  Discovering the underlying physical behavior of complex systems is a crucial,
but less well-understood topic in many engineering disciplines. This study
proposes a finite-difference inspired convolutional neural network framework to
learn hidden partial differential equations from given data and iteratively
estimate future dynamical behavior. The methodology designs the filter sizes
such that they mimic the finite difference between the neighboring points. By
learning the governing equation, the network predicts the future evolution of
the solution by using only a few trainable parameters. In this paper, we
provide numerical results to compare the efficiency of the second-order
Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM
optimizer.
