  Data-driven methods have recently been developed to discover underlying
partial differential equations (PDEs) of physical problems. However, for these
methods, a complete candidate library of potential terms in a PDE are usually
required. To overcome this limitation, we propose a novel framework combining
deep learning and genetic algorithm, called DLGA-PDE, for discovering PDEs. In
the proposed framework, a deep neural network that is trained with available
data of a physical problem is utilized to generate meta-data and calculate
derivatives, and the genetic algorithm is then employed to discover the
underlying PDE. Owing to the merits of the genetic algorithm, such as mutation
and crossover, DLGA-PDE can work with an incomplete candidate library. The
proposed DLGA-PDE is tested for discovery of the Korteweg-de Vries (KdV)
equation, the Burgers equation, the wave equation, and the Chaffee-Infante
equation, respectively, for proof-of-concept. Satisfactory results are obtained
without the need for a complete candidate library, even in the presence of
noisy and limited data.
