  This paper presents the machine learning-based ensemble conditional mean
filter (ML-EnCMF) -- a filtering method based on the conditional mean filter
(CMF) previously introduced in the literature. The updated mean of the CMF
matches that of the posterior, obtained by applying Bayes' rule on the filter's
forecast distribution. Moreover, we show that the CMF's updated covariance
coincides with the expected conditional covariance. Implementing the EnCMF
requires computing the conditional mean (CM). A likelihood-based estimator is
prone to significant errors for small ensemble sizes, causing the filter
divergence. We develop a systematical methodology for integrating machine
learning into the EnCMF based on the CM's orthogonal projection property.
First, we use a combination of an artificial neural network (ANN) and a linear
function, obtained based on the ensemble Kalman filter (EnKF), to approximate
the CM, enabling the ML-EnCMF to inherit EnKF's advantages. Secondly, we apply
a suitable variance reduction technique to reduce statistical errors when
estimating loss function. Lastly, we propose a model selection procedure for
element-wisely selecting the applied filter, i.e., either the EnKF or ML-EnCMF,
at each updating step. We demonstrate the ML-EnCMF performance using the
Lorenz-63 and Lorenz-96 systems and show that the ML-EnCMF outperforms the EnKF
and the likelihood-based EnCMF.
