Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge, especially when encountering noisy observations and no prior knowledge available. This study proposes R-DISCOVER, a framework designed 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 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 superior fits are utilized to iteratively optimize the generator via the RL method 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：揭示非线性动力系统内在控制方程仍然是一项重大挑战，尤其是在观测数据存在噪声且缺乏先验知识的情况下。本研究提出R-DISCOVER框架，旨在从有限且高噪声数据中鲁棒地发现开放形式的偏微分方程。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强化学习引导的混合PDE生成器，高效生成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。拟合优度较高的PDE通过强化学习方法迭代优化生成器，并通过无参数稳定性度量选择表现最佳的PDE。嵌入阶段将发现过程中初步识别的PDE作为物理约束整合到预测模型中，以实现鲁棒训练。通过遍历PDE树结构自动构建计算图并完成嵌入过程，无需人工干预。数值实验表明，该框架能够从非线性动力系统的有限且高噪声数据中揭示控制方程，且性能优于其他基于物理信息神经网络的发现方法。本工作为探索理解有限的真实世界系统开辟了新可能。