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Magnetic flux density (B) is traditionally interpreted as a continuous field whose flux lines form closed loops, as prescribed by Maxwell’s equations. This description is well justified in macroscopic systems, where large ensembles of magnetic dipoles produce statistically smooth fields through spatial averaging. At the nanoscale, however, where only a limited number of dipoles may contribute, the conditions underlying this continuum interpretation become less clear. Here, we reexamine the meaning of magnetic flux density from a classical-statistical perspective, focusing on finite ensembles and isolated magnetic particles. We show that as the number of contributing dipoles decreases, ensemble averaging becomes insufficient to support a statistically stable, coarse-grained field description, even though the underlying electromagnetic fields remain well defined and fully consistent with Maxwell’s equations. In this regime, magnetic flux density retains its formal definition, but its interpretation as a robust macroscopic observable becomes strongly dependent on fluctuations and specific dipole configurations. This framework introduces a quantitative criterion based on a critical particle number and provides a consistent description of the transition from ensemble-averaged magnetostatics to discrete dipole behavior. The results clarify the limits of continuum field interpretations at the nanoscale and offer a unified perspective for understanding isolated nanoparticles, small dipole ensembles, and the emergence of classical magnetic behavior from discrete microscopic sources. 

The relaxation of hot carriers in Si materials is dominated by nonradiative processes. The relaxation dynamics of Agn (n = 0–7) monolayered clusters adsorbed on Si(111) slabs has been computed based on nonadiabatic couplings (NACs) using the Redfield formalism of density matrix theory for dissipative rates, with an electronic basis set generated by density functionals. Here, we present machine learning (ML) models to predict nonradiative relaxation rate constants of charge carriers (ke/h). We construct a data set in which each sample corresponds to ke/h associated with a specific initial photoexcited condition of a given system. The ML feature vector consists of Redfield tensor elements, carrier identity, and atomic structure identity. The ML target label is ke/h obtained from the Redfield formalism. Four ML regressors are evaluated: linear Ridge regression, random forest (RF) regressor, XGBoost, and feed-forward multilayer perceptron (MLP). Model performance is evaluated using stratified 5-fold cross-validation. All models achieve strong predictive performance, with test set R2 values exceeding 0.96. Ridge regression and XGBoost exhibit the highest test set R2 values of 0.99. The small performance gap between Ridge regression and the more complex nonlinear models suggests that much of the structure–property relationship is approximately linear in the transformed feature space. This study shows that complex excited-state relaxation processes can be learned from Redfield tensor elements using relatively simple models. 

PurposeMachine learning to enable precise, non-invasive detection of cerebral amyloid-beta (Aβ) pathology by integrating cognitive assessments, plasma biomarkers, and structural neuroimaging. MethodsWe developed an explainable multimodal machine-learning framework to predict amyloid PET visual read status using plasma biomarkers, cognitive assessments, APOE genotype, demographic variables, and structural MRI measurements. The development cohort consisted of 170 participants from the 1Florida Alzheimer’s Disease Research Center (ADRC) and 129 participants from the ADNI4 cohort. Machine-learning pipelines were evaluated using combinations of five classifiers and multiple feature-selection approaches with Bayesian hyperparameter optimization and stratified five-fold cross-validation. Model interpretability was assessed using SHapley Additive exPlanations (SHAP). ResultsEnsemble methods consistently outperformed linear and distance-based classifiers. The optimal pipeline, XGBoost with mutual-information feature selection, achieved a mean AUC of 0.891 ± 0.048 and a recall of 0.84. SHAP analyses identified the plasma p-tau217/Aβ42 ratio as the most influential predictor, followed by plasma p-tau217, p-tau181, MMSE, and APOE ε4 status. Structural MRI variables provided complementary predictive information. ADNI4 evaluation yielded a mean AUC of 0.545, while training on ADRC and testing on ADNI4 achieved an AUC of 0.636 and an overall accuracy of 63%. Importantly, plasma p-tau217/Aβ42 and p-tau217 remained the dominant predictors across cohorts. ConclusionMultimodal machine learning can predict amyloid PET status while providing biologically interpretable explanations. Plasma tau and amyloid-related biomarkers carried the strongest predictive signal, while cognition, APOE genotype, and MRI refined classification decisions. External validation highlighted the challenges of cohort heterogeneity and domain shift but demonstrated preservation of biologically meaningful biomarker relationships. 

We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the three-dimensional (3D) Ising model. A 3D convolutional autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature $T_c=4.5128\left(58\right)$ , in excellent agreement with the known value. Additionally, we obtain an estimate of the correlation-length critical exponent, $\nu =0.63\left(27\right)$ , also consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.