Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

翻译：揭示非线性动态系统的潜在控制方程仍是一项重大挑战。先验知识不足导致难以确定精确的候选函数库，而噪声观测则引发不精确的评估，进而引入冗余函数项或错误方程。本研究提出一种框架，可从有限且含噪声数据中鲁棒地发现开放形式偏微分方程。该框架通过两个交替更新过程运行：发现与嵌入。发现阶段采用符号表示和新型强化学习引导的混合PDE生成器，高效产生具有树结构的多样化开放形式PDE。基于神经网络的预测模型拟合系统响应，并作为生成PDE的奖励评估器。利用高奖励PDE通过RL策略迭代优化生成器，通过无参数稳定性指标选择最优PDE。嵌入阶段将发现过程初步识别的PDE作为物理约束集成到预测模型中进行鲁棒训练。PDE树的遍历自动完成计算图构建与嵌入过程，无需人工干预。数值实验表明，该框架能从有限高噪声数据的非线性动态系统中发现控制方程，且性能优于其他基于物理信息神经网络的发现方法。本工作为探索先验知识有限的真实系统开辟了新潜力。