Chaos, Solitons and Fractals 177 (2023) 114302 Available online 24 November 2023 0960-0779/© 2023 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model İbrahim Avcı ∗ , Hüseyin Lort, Buğce E. Tatlıcıoğlu Department of Computer Engineering, Faculty of Engineering, Final International University, Kyrenia, 99300, Northern Cyprus, via Mersin 10, Turkey ARTICLE INFO MSC: 26A33 34A08 34K40 Keywords: Deep learning Fractional differential equations Fractal–fractional derivative Numerical analysis Neural network ABSTRACT This paper investigates the dynamics of Hopfield neural networks involving fractal–fractional derivatives. The incorporation of fractal–fractional derivatives in the neural network framework brings forth novel modeling capabilities, capturing nonlocal dependencies, complex scaling behaviors, and memory effects. The aim of this study is to provide a comprehensive analysis of the dynamics of Hopfield neural networks with fractal– fractional derivatives, including the existence and uniqueness of solutions, stability properties, and numerical analysis techniques. Numerical analysis techniques, including the Adams–Bashforth method, are employed to accurately simulate the fractal–fractional Hopfield neural network system. Moreover, the obtained numerical data serves as validation for developing predictions using Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) neural network methods. The findings contribute to the advancement of both fractional calculus and neural network theory, providing valuable insights for theoretical investigations and practical applications in complex systems analysis. 1. Introduction Investigating dynamical systems involving fractal–fractional deriva- tive operators has emerged as a captivating and promising research direction in recent years [1–4]. The incorporation of fractal–fractional derivative operators introduces a new level of complexity and richness to the dynamics of the systems. Unlike traditional derivatives, fractal– fractional derivatives provide a powerful tool to describe and analyze phenomena with memory effects, nonlocality, and complex scaling behaviors [5]. This novel approach offers several advantages, such as capturing long-range dependencies, modeling systems with non- Markovian dynamics, and incorporating fractional order dynamics in a fractal framework. The exploration of dynamical systems involv- ing fractal–fractional derivative operators has led to groundbreaking insights and opened up new avenues for understanding complex behav- iors in various fields, including physics, engineering, biology, finance, and beyond [6–8]. By delving into the intricate interplay between fractal geometry and fractional calculus, researchers are uncovering the hidden dynamics, stability properties, chaos, and synchronization phenomena within these systems. The investigation of such systems holds immense potential to advance our understanding of complex phenomena and foster the development of innovative modeling ap- proaches, analysis techniques, and applications in diverse scientific and technological domains. ∗ Corresponding author. E-mail address: ibrahim.avci@final.edu.tr (İ. Avcı). Although artificial neural networks (ANN) have been known in the scientific community since the 1940s [9], they had not achieved sufficient popularity due to the capacity of computing machines. ANNs have emerged as a powerful computational paradigm inspired by the intricate workings of the human brain. With improvements in data storage and computer processing [10], artificial neural networks have evolved into information-processing systems that share certain perfor- mance characteristics with biological neural networks [11]. Among the various types of ANNs, Hopfield neural networks (HNNs) hold a significant place. Developed by John Hopfield in the early 1980s, HNNs were among the pioneering models of recurrent neural networks. They gained prominence for their ability to store and retrieve patterns, ex- hibit associative memory capabilities, and solve optimization problems. Over the years, extensive research has been conducted to enhance the understanding of HNNs, explore their theoretical properties, and develop efficient learning algorithms [12–15]. These efforts have led to a deeper comprehension of the dynamics, stability, and convergence properties of HNNs. Moreover, HNNs have found diverse applications in areas such as pattern recognition, image processing, optimization, data compression, and combinatorial optimization [16–18]. In [19], the application of fractional-order Hopfield neural networks for solving optimization problems was explored. This work employed a semi- analytical method based on Adomian decomposition, and it showcased https://doi.org/10.1016/j.chaos.2023.114302 Received 23 August 2023; Received in revised form 15 November 2023; Accepted 21 November 2023