  Data-driven discovery of partial differential equations (PDEs) has attracted
increasing attention in recent years. Although significant progress has been
made, certain unresolved issues remain. For example, for PDEs with high-order
derivatives, the performance of existing methods is unsatisfactory, especially
when the data are sparse and noisy. It is also difficult to discover
heterogeneous parametric PDEs where heterogeneous parameters are embedded in
the partial differential operators. In this work, a new framework combining
deep-learning and integral form is proposed to handle the above-mentioned
problems simultaneously, and improve the accuracy and stability of PDE
discovery. In the framework, a deep neural network is firstly trained with
observation data to generate meta-data and calculate derivatives. Then, a
unified integral form is defined, and the genetic algorithm is employed to
discover the best structure. Finally, the value of parameters is calculated,
and whether the parameters are constants or variables is identified. Numerical
experiments proved that our proposed algorithm is more robust to noise and more
accurate compared with existing methods due to the utilization of integral
form. Our proposed algorithm is also able to discover PDEs with high-order
derivatives or heterogeneous parameters accurately with sparse and noisy data.
