Hybrid Modeling Approaches: Deep Learning and Fractional Differential Equations in Complex Systems

Dear Colleagues,

This Special Issue will advance research at the intersection of deep learning and fractional differential equations (FDEs), with a particular emphasis on hybrid modeling techniques that fuse the interpretability of mathematical theory with the predictive capabilities of modern AI. Fractional-order models are known for their ability to capture memory effects and hereditary properties in complex systems—phenomena often inadequately described using classical differential equations. By integrating these models with machine learning frameworks, especially deep learning, researchers can address real-world dynamics more accurately and robustly.

We invite contributions that explore novel theoretical developments, computational techniques, and interdisciplinary applications centered on fractional-order systems. Priority will be given to studies that demonstrate rigorous mathematical grounding, innovative learning architectures, and real-world relevance.

The scope of this Special Issue includes, but is not limited to, the following:

• Physics-Informed Neural Networks (PINNs) and Fractional PINNs (fPINNs) tailored to FDEs;
• Operator learning techniques (e.g., DeepONets, Fourier Neural Operators) for learning solution operators of FDEs;
• Deep neural solvers for time-fractional, space-fractional, and variable-order FDEs;
• Learning-based methods for inverse and ill-posed problems governed by fractional models;
• Data-driven discovery of fractional dynamics from empirical observations;
• Scientific machine learning frameworks for modeling nonlocal, memory-dependent, and multiscale phenomena;
• Applications in biology, control systems, materials science, finance, neuroscience, and complex dynamical systems.

Dr. Ali Turab
Prof. Dr. Josue Antonio Nescolarde Selva
Prof. Dr. Andres Montoyo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• fractional differential equations (FDEs)
• fractional difference equations
• fractional physics-informed neural networks (fPINNs)
• deep learning for dynamical systems
• Fourier neural operators (FNOs)
• nonlocal and memory-dependent models
• variable-order differential equations
• fractal-fractional order mathematical models
• time-fractional and space-fractional models
• hybrid neural–stochastic modeling
• inverse problems in fractional systems
• reinforcement learning in fractional systems
• cognitive modeling with FDEs
• multiscale modeling in complex systems
• deep neural network solvers for FDEs
• applications in biology and neuroscience
• computational and stability methods in fractional calculus