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The research demonstrates that combining symbolic learning, neural differential equations, and neural operator learning leads to models that are not only highly accurate and efficient but also generalize far beyond traditional approaches. Key findings reveal that symbolic modeling can automatically discover closed-form dynamical equations with superior speed, neural differential equations can predict complex hysteresis loops from only limited data, and neural operators can successfully generalize to entirely unseen excitations, achieving improvements that surpass conventional models by orders of magnitude. Together, these results point toward a new generation of hysteresis surrogates that are efficient, interpretable, scalable, and capable of supporting the design of more energy-efficient and sustainable electromechanical systems.

Understanding why hysteresis matters

Hysteresis arises when a material鈥檚 response depends not only on the current electric or magnetic field but also on its past exposure. This memory effect reflects microstructural changes that unfold over time and do not instantly reverse. In practice, this causes nonlinear and history-dependent behavior that complicates predictions in engineering systems. Piezoelectric and soft magnetic materials display these effects strongly, which means that actuators, motors, sensors, and precision devices built from them may exhibit unexpected delays, energy losses, or reduced efficiency. Engineers often struggle with this because traditional control strategies must compensate for behavior that is difficult to model accurately. This is why a deeper understanding of hysteresis is so important: it directly influences system reliability, energy consumption, and performance across a wide range of technologies.

The limits of traditional approaches

For decades, hysteresis was typically modeled with domain-specific formulations, such as Preisach-type operators or phenomenological models developed within magnetics or piezoelectricity. These methods provided valuable insights but rarely generalized well from one system or excitation type to another. They often required expert intuition, limiting their adaptability and making cross-domain transfer difficult. As engineering systems grow more complex and as data becomes more plentiful, these rigid approaches are no longer sufficient. A more flexible, scalable, and data-driven strategy is needed to advance both scientific understanding and technological development.

Machine Learning as a new pathway

The research of proposes three complementary machine-learning-based frameworks that together provide a significant advancement in modeling hysteresis. The first approach, symbolic modeling, uses sparse regression to discover interpretable, closed-form dynamical equations directly from measurement data. Instead of relying on black-box neural networks, this method recovers transparent expressions that match or surpass the accuracy of existing deep learning models while computing results much faster. This restores interpretability to hysteresis modeling, allowing engineers and scientists to understand why a model behaves as it does.

The second contribution centers on neural differential equations, which learn the underlying dynamics of hysteresis loops and demonstrate exceptional generalization capabilities. Remarkably, these models can reconstruct minor loops and first-order reversal loops even when trained only on major loop data. They provide significantly higher accuracy than conventional recurrent neural networks while requiring far less computational effort. This opens the door to effective modeling even when experimental data is limited or costly to obtain.

The third framework focuses on neural operator learning, which predicts material behavior under completely new input signals that were never seen during training. This represents one of the most difficult challenges in hysteresis modeling. The neural operator developed in this work outperforms traditional models by several orders of magnitude and, with the addition of a neuro-symbolic extension, achieves even greater accuracy and interpretability. The resulting operator can describe complex hysteresis behavior under a wide variety of excitation scenarios, providing robustness and flexibility that traditional models cannot match.

The methodologies developed here lay the foundation for a new paradigm in the design and modeling of electromechanical materials. As these approaches mature and begin to influence real-world technologies, such as more efficient electric motors, smarter robotics, and improved energy systems, their societal impact will become much more visible.

Toward more efficient and sustainable technology

The overarching conclusion of this research is that integrating symbolic learning, neural differential equations, and operator learning enables the creation of hysteresis surrogates that are accurate, efficient, interpretable, and broadly generalizable. These advances support better modeling of piezoelectric and soft magnetic materials, which in turn contributes to improved electric machine design, reduced energy losses, and more reliable electromechanical systems. By bridging machine learning with material science and electromechanical engineering, the work presented in this thesis offers new possibilities for sustainable technological development and provides a solid foundation for future innovations in the field.

Title of PhD thesis: . Supervisors: Prof. Elena Lomonova, Dr. Mitrofan Curti, and Dr. Koen Tiels.