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The working mechanisms of complex natural systems tend to abide by conciseand profound partial differential equations (PDEs). Methods that directly mineequations from data are called PDE discovery, which reveals consistent physicallaws and facilitates our adaptive interaction with the natural world. In thispaper, an enhanced deep reinforcement-learning framework is proposed to uncoversymbolically concise open-form PDEs with little prior knowledge. Particularly,based on a symbol library of basic operators and operands, a structure-awarerecurrent neural network agent is designed and seamlessly combined with thesparse regression method to generate concise and open-form PDE expressions. Allof the generated PDEs are evaluated by a meticulously designed reward functionby balancing fitness to data and parsimony, and updated by the model-basedreinforcement learning in an efficient way. Customized constraints andregulations are formulated to guarantee the rationality of PDEs in terms ofphysics and mathematics. The experiments demonstrate that our framework iscapable of mining open-form governing equations of several dynamic systems,even with compound equation terms, fractional structure, and high-orderderivatives, with excellent efficiency. Without the need for prior knowledge,this method shows great potential for knowledge discovery in more complicatedcircumstances with exceptional efficiency and scalability.

DISCOVER: Deep identification of symbolically concise open-form PDEs via enhanced reinforcement-learning