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Accurately estimating the magnetization distribution in permanent magnets is critical for optimising their performance in various applications, such as electric motors, generators and magnetic sensors, where precise magnetic field control is essential. A physics-informed neural network (PINN) is demonstrated to solve the inverse problem of magnetization distribution within the volume of permanent magnets. A neural network is constructed to model the spatially dependent magnetization in the magnet. The physical model, based on the Biot–Savart law, is integrated into the PINN framework. The discrepancy between the magnetic field calculated by the physical model and the externally observed one is used to guide the network training, exhibiting both the model-based and data-driven characteristics of the PINN. The accuracy and robustness of the proposed PINN are demonstrated through numerical experiments with both uniform and nonuniform magnetization scenarios, as well as both noise-free and noisy observation data. This study provides a new approach for solving magnetization distribution estimation problems, benefiting the development of high-quality permanent magnets for electrical engineering applications.

The data assimilation and state estimation play an important role in the digital twin (DT) technology. The parameterized-background data-weak (PBDW) method is an emerging non-intrusive data assimilation algorithm which reconstructs the field distribution in the physical system by using the measured data combined with the best-knowledge model, and the reconstructed field can be further post-processed to extract state parameters of interest for the state estimation of the physical system. However, the conventional PBDW method still faces great challenges for practical applications in electrical engineering. The main difficulties are the large number of required sensors to ensure the well-posedness of the PBDW system, the difficulty accessing the optimal sensor locations, and the handling of the noisy measurement data. To address these problems, a real-virtual sensor PBDW (RV-PBDW) framework is proposed in this work. Virtual sensors with readings mapped from real sensors using a neural network (NN) are proposed to reach the optimal sensor locations and significantly reduce the number of required real sensors placed in the practical physical system. Furthermore, the data augmentation techniques are integrated with the NN to enhance the performance on handling noisy measurement data, and the regularization parameter which is hard to decide in the conventional PBDW is not needed. The performance of the proposed framework is verified and explored through an EI-core inductor and a rotating electrical machine case studies. The field distribution reconstructed by the RV-PBDW method can be further used in the post-processing and facilitates various functions of DT system such as the sensing, detection, and diagnosis, etc., enabling more diverse, in-depth state estimation of the physical system. This work significantly extends the feasibility of the conventional PBDW method in the context of the DT technology for practical electrical engineering applications.