  In this work, we present an adjoint-based method for discovering the
underlying governing partial differential equations (PDEs) given data. The idea
is to consider a parameterized PDE in a general form and formulate a
PDE-constrained optimization problem aimed at minimizing the error of the PDE
solution from data. Using variational calculus, we obtain an evolution equation
for the Lagrange multipliers (adjoint equations) allowing us to compute the
gradient of the objective function with respect to the parameters of PDEs given
data in a straightforward manner. In particular, we consider a family of
parameterized PDEs encompassing linear, nonlinear, and spatial derivative
candidate terms, and elegantly derive the corresponding adjoint equations. We
show the efficacy of the proposed approach in identifying the form of the PDE
up to machine accuracy, enabling the accurate discovery of PDEs from data. We
also compare its performance with the famous PDE Functional Identification of
Nonlinear Dynamics method known as PDE-FIND (Rudy et al., 2017), on both smooth
and noisy data sets. Even though the proposed adjoint method relies on
forward/backward solvers, it outperforms PDE-FIND for large data sets thanks to
the analytic expressions for gradients of the cost function with respect to
each PDE parameter.
