Convergence of neural network weights for stochastic systems

This paper considers the convergence of neural network weights in identification of a deterministic system dx=φ(x, u)*dt with stochastic observation y=x+ξ. Since the backpropagation learning algorithm is based on the error between the output of the neural network and the desired output (i.e. the observation in our system), and the observation is stochastic in our system, it is not possible to use the steepest descent to estimate the weights directly. Therefore we consider the use of the stochastic approximation method to estimate the weights, and prove its convergence in the presence of noise