Regression-based explainable deep learning for estimating Hamiltonian parameters from magnetic nanodot images

Magnetic nanodots host topological spin textures whose stability is dictated by an extended Heisenberg Hamiltonian combining symmetric exchange, the Dzyaloshinskii–Moriya interaction, magnetocrystalline anisotropy, and the Zeeman field. Inferring these atomistic parameters from two-dimensional magnetization images underpins the design of functional magnetic materials, yet it remains an ill-posed inverse problem: under certain external conditions and transient states, configurations relax into noisy, weakly discriminative textures that render the dataset ill-conditioned. Deep-learning estimators act as opaque black boxes and rely on slow atomistic simulations. We propose an explainable regression methodology whose novelty lies in data generation and interpretability: a matrix decomposition of the extended Hamiltonian enables GPU-accelerated Metropolis–Hastings Monte Carlo with simulated annealing, accelerating computation without altering the underlying physics, while Regression Activation Maps (RAMs), a novel error-based attribution metric, turn the network into a traceable gray-box model. Trained on several configurations, convolutional and Vision Transformer regressors reliably recover the physical parameters from simulated nanodot images, while the higher-order exchange constants remain bounded by the intrinsic identifiability limits of the s z projection. Comparing both families through their attention maps and accuracy-versus-size trade-off, RAMs isolate the chiral domain walls and skyrmion cores driving predictions.