  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.
