Hysteresis is a nonlinear phenomenon with memory effects, where a system's output depends on both its current state and past states. It is prevalent in various physical and mechanical systems, such as yielding structures under seismic excitation, ferromagnetic materials, and piezoelectric actuators. Analytical models like the Bouc-Wen model are often employed but rely on idealized assumptions and careful parameter calibration, limiting their applicability to diverse or mechanism-unknown behaviors. Existing equation discovery approaches for hysteresis are often system-specific or rely on predefined model libraries, which limit their flexibility and ability to capture the hidden mechanisms. To address these challenges, this research classifies equation discovery problems for hysteretic systems and develops a unified framework in which the state-space form is reformulated, and hysteretic variables are treated as trainable parameters from data. The framework further employs symbolic regression (SR) to automatically recover explicit governing equations without relying on predefined libraries, unlike methods such as sparse identification of nonlinear dynamics (SINDy). Experimental results demonstrate that the proposed method is effective in recovering governing equations for hysteretic systems, even in a challenging Full Equation Discovery setting, where prior information is extremely limited, and solving the equations naturally enables the dynamic prediction of hysteretic systems.

翻译：迟滞是一种具有记忆效应的非线性现象，系统输出不仅取决于当前状态，还依赖于历史状态。这种现象广泛存在于各类物理和机械系统中，例如地震激励下的屈服结构、铁磁材料以及压电致动器。尽管Bouc-Wen等解析模型常被采用，但它们依赖于理想化假设和精细的参数标定，限制了其对多样性或机理未知行为的适用性。现有针对迟滞的方程发现方法通常具有系统特异性，或依赖于预定义模型库，这限制了其灵活性和捕捉隐藏机理的能力。为解决这些挑战，本研究对迟滞系统的方程发现问题进行了分类，并开发了一个统一框架，在该框架中重构了状态空间形式，并将迟滞变量视为从数据中学习可训练参数。该框架进一步采用符号回归(SR)自动恢复显式控制方程，无需依赖预定义库，这与稀疏非线性动力学辨识(SINDy)等方法不同。实验结果表明，所提出的方法能有效恢复迟滞系统的控制方程，即使在极具挑战性的完整方程发现设定下——其中先验信息极其有限——求解这些方程也能自然实现迟滞系统的动态预测。