  In recent years, data-driven methods have been developed to learn dynamical
systems and partial differential equations (PDE). The goal of such work is
discovering unknown physics and the corresponding equations. However, prior to
achieving this goal, major challenges remain to be resolved, including learning
PDE under noisy data and limited discrete data. To overcome these challenges,
in this work, a deep-learning based data-driven method, called DL-PDE, is
developed to discover the governing PDEs of underlying physical processes. The
DL-PDE method combines deep learning via neural networks and data-driven
discovery of PDE via sparse regressions. In the DL-PDE, a neural network is
first trained, and then a large amount of meta-data is generated, and the
required derivatives are calculated by automatic differentiation. Finally, the
form of PDE is discovered by sparse regression. The proposed method is tested
with physical processes, governed by groundwater flow equation,
convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV)
equation, for proof-of-concept and applications in real-world engineering
settings. The proposed method achieves satisfactory results when data are noisy
and limited.
