Machine learning based conditional mean filter: a non-linear extension of the ensemble Kalman filter

Abstract

Filtering is an uncertainty quantification technique that refers to the inference of the states of dynamical systems from noisy observations. This work proposes a machine learning-based filtering method for tracking the high-dimensional non-Gaussian state-space models with non-linear dynamics and sparse observations. Our filter method is based on the conditional expectation mean filter and uses machine-learning techniques to approximate the conditional mean (CM). The contribution of this work is twofolds: (i) we demonstrate theoretically that the assimilated ensembles obtained using the ensemble conditional mean filter (EnCMF) provide a correct prediction of the posterior mean and have the optimal variance, and (ii) we implement the EnCMF using artificial neural networks, which has a significant advantage in representing non-linear functions that map between high-dimensionality domains, such as the CM. We implement the machine learning-based EnCMF for tracking the states of the Lorenz-63 and 96 systems under the chaotic regime. Numerical results show that the EnCMF outperforms the ensemble Kalman filter.

Authors 3

1. Truong‐Vinh Hoang Aachen Chair of Mathematics for Uncertainty Quantification Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   RWTH-AAchen University, Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, Germany (hoang@uq.rwth-aachen.de)

2. Sebastian Krumscheid Aachen Chair of Mathematics for Uncertainty Quantification Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   RWTH-AAchen University, Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, Germany (hoang@uq.rwth-aachen.de)

3. Raul F. Tempone Aachen Chair of Mathematics for Uncertainty Quantification Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   RWTH-AAchen University, Chair of Mathematics for Uncertainty Quantification, Department of Mathematics, Germany (hoang@uq.rwth-aachen.de)