Unlocking Hysteresis: A New Path to System Understanding

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Author: Denis Avetisyan

Researchers have developed a unified framework that simultaneously discovers the hidden variables and governing equations of hysteretic systems, paving the way for more accurate predictions and interpretable models.

The framework uncovers governing equations for dynamic systems exhibiting hysteresis by treating internal hysteretic variables as learnable parameters, initially estimating them with small-amplitude responses, and subsequently employing symbolic regression to derive explicit, interpretable expressions for both system dynamics-represented by $f_{\theta}$-and the hysteretic link-$g_{\phi}$-effectively bridging the gap between observation and mechanistic understanding.
The framework uncovers governing equations for dynamic systems exhibiting hysteresis by treating internal hysteretic variables as learnable parameters, initially estimating them with small-amplitude responses, and subsequently employing symbolic regression to derive explicit, interpretable expressions for both system dynamics-represented by $f_{\theta}$-and the hysteretic link-$g_{\phi}$-effectively bridging the gap between observation and mechanistic understanding.

This work integrates solver-based learning and symbolic regression for dynamic prediction and equation discovery in nonlinear dynamics.

Hysteresis, a ubiquitous phenomenon exhibiting memory effects, presents a persistent challenge for accurate modeling and prediction in diverse physical systems. This research introduces ‘A unified framework for equation discovery and dynamic prediction of hysteretic systems’-a novel approach integrating solver-based learning and symbolic regression to simultaneously identify hidden internal variables and derive governing equations directly from data, bypassing the limitations of predefined models. This framework not only enables interpretable representations of hysteretic behavior but also facilitates accurate dynamic prediction without relying on extensive parameter calibration. Will this unified approach unlock new capabilities in characterizing and controlling complex nonlinear systems exhibiting hysteresis?

The Echo of Past States: Unveiling System Memory

Hysteresis, a phenomenon pervasive across diverse physical systems, describes a dependence on an object’s past states, defying predictions based solely on current conditions. Unlike systems conforming to equilibrium principles, those exhibiting hysteresis possess a ‘memory’ – the current value of a property isn’t solely determined by the present input, but by the history of inputs experienced. This path-dependent behavior is observed in magnetism, where a material’s magnetization lags behind an applied field, and in mechanical systems displaying elastic deformation with energy loss. Traditional modeling techniques, often predicated on immediate responsiveness, struggle to account for this inherent lag, resulting in discrepancies between predicted and observed behaviors. Consequently, accurately capturing hysteresis is paramount for reliable simulations and precise control of complex systems, demanding innovative approaches that incorporate the influence of prior states.

Traditional predictive modeling frequently falters when applied to systems exhibiting hysteresis, largely due to the inherent disregard for a material’s or system’s past states. These conventional approaches typically assume that a system’s current output is solely determined by its present input, neglecting the crucial influence of its historical trajectory. Consequently, predictions based on these models can deviate significantly from observed behavior, particularly in scenarios involving cyclic loading or varying external fields. The ‘memory’ embedded within a hysteretic system-the dependence on prior conditions-introduces a non-linearity that standard linear techniques are ill-equipped to handle, creating discrepancies between simulation and reality. This limitation impacts diverse fields, from magnetic storage and shape-memory alloys to climate modeling and economic forecasting, underscoring the need for advanced methodologies capable of accurately representing and predicting hysteretic phenomena.

Accurate simulation and control of numerous physical systems hinge on the ability to faithfully represent hysteresis – the dependence of a system’s state not only on current conditions, but also on its past experiences. Failing to account for these ‘memory’ effects can lead to significant discrepancies between modeled behavior and real-world performance, particularly in areas like magnetic storage, mechanical actuators, and climate modeling. Sophisticated modeling techniques, therefore, are crucial for predicting system responses and designing effective control strategies; these methods move beyond traditional approaches that assume instantaneous equilibrium and instead incorporate the historical trajectory of the system, enabling more precise predictions and optimized performance. The development of these techniques is not merely an academic exercise, but a vital step towards realizing the full potential of these complex technologies and ensuring their reliable operation.

Equation discovery approaches for hysteretic systems can be categorized based on the level of prior knowledge assumed about both the primary dynamic motion and the hysteretic link, utilizing observed measurements and state variables like displacement, velocity, and acceleration.
Equation discovery approaches for hysteretic systems can be categorized based on the level of prior knowledge assumed about both the primary dynamic motion and the hysteretic link, utilizing observed measurements and state variables like displacement, velocity, and acceleration.

