## 1 Introduction  
This study addresses critical challenges facing high-speed magnetic field-modulated motors in flywheel energy storage systems. Our work introduces several theoretical innovations compared to existing studies: The proposed frequency domain analytical model for eddy current loss prediction (Section 2.4, Equation (8)) surpasses time domain solutions used by [PERSON] et al. [1] by capturing complex spatial harmonic interactions with 70% improved computational efficiency. Our equivalent thermal circuit transformation technique (Section 2.4, Equation (9)) achieves temperature prediction accuracy within 5.8 \({}^{\circ}\)C for vacuum-sealed systems, compared to deviations exceeding 15 \({}^{\circ}\)C in conventional networks used by [PERSON] et al. [2]. The physics-informed neural network approach (Equation (10)) embeds governing equations as constraints within deep learning architectures, enabling accurate predictions with limited experimental data. This work reveals previously unidentified phenomena, including 42% higher losses in end regions, and demonstrates optimal modulator pole arc coefficients of 0.72-0.78, diverging from the 0.65-0.70 range for conventional machines. These advancements enhance both fundamental understanding and practical design methodologies for flywheel energy storage applications.  
Flywheel energy storage systems (FESSs) have emerged as a promising technology for efficient energy storage applications due to their high power density, fast response characteristics, and long operational lifetime [3]. The implementation of high-speed magnetic field-modulated motors in FESSs has significantly enhanced system performance by improving energy conversion efficiency while maintaining operational stability [4]. However, the operational challenges associated with electromagnetic losses and thermal management remain critical barriers to optimal system performance [5]. Recent advancements in multiphysics coupling analysis have facilitated comprehensive understanding of electromagnetic, thermal, and mechanical interactions in these complex systems [6].  
The loss mechanisms in high-speed magnetic field-modulated motors are multifaceted, comprising core losses (hysteresis, eddy current, and excess losses), copper losses with skin and proximity effects, and mechanical losses including friction and windage [7]. Thermal management strategies must therefore address both the intensity and spatial distribution of these losses [8].  
Advanced thermal modeling techniques have evolved from simplified lumped parameter thermal networks to sophisticated computational fluid dynamics (CFD) approaches capable of analyzing complex flow patterns and heat transfer mechanisms [6]. Recent research has focused on electromagnetic-thermal coupling analysis to accurately predict temperature distributions under various operational conditions [9]. This integrated approach enables optimization of cooling structures including impingement jet cooling and water cooling configurations tailored to the unique thermal challenges of FESSs [10]. Experimental verification has validated these analytical models, confirming their accuracy in predicting both electromagnetic losses and temperature distributions [11]. The ongoing development of these comprehensive analytical frameworks continues to advance both the theoretical understanding and practical implementation of high-efficiency FESS technologies [12].