The optimality of Bayesian filtering relies on the completeness of prior
models, while deep learning holds a distinct advantage in learning models from
offline data. Nevertheless, the current fusion of these two methodologies
remains largely ad hoc, lacking a theoretical foundation. This paper presents a
novel solution, namely a multi-level gated Bayesian recurrent neural network
specifically designed to state estimation under model mismatches. Firstly, we
transform the non-Markov state-space model into an equivalent first-order
Markov model with memory. It is a generalized transformation that overcomes the
limitations of the first-order Markov property and enables recursive filtering.
Secondly, by deriving a data-assisted joint state-memory-mismatch Bayesian
filtering, we design a Bayesian multi-level gated framework that includes a
memory update gate for capturing the temporal regularities in state evolution,
a state prediction gate with the evolution mismatch compensation, and a state
update gate with the observation mismatch compensation. The Gaussian
approximation implementation of the filtering process within the gated
framework is derived, taking into account the computational efficiency.
Finally, the corresponding internal neural network structures and end-to-end
training methods are designed. The Bayesian filtering theory enhances the
interpretability of the proposed gated network, enabling the effective
integration of offline data and prior models within functionally explicit gated
units. In comprehensive experiments, including simulations and real-world
datasets, the proposed gated network demonstrates superior estimation
performance compared to benchmark filters and state-of-the-art deep learning
filtering methods