**Mathematical Complexity Modeling and Artificial Intelligence of Complex Systems and Nonlinear Sciences: Fractal-Fractional, Variational, Stochastic and Quantum Calculus’ Analyses and Applications**
Nonlinear dynamical systems, reflected as a complex archive of different behaviors, overflow with models of varying phenomena in complex systems. Thus, nonlinearity signifies a relationship that cannot be explained or modeled by a linear algebraic or differential combination of input or state variables. Across this line of thought, nonlinear science serves to reveal the nonlinear descriptions of generally different systems having profound impacts on complex dynamics. Fractional calculus, in that regard, is a powerful tool for system identification including the capability of approximating nonlinear functions owing to nonlinear activation functions and diverse inputs as well as the employment of outputs employment, which enhances the processing and control of complex, chaotic and heterogeneous elements in complex, nonlinear and dynamic systems and systems sciences. Fractal calculus as an algorithmic and constructive approach to the modeling of natural processes cannot be feasible through the use of smooth differentiable structures and conventional modeling means like differential equations. With these benefits, fractal calculus can lend a framework for exploring derivatives and integration of functions with the support of fractals that provide a unified viewpoint along with the varied trajectories of complexities in the natural world, medicine and biology by presenting innovative dimensions with multiple strata. Variational calculus, a.k.a. the calculus of variations, refers to the domain of mathematics that is concerned with the optimization of functionals as mathematical objects with its focal point oriented towards obtaining the maxima or minima of functionals, known as the extrema. Variational calculus is geared towards the infinitesimally slight alterations in the function per se, called variations. The methods of variational calculus constitute advanced mathematical techniques which are employed for finding the extremum points of functionals within a solution-oriented framework for the optimization problems. Stochastic calculus, on the other hand, is employed for modeling analyses to entangle random processes and their impacts on the systems by enabling the analysis of behaviors of a stochastic process over time, which is possible by different mathematical methods like integration and differential equations. Last but not least, quantum calculus, known as calculus without limits, ensures the extension of traditional calculus through the elimination of concept of limits and bridging the gap between continuous and discrete analysis and presenting a sort of non-standard analysis. Mathematical complexity modeling, on the other hand, signifies the level of intricacy in the optimization and design of a deep learning model, constituting model framework, optimization process size as well as the complexity pertaining to the data at hand and employed. The importance of model complexity lies in terms of theory of complexity which makes distinctions precisely through the proposal of formal criteria concerning the mathematical problem rendering it to be feasibly decidable so that it can be solved, and interpretability in applications become important for decision-making processes. Computational resources come to the foreground as complex models entail further computational power and time. The relationship between complexity and mathematical modeling requires manifold aspects so that the real-world phenomena can be represented as precisely and accurately as possible. In that regard, balance between complexity and simplicity is about trade-offs, with more complex models’ overfitting the data and capturing noise while simple models may be underfitting, which causes the missing of important patterns and details. Regarding the iterative processes, starting with a simple model evolves into increase in complexity gradually, striking a balance between interpretability and accuracy. Given these, mathematical modeling is an abstract description of a concrete system through mathematical concepts, and developing a mathematical model as a process refers to the progression of formulating a mathematical representation for real-world phenomena and systems for the purpose of understanding, analyzing or predicting the impacts and outcomes emerging from them. Mathematics, as applied to a broad range of real-world problems, entails modeling presenting it as a robust means to understand complex systems amidst the advancing landscape of technology necessitating the generation of timely and effective solutions. Models are generated by means of different tools including neural networks, probability distributions, differential equations, game theoretic models, dynamical systems, statistical models, and so forth. Artificial Intelligence (AI) and mathematical modeling based on AI are employed to tackle challenging problems with accuracy through the incorporation of machine learning, deep learning algorithms and conventional methods in a way that could leverage computing resources equipped with high resolution image analysis techniques. All these affordances enable the prediction about the way systems will behave under certain conditions. Besides prediction, AI can facilitate with building classification models so that new data can be categorized through learning the patterns in the data, which is important since it could be possible to respond and adapt to uncertainties and changes in the system beforehand. Regarding decision-making and optimization, AI is utilized for finding optimal decisions and strategies through learning about the dynamics and functions of the systems. In view of these and many other related points with mathematical, numerical and computational modeling aspects, the importance of generating applicable solutions to problems for various domains including engineering, medicine, biology, mathematical science, computational science and applied sciences, amongst many others become critical to address predictability, accuracy, interpretability and reliability at times with AI and machine learning situated at the intersection of multiple realms marked by chaotic, nonlinear, dynamic, transient and complex components to corroborate the significance of optimized approaches. Based on these sophisticated integrative and multiscale approaches with computer-assisted applications, our special issue resting upon interdisciplinary perspectives aims at paving the way for new crossroads in real systems and in other related realms. **The potential topics include but are not limited to:**
- Control, synchronization and machine learning
- Stochastic analysis, modeling and stochastic Markov processes
- High dimensional Bayesian inference problems
- Applied probability
- Differential / difference equations for modeling real-world phenomena
- Fractional variational calculus principles and transversality conditions with applications
- Operators’ theory, integral and differential operators, integral transforms
- Dynamic equations on time scales
- Signal / image processing
- AI for cybersecurity and risk management
- Fractional and fractal signals
- Network theory and classification using differential equations (i.e. PDEs, ODEs)
- Fractal-Fractional calculus AI and applications
- Mathematical and computational statistics
- Quantum mechanics and processes
- Quantum algorithms and complexity
- Quantum AI and machine learning methods
- Entropy and wavelet with applications
- Bifurcation, attractors and chaos
- Optimization algorithms
- Data-driven science, optimization, model-building, time series and applications
- Fractional calculus and its interdisciplinary applications
- Fractal geometry
- Cantor sets and fractal curves
- Fractional-order differential and integral equations
- Fractional dynamics and biological systems
- Uncertainty quantification
- Quantification and approximate symmetries in biological systems
- Mathematical and computational biology, biomedicine and bioengineering
- Theories and / or applications of advanced AI (i.e. machine learning, deep learning)
- Algorithmic (computational) complexity
- Complex adaptive systems
- Historical and philosophical aspects of complex systems
- Complexity and networks, modeling and dynamics
- Applied complex systems and nonlinear dynamics
**Important deadlines for submission:**
- Initial submission of articles: November 1, 2024
- Closing date for initial submission: December 30, 2025
- Deadline for final decision notification: February 28, 2026
**Editors of the Special Issue: **
**Prof. Yeliz Karaca,** University of Massachusetts Chan Medical School, Worcester, USA E-mail: yeliz.karaca@ieee.org
**Prof. Dumitru Baleanu**, Lebanese American University, Beirut, Lebanon and Institute of Space Science, Magurele, Ilfov, Romania E-mail: dumitru.baleanu@gmail.com
**Prof. Mahmoud Abdel-Aty,** Sohag University, Sohag, Egypt E-mail: chairman@natural-s-publishing.com
**Prof. Martin Bohner,** Missouri University, Columbia, USA E-mail: bohner@mst.edu
**Prof. Yu-Dong Zhang,** The University of Leicester, Leicester, UK E-mail: yudongzhang@ieee.org
**Prof. Shaher Momani,**** **Ajman University, Ajman, UAE E-mail: s.momani@ajman.ac.ae
Our special issue will have the papers to be published at NSP with the following journals: |