A multiscale review of magnetic permeability in polymer composites: Theory, simulation, and machine learning frontiers

Polymer-based magnetic composites (PMCs) are vital in applications such as sensors, EMI shielding, and inductors, yet predicting their effective magnetic permeability (μ eff ) remains challenging. This review critically analyzes classical models like Maxwell Garnett and Bruggeman, which offer foundational insights but fall short in systems with clustering, nonlinearity, or high filler fractions. Advanced techniques—such as finite element methods, phase-field modeling, and micromechanical frameworks—address these gaps by accounting for real geometries and multiphysics interactions, including magnetoelastic coupling and domain evolution. The role of elasticity networks and interfacial dynamics is emphasized, alongside the rise of machine learning for modeling complex structure-property relationships and enabling multi-objective optimization. Despite progress, challenges persist in dynamic hysteresis modeling, interphase parameterization, and morphology-percolation linkage. This review advocates for hybrid modeling approaches that integrate physics-based and data-driven methods to design next-generation PMCs with predictive accuracy and enhanced functionality across electromagnetic and multifunctional domains.