DISCOVER: Deep identification of symbolically concise open-form partial differential equations via enhanced reinforcement learning

The working mechanisms of complex natural systems tend to abide by concise partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our interactions with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a PDE can be represented by a tree structure. A structure-aware recurrent neural network agent is designed to capture structured information, and is seamlessly combined with the sparse regression method to generate open-form PDE expressions. All of the generated PDEs are evaluated by a meticulously designed reward function by balancing fitness to data and parsimony, and updated by the model-based reinforcement learning. Customized constraints and regulations are formulated to guarantee the rationality of PDEs in terms of physics and mathematics. Numerical experiments demonstrate that our framework is capable of mining open-form governing equations of several dynamic systems, even with compound equation terms, fractional structure, and high-order derivatives. This method is also applied to a real-world problem of the oceanographic system and demonstrates great potential for knowledge discovery in more complicated circumstances with exceptional efficiency and scalability.