  Partial differential equations (PDEs) are concise and understandable
representations of domain knowledge, which are essential for deepening our
understanding of physical processes and predicting future responses. However,
the PDEs of many real-world problems are uncertain, which calls for PDE
discovery. We propose the symbolic genetic algorithm (SGA-PDE) to discover
open-form PDEs directly from data without prior knowledge about the equation
structure. SGA-PDE focuses on the representation and optimization of PDE.
Firstly, SGA-PDE uses symbolic mathematics to realize the flexible
representation of any given PDE, transforms a PDE into a forest, and converts
each function term into a binary tree. Secondly, SGA-PDE adopts a specially
designed genetic algorithm to efficiently optimize the binary trees by
iteratively updating the tree topology and node attributes. The SGA-PDE is
gradient-free, which is a desirable characteristic in PDE discovery since it is
difficult to obtain the gradient between the PDE loss and the PDE structure. In
the experiment, SGA-PDE not only successfully discovered nonlinear Burgers'
equation, Korteweg-de Vries (KdV) equation, and Chafee-Infante equation, but
also handled PDEs with fractional structure and compound functions that cannot
be solved by conventional PDE discovery methods.
