  We propose a new approach to constructing a neural network for predicting
expectations of stochastic differential equations. The proposed method does not
need data sets of inputs and outputs; instead, the information obtained from
the time-evolution equations, i.e., the corresponding dual process, is directly
compared with the weights in the neural network. As a demonstration, we
construct neural networks for the Ornstein-Uhlenbeck process and the noisy van
der Pol system. The remarkable feature of learned networks with the proposed
method is the accuracy of inputs near the origin. Hence, it would be possible
to avoid the overfitting problem because the learned network does not depend on
training data sets.