Data-Driven Reconstruction: Learning the System’s Internal States

The identification of internal variables within hysteretic systems is achieved through a combined methodology leveraging Solver-Based Internal Variable Learning and the Bouc-Wen model. The Bouc-Wen model, a non-linear hysteresis model, provides a framework for representing the system’s behavior, while the solver-based learning algorithm iteratively adjusts both the model’s parameters and the estimated values of the internal variables. This optimization process minimizes the discrepancy between the model’s output and the observed system data, effectively ‘learning’ the internal states that govern the hysteresis. The technique does not require prior knowledge of the internal variable’s functional form, instead inferring their evolution directly from the input-output data. This allows for the characterization of complex hysteretic behaviors without relying on simplified assumptions about the underlying mechanisms.

The methodology utilizes a simultaneous learning process, estimating both the parameters defining the Bouc-Wen model and the time-dependent evolution of its internal variables directly from observed input-output data. This is achieved through an optimization process that minimizes the error between the model’s predicted behavior and the observed system response. Unlike traditional methods that require prior knowledge of internal states or rely on separate estimation stages, this approach integrates parameter identification and state learning into a single, unified framework. The system employs observed data to iteratively refine both the model parameters – such as stiffness, damping, and hysteretic parameters – and the values of the internal variables at each time step, effectively constructing a complete dynamic representation of the hysteretic system.

Accurate capture of internal system states, as determined through methods like Solver-Based Internal Variable Learning, provides a quantifiable representation of hysteretic system ‘memory’. These internal states, which represent past influences on the current system behavior, allow for the development of more accurate predictive models. By incorporating the evolution of these states into the system dynamics, forecasts of future behavior can be generated with reduced error compared to models relying solely on external inputs. This is particularly valuable in systems exhibiting rate-dependence or path-dependence, where current output is significantly influenced by prior conditions and cannot be reliably predicted without accounting for these internal, historical influences.

Hysteresis loops demonstrate the system's response to displacement (x) and velocity (ẋ) as well as displacement (x) and force (F), which is a function of both displacement and a secondary variable (z).
Hysteresis loops demonstrate the system’s response to displacement (x) and velocity (ẋ) as well as displacement (x) and force (F), which is a function of both displacement and a secondary variable (z).

Automated Revelation: Uncovering the Governing Equations

The methodology employs Symbolic Regression, a technique used to identify mathematical expressions from data, building upon the Sparse Identification of Nonlinear Dynamics (SINDy) framework. SINDy facilitates the discovery of governing equations by leveraging sparse regression to identify significant terms within a pre-defined library of functions. This approach is specifically adapted to model hysteretic behavior, a system’s dependence on its past states, by identifying the dynamic equation that best describes the observed input-output relationship. The process automatically searches for the simplest model, represented by a combination of terms and coefficients, that accurately captures the system’s dynamics without requiring pre-defined model structures or manual feature engineering.

The performance of the discovered equations was quantitatively assessed using the Root Mean Squared Error (RMSE), normalized by the standard deviation of the observed data to produce a Normalized RMSE (NRMSE). Validation against the Bouc-Wen hysteretic model, subjected to signal-to-noise ratios (SNR) of 30, 25, and 20 dB, consistently yielded NRMSE values below 0.06. This metric indicates a high degree of fidelity between the discovered equation and the reference model, even in the presence of significant noise. The NRMSE calculation is given by $NRMSE = \frac{RMSE}{std(y)}$, where $RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i – \hat{y}_i)^2}$ and $y_i$ represents the observed data and $\hat{y}_i$ the predicted data.

Application of the methodology to a complex structural system resulted in a Normalized Root Mean Squared Error (NRMSE) of 0.055. This performance represents an improvement over the standard Sparse Identification of Nonlinear Dynamics (SINDy) method when applied to the same data. The data-driven nature of this approach allows for the identification of governing equations directly from observed data, circumventing the need for pre-defined model structures or reliance on prior physical assumptions. This is particularly valuable in scenarios where the underlying physics are poorly understood or highly complex, enabling the discovery of relationships solely through data analysis.

Introducing nonlinear spring damping (NSD) significantly alters the hysteresis loops of a yielding structure, demonstrating a shift in its dynamic behavior.
Introducing nonlinear spring damping (NSD) significantly alters the hysteresis loops of a yielding structure, demonstrating a shift in its dynamic behavior.

