Training data studies for the cell-average-based neural network method for linear hyperbolic and parabolic equations

In this paper, we conduct training data studies for the cell-averagebased neural network (CANN) method for linear hyperbolic and parabolic equations.We enlarge the training data set to include representative solution trajectories such as Fourier basis functions and determine whether the method will be improved.A perturbed stochastic gradient descent method is proposed and is verified to improve the method's stability after long-time simulations.The robustness of the method to contaminated training data is investigated.For the linear parabolic equation, the CANN method can be generalized to solve a large group of initial value problems, especially those involving high-frequency modes, while being an explicit method with a large time step size applied to evolve the solution forward in time.