021}, abstract = {Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data -driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems. (C) 2021 Elsevier B.V. All rights reserved.}, language = {en} } @article{EngbertRabeSchwetlicketal.2022, author = {Engbert, Ralf and Rabe, Maximilian Michael and Schwetlick, Lisa and Seelig, Stefan A. and Reich, Sebastian and Vasishth, Shravan}, title = {Data assimilation in dynamical cognitive science}, series = {Trends in cognitive sciences}, volume = {26}, journal = {Trends in cognitive sciences}, number = {2}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1364-6613}, doi = {10.1016/j.tics.2021.11.006}, pages = {99 -- 102}, year = {2022}, abstract = {Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models.}, language = {en} } @article{ReichWeissmann2021, author = {Reich, Sebastian and Weissmann, Simon}, title = {Fokker-Planck particle systems for Bayesian inference: computational approaches}, series = {SIAM ASA journal on uncertainty quantification}, volume = {9}, journal = {SIAM ASA journal on uncertainty quantification}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {2166-2525}, doi = {10.1137/19M1303162}, pages = {446 -- 482}, year = {2021}, abstract = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.}, language = {en} }