A unified framework for equation discovery and dynamic prediction of hysteretic systems

TL;DR

Hysteresis exhibits memory effects that complicate modeling and prediction. The paper introduces a unified data-driven framework that simultaneously learns unobservable internal hysteretic variables and discovers explicit governing equations using symbolic regression, without relying on predefined model libraries. The approach is validated on Bouc-Wen benchmark data, a complex high-order structure with fractional terms, and shake-table experiments, demonstrating accurate forward prediction and interpretable, physically plausible equations. This framework advances data-driven hysteretic modeling and has potential applications in structural dynamics, smart materials, and actuation systems where memory-dependent nonlinearity is important.

Abstract

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.

Figures (15)

• Figure 1: Classification of equation discovery approaches for hysteretic systems according to the assumed knowledge of the primary dynamics motion equation ($f_\theta$) and the hysteretic link equation ($g_\phi$). The state variables $x, \dot{x}, \ddot{x}, z \in \mathbb{R}^{1}$ denote the scalar displacement, velocity, acceleration, and internal hysteretic variable, respectively. $\mathbf{y}\in\mathbb{R}^{d}$ denotes the available measurements with selection matrix $\mathbf{S}$ specifying the observed quantities. All these variables are dependent on $t$, which is omitted here for brevity. Each quadrant represents a level of prior structural specification.
• Figure 2: The proposed unified framework for equation discovery and dynamic prediction of hysteretic systems. Problem definition and formulation: The state-space form and learnable hysteretic model are established to describe the system dynamics. Two cases in Figure \ref{['fig:structure']} are considered as typical examples, where the hysteretic link equation ($g_\phi$) is unknown. The internal hysteretic variable $z$ is treated as a learnable parameter and jointly optimized during training. Its time derivative is computed via finite differences, denoted Diff$(z)$. Step 1: Small-amplitude responses are employed to estimate the initial parameter values. A differentiable ODE solver is embedded to learn internal variable $z$ and system parameters, with the selection matrix $\mathbf{S}$ (as in Fig. \ref{['fig:structure']}) specifying the observed components in the loss function. Step 2: Symbolic regression (SR) extracts explicit and interpretable governing equations for both the dynamics motion equation ($f_\theta$) (if Case 2) and hysteretic link equation ($g_\phi$).
• Figure 3: Different types of external excitation of benchmark data. The sinesweep signal (left) is used as the training input, while the multisine signal (right) serves as the testing input.
• Figure 4: Hysteresis loops of benchmark data. (a) $x-\dot{x}$ hysteresis loop. (b) $x-z$ hysteresis loop. (c) $x-F (F = kx + \alpha z)$ hysteresis loop.
• Figure 5: Displacement results of benchmark data. (a) Training results. (b) Testing 1 (same excitation) results. (c) Testing 2 (different excitation) results.