Multifractal-spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes pose a unique challenge in classification and characterization. Previously (Mangalam et al., 2023, Physical Review Research 5, 023144), we established a framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous time random walk, annealed transient time motion, and L\'evy walk. To accomplish this, we generate extensive datasets comprising $10^6$ trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. Furthermore, we explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and $p$-variation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. Our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing classification of anomalous diffusion.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

Single-particle traces of the diffusive motion of molecules, cells, or animals are by-now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics is vital in understanding the observed systems. Typically, the task is to decipher the exact type of diffusion and/or to determine system parameters. The tools used in this endeavor are currently revolutionized by modern machine-learning techniques. In this Perspective we provide an overview over recently introduced methods in machine-learning for diffusive time series, most notably, those successfully competing in the Anomalous-Diffusion-Challenge. As such methods are often criticized for their lack of interpretability, we focus on means to include uncertainty estimates and feature-based approaches, both improving interpretability and providing concrete insight into the learning process of the machine. We expand the discussion by examining predictions on different out-of-distribution data. We also comment on expected future developments.