Dynamic hysteresis model of grain-oriented ferromagnetic material using neural operators

TL;DR

This work presents a neural-operator approach to dynamic hysteresis in grain-oriented (GO) steel by learning the mapping from the magnetic flux input $B(t)$ to the magnetic field output $H(t)$, capturing peak values, frequency effects, and phase shifts. It situates the problem within Bertotti's loss-separation framework and uses the dynamic relation $H(t)=H_{hys}(B(t)) + \frac{m^2}{12\rho}\frac{dB}{dt} + g(B)\delta|\frac{dB}{dt}|^\alpha$ to motivate operator learning. Three neural-operator architectures—Fourier Neural Operator (FNO), its U-Net augmented variant (U-FNO), and DeepONet—are evaluated on GO steel DHL data, with data augmentation (cyclic rolling and Gaussian data augmentation) enhancing robustness to phase shifts and input noise. Across experiments, FNO consistently achieves the best accuracy and training efficiency, while augmentation strategies substantially reduce mean relative errors in predicted core losses. The results highlight neural operators as effective, scalable tools for accurate dynamic hysteresis modeling in electromagnetic applications.

Abstract

Accurately capturing the behavior of grain-oriented (GO) ferromagnetic materials is crucial for modeling the electromagnetic devices. In this paper, neural operator models, including Fourier neural operator (FNO), U-net combined FNO (U-FNO) and Deep operator network (DeepONet) are used to approximate the dynamic hysteresis models of GO steel. Furthermore, two types of data augmentation strategies including cyclic rolling augmentation and Gaussian data augmentation (GDA) are implemented to enhance the learning ability of models. With the inclusion of these augmentation techniques, the optimized models account for not only the peak values of the magnetic flux density but also the effects of different frequencies and phase shifts. The accuracy of all models is assessed using the L2-norm of the test data and the mean relative error (MRE) of calculated core losses. Each model performs well in different scenarios, but FNO consistently achieves the best performance across all cases.

Figures (8)

• Figure 1: FNO structure and unit.
• Figure 2: U-net structure and U-FNO unit.
• Figure 3: DeepONet structure. Branch net with $B$ and $c_k(B)$ as input and output, trunk net with the coordinates $t$ and $\xi_k{t}$ as the input and output, $b$ as the bias, $\mathcal{G}(B)(t)$ as the approximated output $H$.
• Figure 4: The measuring system. It comprises a personal computer, a NI PCI‐6120 data acquisition (DAQ) card, an audio power amplifier, and an air‐flux compensated single strip tester (SST).
• Figure 5: $B-H$ predictions by FNO, U-FNO, and DeepONet, comparing their performance on 4 test loops.