  This paper develops a deep learning method for linear and nonlinear
filtering. The idea is to start with a nominal dynamic model and generate Monte
Carlo sample paths. Then these samples are used to train a deep neutral
network. A least square error is used as a loss function for network training.
Then the resulting weights are applied to Monte Carlo sampl\ es from an actual
dynamic model. The deep filter obtained in such a way compares favorably to the
traditional Kalman filter in linear cases and the extended Kalman filter in
nonlinear cases. Moreover, a switching model with jumps is studied to show the
adaptiveness and power of our deep filtering method.
  A main advantage of deep filtering is its robustness when the nominal model
and actual model differ. Another advantage of deep filtering is that real data
can be used directly to train the deep neutral network. Therefore, one does not
need to calibrate the model.
