State, parameters and hidden dynamics estimation with the Deep Kalman Filter: Regularization strategies

In this paper we present in detail the various regularization strategies adopted for a novel scientific machine learning extension of the well known Kalman Filter (KF) that we call the Deep Kalman Filter (DKF), briefly presented in the conference paper (Chinellato and Marcuzzi 2024) . It is based on a recent scientific machine learning paradigm, called algorithm unfolding/unrolling, that allows to create a neural network from the algebraic structure dictated by an analytical method of scientific computing. We show the interpretable consistency of DKF with the classic KF when this is optimal, and its improvements against the KF with both linear and nonlinear models in general. Indeed, the DKF learns parameters of a quite general reference model, comprising: corrector gains, predictor model parameters and eventual unmodeled dynamics. This goes well beyond the ability of the KF and its known extensions.