The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction 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 structure-aware recurrent neural network agent is designed and seamlessly combined with the sparse regression method to generate concise and 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 in an efficient way. Customized constraints and regulations are formulated to guarantee the rationality of PDEs in terms of physics and mathematics. The 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, with excellent efficiency. Without the need for prior knowledge, this method shows great potential for knowledge discovery in more complicated circumstances with exceptional efficiency and scalability.

翻译：复杂自然系统的工作机制往往遵循简洁而深刻的偏微分方程。直接从数据中挖掘方程的方法被称为偏微分方程发现，它揭示了恒定的物理规律，并促进了我们与自然界的自适应交互。本文提出了一种增强的深度强化学习框架，用于在较少先验知识的情况下挖掘符号简洁的开式偏微分方程。具体而言，基于基本运算符和操作数的符号库，设计了一种结构感知的循环神经网络智能体，并与稀疏回归方法无缝结合，以生成简洁且开式的偏微分方程表达式。所有生成的偏微分方程通过精心设计的奖励函数进行评估，平衡了对数据的拟合度与简洁性，并通过基于模型的强化学习高效更新。还制定了定制化的约束和规则，以确保偏微分方程在物理和数学上的合理性。实验表明，我们的框架能够高效挖掘多个动态系统的开式控制方程，即使包含复合方程项、分式结构和高阶导数。该方法无需先验知识，在更复杂的情境下展现出卓越的知识发现潜力，兼具高效性与可扩展性。