  In recent years, the prosperity of deep learning has revolutionized the
Artificial Neural Networks. However, the dependence of gradients and the
offline training mechanism in the learning algorithms prevents the ANN for
further improvement. In this study, a gradient-free training framework based on
data assimilation is proposed to avoid the calculation of gradients. In data
assimilation algorithms, the error covariance between the forecasts and
observations is used to optimize the parameters. Feedforward Neural Networks
(FNNs) are trained by gradient decent, data assimilation algorithms (Ensemble
Kalman Filter (EnKF) and Ensemble Smoother with Multiple Data Assimilation
(ESMDA)), respectively. ESMDA trains FNN with pre-defined iterations by
updating the parameters using all the available observations which can be
regard as offline learning. EnKF optimize FNN when new observation available by
updating parameters which can be regard as online learning. Two synthetic cases
with the regression of a Sine Function and a Mexican Hat function are assumed
to validate the effectiveness of the proposed framework. The Root Mean Square
Error (RMSE) and coefficient of determination (R2) are used as criteria to
assess the performance of different methods. The results show that the proposed
training framework performed better than the gradient decent method. The
proposed framework provides alternatives for online/offline training the
existing ANNs (e.g., Convolutional Neural Networks, Recurrent Neural Networks)
without the dependence of gradients.
