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 interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolic open-form PDEs with little prior knowledge. Specifically, (1) we first build a symbol library and define that a PDE can be represented as a tree structure. Then, (2) we design a structure-aware recurrent neural network agent by combining structured inputs and monotonic attention to generate the pre-order traversal of PDE expression trees. The expression trees are then split into function terms, and their coefficients can be calculated by the sparse regression method. (3) All of the generated PDE candidates are first filtered by some physical and mathematical constraints, and then evaluated by a meticulously designed reward function considering the fitness to data and the parsimony of the equation. (4) We adopt the risk-seeking policy gradient to iteratively update the agent to improve the best-case performance. The experiment demonstrates that our framework is capable of mining the governing equations of several canonical systems with great efficiency and scalability.

翻译：复杂的自然系统的工作机制往往符合简明和深刻的部分差异方程式(PDEs) 数据直接利用等式的方法称为PDE发现法,这种方法揭示了连贯的物理法则,便利了我们与自然界的互动。在本文件中,提议了一个强化的深强化学习框架,以发现具有象征意义的开放式PDE,而事先没有多少知识。具体地说,(1) 我们首先建立一个符号库,确定PDE可以作为树结构结构结构来代表。(2) 然后,我们设计一个结构化的、有意识的经常性神经网络代理,将结构化的投入和单调的关注结合起来,以生成PDE表达式树的顺序前跨行。然后,表达式树被分割成功能术语,其系数可以通过稀薄的回归法计算。(3) 所有产生的PDE候选人首先受到一些物理和数学限制的过滤,然后通过精心设计的奖励功能进行评估,考虑数据是否适合和等式的微调。(4) 我们采用风险政策梯度来反复更新代理,以改进最佳业绩。实验表明,我们的框架能够利用多种系统的巨大等式效率。