A dynamical neural Galerkin scheme for filtering problems

¹ IRMA, Université de Strasbourg, CNRS UMR 7501, 7 rue René Descartes, 67084 Strasbourg, France
² Université de Strasbourg, CNRS, Inria, IRMA, F-67000 Strasbourg, France
³ Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
⁴ Centrale Nantes, Nantes Université, LMJL UMR CNRS 6629, Nantes, France
⁵ Sorbonne Université, Inria, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France
⁶ Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600 Eindhoven, Netherlands
⁷ INRIA-Paris Project Team ANGE, Sorbonne Université (LJLL), 2 Rue Simone Iff, 75012 Paris, France

Abstract

This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model with unknown parameters. We present a filtering algorithm where the reconstruction of the dynamics is done with neural network approximations whose weights are dynamically updated using observational data. In addition to the estimate of the state, we also obtain time-dependent parameter estimations of the PDE parameters governing the observed evolution. We illustrate the behavior of the method in a one-dimensional KdV equation involving the transport of solutions with local support. Our numerical investigation reveals the importance of the location and number of the observations. In particular, it suggests to consider dynamical sensor placement.