Data-driven and physics-informed estimation of magnetic nanoparticle properties via stochastic Langevin model

Abstract The dynamic magnetization of magnetic nanoparticles (MNPs) arises from coupled Néel and Brownian relaxations, which are influenced by intrinsic particle properties such as size, saturation magnetization, magnetic anisotropy, and damping. While experimental AC magnetization measurements can reveal the collective dynamic behavior of MNP ensembles, extracting accurate nanoparticle-specific parameters from such data remains a challenge due to experimental limitations and model oversimplifications. To address this, we apply a stochastic Langevin model that explicitly captures the time-dependent magnetization response of MNPs under alternating magnetic fields by incorporating both thermal fluctuations and stochastic relaxation processes. This model provides a physically grounded framework for simulating magnetization hysteresis under experimental conditions, enabling parameter estimation through direct data fitting. In this work, we fit the stochastic Langevin model to experimentally measured hysteresis loops of different MNPs collected under a 20 mT, 5 kHz AC field. By coupling the model with Bayesian optimization and Gaussian process regression, we identify optimal values of key magnetic parameters: saturation magnetization ( {M}_s ), effective anisotropy ( {K}_a ), and Gilbert damping parameter ( α ). Furthermore, the {M}_s is experimentally measured and employed as a validation parameter. Accordingly, the determination of the α and the {K}_a is based on two complementary criteria: (1) the best agreement between the simulated and experimental AC response magnetization hysteresis loops, quantified by the coefficient of determination ( {R}² ), and (2) the closest correspondence between the estimated and experimentally measured {M}_s values, evaluated using the mean absolute percentage error. Our approach is validated on four commercial MNP products (SHS30, IPG30, SHP25, and SHP15, from Ocean Nanotech, LLC), yielding high-fidelity fits to experimental data and robust estimation of their magnetic properties.