  We study the problem of identifying unknown processes embedded in
time-dependent partial differential equation (PDE) using observational data,
with an application to advection-diffusion type PDE. We first conduct
theoretical analysis and derive conditions to ensure the solvability of the
problem. We then present a set of numerical approaches, including Galerkin type
algorithm and collocation type algorithm. Analysis of the algorithms are
presented, along with their implementation detail. The Galerkin algorithm is
more suitable for practical situations, particularly those with noisy data, as
it avoids using derivative/gradient data. Various numerical examples are then
presented to demonstrate the performance and properties of the numerical
methods.
