  We present a framework for recovering/approximating unknown time-dependent
partial differential equation (PDE) using its solution data. Instead of
identifying the terms in the underlying PDE, we seek to approximate the
evolution operator of the underlying PDE numerically. The evolution operator of
the PDE, defined in infinite-dimensional space, maps the solution from a
current time to a future time and completely characterizes the solution
evolution of the underlying unknown PDE. Our recovery strategy relies on
approximation of the evolution operator in a properly defined modal space,
i.e., generalized Fourier space, in order to reduce the problem to finite
dimensions. The finite dimensional approximation is then accomplished by
training a deep neural network structure, which is based on residual network
(ResNet), using the given data. Error analysis is provided to illustrate the
predictive accuracy of the proposed method. A set of examples of different
types of PDEs, including inviscid Burgers' equation that develops discontinuity
in its solution, are presented to demonstrate the effectiveness of the proposed
method.
