Hybrid Electromagnetic Modelling of Tubular Permanent Magnet Linear Motors Based on Transfer Learning Physics-Informed Neural Networks

Abstract

Due to the inherent nonlinearity and saturation in the magnetic circuits of tubular permanent magnet linear motors, the analytical method (AM), while computationally efficient, often fails to capture complex electromagnetic behaviours accurately. In contrast, the finite element analysis (FEA) offers high precision but is time consuming. The nonlinearity of magnetic materials introduces strong input–output coupling, while saturation leads to localised deviations in field distributions, both of which reduce the effectiveness and generalisability of conventional modelling approaches. To overcome these challenges, a physics-informed, data-driven modelling approach is proposed. Initially, a novel hybrid modelling framework based on physics-informed neural networks (PINNs) is introduced. In this framework, AM is incorporated into both the input-output layers and the relevant variables, thereby enabling the direct embedding of physical constraints into the loss function. Consequently, the network's training process is rigorously guided in accordance with established physical principles. To further enhance prediction accuracy and generalisation, a transfer learning framework is integrated into PINN, utilising pre-trained datasets from AM and fine-tuning the model using high-accuracy datasets derived from FEA. Additionally, to optimise the physical information-related hyperparameters that impact model accuracy, functional analysis of variance is employed to quantitatively assess their importance and determine the optimal hyperparameter values. Experimental results show that, with training sample sizes representing only 5% of the FEA data, TL-PINN achieves significant improvements over DNN, including a 74.25% reduction in (1 − R²), a 49.51% reduction in RMSE, and a 50.46% reduction in MAE. These findings demonstrate that TL-PINN delivers superior accuracy while utilising substantially fewer FEA datasets.