<p>{Fractional calculus (FC) has been shown to help us understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase optimization performance, and even extend the enabling of the potential for creativity. Renormalization Group (RG) allows one to investigate the changes of a physical system at different scales. For example, in quantum field theory, divergent parts of a calculation can lead to nonsensical infinite research results. By applying RG, the divergent parts can be adsorbed into fewer measurable quantities, yielding finite results. To date, Machine Learning (ML) is a hot research topic and will probably remain so in the future. How a model can learn efficiently (optimally) is always essential. The key to learnability is designing efficient optimization methods. Although extensive research has been carried out on the three topics separately, few studies have investigated the association triangle between the FC, RG, and ML. Therefore, in this article, the authors discussed the critical connections between them (Fig.~\ref{fa}). For FC and RG, scaling laws and complexity were discussed. The authors pointed out that FC and RG's critical connection is the inverse power law (IPL). For FC and ML, the critical connections in big data, variability, optimization, and nonlocal models were described. The authors introduced the derivative-free and gradient-based optimization methods and explained how the FC could contribute to these study areas. In the end, the association between the RG and ML was explained. The mutual information, feature extraction, and locality were discussed in this section. Many of the cross-sectional studies suggested a connection between the RG and ML. The RG has a superficial similarity to deep neural networks structure in which one marginalizes over hidden degrees of freedom. Then, the authors draw conclusive remarks about the association triangle between FC, RG, and ML, which will be necessary for a wide range of research topics in the future.}</p>

A New Triangle: Fractional Calculus, Renormalization Group, and Machine Learning

Paper Proceedings: https://asmedigitalcollection.asme.org/IDETC-CIE/IDETC-CIE2021/volume/85369

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