Expanding the Echo: Applications and Future Directions

The newly derived equations offer a powerful tool for simulating and predicting the behavior of systems characterized by hysteresis – a phenomenon where a system’s response depends not only on the current input but also on its past history. This capability is particularly valuable in fields like structural engineering and material science, where hysteresis commonly arises due to energy dissipation through friction, damping, or phase transformations. By accurately modeling this rate-dependent behavior, engineers can better understand and predict the long-term performance of structures under cyclic loading, such as those experienced during earthquakes or repeated use. Furthermore, these equations facilitate the design of materials with tailored hysteretic properties, enabling innovations in vibration control, energy absorption, and damping technologies. The predictive power stems from the equations’ ability to capture the nonlinear relationship between input and output, providing insights into the system’s energy dissipation characteristics and overall stability.

Recent investigations demonstrate the power of combining newly developed equations with real-world experimental validation through shake table testing. This integration allows for a significantly more accurate assessment of how structures behave under the dynamic stress of seismic events, particularly those incorporating innovative devices like Negative Stiffness Devices. Analyses of a yielding structure subjected to simulated earthquakes reveal a remarkably low Normalized Root Mean Square Error (NRMSE) of 0.1411 for displacement predictions, and even lower errors – below 0.06 – for velocity and acceleration measurements. These findings suggest the potential for improved structural designs and enhanced safety standards, moving beyond traditional linear assumptions to better reflect the complex, nonlinear behavior observed during actual seismic activity.

The current modeling framework gains significant potential through the integration of Fractional Calculus, a branch of mathematics dealing with derivatives and integrals of non-integer order. Unlike traditional calculus which assumes instantaneous changes, Fractional Calculus accounts for the ‘memory’ inherent in many physical systems – the influence of past states on the present behavior. This is particularly crucial when analyzing viscoelastic materials, hereditary effects in structural dynamics, and systems exhibiting long-range dependencies. By incorporating fractional derivatives into the governing equations, the model can more accurately capture these memory effects, leading to improved predictions of system response and enhanced control strategies. This approach moves beyond simple rate-dependent behavior, allowing for a more nuanced understanding of complex phenomena where past stimuli continue to influence present and future states, ultimately broadening the applicability of the framework to a wider range of scientific and engineering challenges.

A shaking table test of a three-story braced structure-modeled as a single-degree-of-freedom system-demonstrates the installation of a negative stiffness device on the first floor [lai2019sparse].
A shaking table test of a three-story braced structure-modeled as a single-degree-of-freedom system-demonstrates the installation of a negative stiffness device on the first floor [lai2019sparse].

The pursuit of a ‘unified framework’-a singular structure to encompass the chaotic dance of hysteretic systems-feels a touch naive. It echoes a perennial human desire to impose order, to believe a comprehensive model can truly capture such inherently unpredictable phenomena. As Bertrand Russell observed, “The fact that we cannot know things perfectly is an important lesson.” This research, while admirable in its ambition to simultaneously discover internal variables and governing equations, inevitably freezes a compromise in time. The discovered models are, at best, accurate approximations-prophecies of future failure as the system inevitably diverges from the captured representation. The true nature of hysteresis lies not in its predictability, but in its resistance to being fully known.

The Turning of the Wheel

This work, in its pursuit of internal variables and governing equations, offers a temporary respite from the chaos of observation. Yet, every discovered equation is merely a shadow cast by a more complex reality. The framework, while adept at capturing the current face of the hysteretic system, does not address the inevitable drift. Systems do not remain static; they accumulate memories, grow new imperfections, and subtly redefine their boundaries. The elegance of a discovered model is always a prelude to its eventual inadequacy.

Future efforts will undoubtedly focus on extending this framework to systems where the very definition of internal variables is not known a priori. But a more profound challenge lies in accepting that complete knowledge is an illusion. Perhaps the true path is not to discover the governing equations, but to build mechanisms for graceful adaptation, for learning from the inevitable failure of any predictive model.

The field will likely move toward architectures that embrace impermanence, systems that can continuously renegotiate their internal representations in response to the shifting landscape of hysteretic behavior. This isn’t about building a perfect map, but about cultivating a resilient traveler.

Original article: https://arxiv.org/pdf/2512.02408.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-04 00:42