@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{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     = {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}
}