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, this research develops a unified framework that integrates learning of internal variables (commonly used in modeling hysteresis) and symbolic regression to automatically extract internal hysteretic variable, and discover explicit governing equations directly from data without predefined libraries as required by methods such as sparse identification of nonlinear dynamics (SINDy). Solving the discovered equations naturally enables prediction of the dynamic responses of hysteretic systems. This work provides a systematic view and approach for both equation discovery and characterization of hysteretic dynamics, defining a unified framework for these types of problems.

翻译：滞回是一种具有记忆效应的非线性现象，系统的输出不仅取决于当前状态，还依赖于其历史状态。该现象广泛存在于各类物理与机械系统中，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模型等解析模型常被采用，但其依赖于理想化假设与精细的参数标定，限制了其在多样化或机理未知行为中的适用性。现有针对滞回的方程发现方法往往针对特定系统或依赖预定义的模型库，这制约了其灵活性及捕捉隐藏机理的能力。为解决这些问题，本研究开发了一个统一框架，该框架融合了内部变量（常用于滞回建模）的学习与符号回归技术，能够从数据中自动提取内部滞回变量，并在无需如稀疏辨识非线性动力学（SINDy）等方法所要求的预定义模型库的情况下，直接发现显式控制方程。求解所发现的方程自然能够实现对滞回系统动态响应的预测。本工作为滞回动力学的方程发现与特性表征提供了系统化的视角与方法，为此类问题定义了一个统一的框架。