AI 中文总结
本文提出SPIRAL-PO框架，用于从含隐藏状态的部分观测数据中识别非线性动力学控制方程，在旋转机械案例中成功恢复耦合项与非线性特性，实现隐藏轨迹重建。
AI 中文摘要
许多工程系统具有无法直接测量的状态，如旋转机械的倾斜角、内部流动变量、气动弹性模态等，这些隐藏状态会耦合到可测量的输出中。本文研究如何从这类部分观测数据中识别系统的控制方程。我们提出了SPIRA-PO（Symbolic Physics-Informed Residual Augmentation Loop-Partially Observed）框架，该框架将隐藏状态的影响视为可观测运动方程中具有物理可解释性的结构化特征，而非需要消除的干扰项。基于测量坐标中的最小物理种子，该方法针对投影残差拟合多输出残差网络，将学习到的结构投影到物理约束的候选库中，并通过顺序统计门控协议引入相关项。我们给出了隐藏耦合系数唯一恢复的充分条件、一个匹配的不可能结果（表明激励不足会导致任何估计器都无法实现恢复），以及闭式样本复杂度界。这些保证针对的是投影后的隐藏耦合系数，而非隐藏轨迹本身，隐藏轨迹可通过状态估计器单独重建。该框架在带有杜芬（Duffing）支撑的垂直柔性转子上进行了验证，其中仅测量横向位移，而倾斜角保持隐藏状态。恒速运行被证明是不可识别的，而转速扫描则恢复了可识别性；从带噪声的滑行数据中，SPIRAL-PO成功恢复了陀螺耦合、杜芬非线性以及平动-倾斜交叉耦合，每项参数均给出了标准误差和t统计量。基于已验证模型构建的扩展卡尔曼滤波器（Extended Kalman Filter）随后仅通过观测到的位移即可重建隐藏的倾斜轨迹。
英文摘要
Many engineering systems have states that cannot be directly measured-tilt angles in rotating machines, internal flow variables, aeroelastic modes-yet these hidden states couple into the measured outputs. This paper addresses identifying the governing equations from such partial observations. We present SPIRA-PO (Symbolic Physics-Informed Residual Augmentation Loop-Partially Observed), a framework that treats hidden-state effects not as nuisances to be eliminated but as structured, physically interpretable signatures in the observable equations of motion. From a minimal physics seed in measured coordinates, the method fits a multi-output residual network to the projection residual, projects the learned structure onto a physics-constrained candidate library, and admits terms through a sequential statistical gating protocol. We give sufficient conditions for unique recovery of the hidden-coupling coefficients, a matching impossibility result showing that insufficiently rich excitation makes recovery impossible for any estimator, and a closed-form sample-complexity bound. These guarantees concern the projected hidden-coupling coefficients, not the hidden trajectory itself, which is reconstructed separately by a state estimator. The framework is demonstrated on a vertical flexible rotor with Duffing supports, where only the lateral displacements are measured while the tilt angles remain hidden. A constant-speed run is provably unidentifiable, whereas a speed sweep restores identifiability; from noisy coast-down data, SPIRAL-PO recovers the gyroscopic coupling, the Duffing nonlinearity, and the translation-tilt cross-coupling, each with a standard error and t-statistic. An Extended Kalman Filter built on the validated model then reconstructs the hidden tilt trajectory from observed displacements alone.
Comments32 pages, 6 figures. Submitted to Nonlinear Dynamics (Springer)
