  We present a numerical framework for deep neural network (DNN) modeling of
unknown time-dependent partial differential equations (PDE) using their
trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020],
where the learning takes place in modal/Fourier space, the current method
conducts the learning and modeling in physical space and uses measurement data
as nodal values. We present a DNN structure that has a direct correspondence to
the evolution operator of the underlying PDE, thus establishing the existence
of the DNN model. The DNN model also does not require any geometric information
of the data nodes. Consequently, a trained DNN defines a predictive model for
the underlying unknown PDE over structureless grids. A set of examples,
including linear and nonlinear scalar PDE, system of PDEs, in both one
dimension and two dimensions, over structured and unstructured grids, are
presented to demonstrate the effectiveness of the proposed DNN modeling.
Extension to other equations such as differential-integral equations is also
discussed.
