# An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks
## 1 Introduction  
Integer-order differentiation is a powerful tool for solving problems involving rates of change. Its roots are in [PERSON] and [PERSON]'s research and it has evolved into a fundamental branch of calculus. In addition, it possesses several excellent characteristics including ease of understanding, reliable numerical stability, and physical solvability, which make it highly applicable in real-world situations [1]. Meanwhile, fractional-order differentiation is an extended form of integer-order differentiation that enables the description of more complex phenomena such as long-term memory effects and nonlocality [2]. In the 1960s, it was first used to describe how heat energy moves through materials in the study of heat conduction and diffusion [3]. Over time, fractional-order differentiation spread to numerous other fields. In recent years, many researchers have established various physical models via fractional-order differentiation, leading to high-quality approximate solutions. For example, [PERSON] et al. [4] delved into the application of fractional-order differential calculus in modeling viscoelastic systems. [PERSON] et al. [5] introduced fractional-order Legendre-Laguerre functions and explored their applications in solving fractional partial differential equations. [PERSON] et al. [6] discussed the variable order fractional grey model. [PERSON] et al. [7] focused on the application of fractional-order variable-gain supertwisting control to stabilize and control wafer stages in photolithography systems. [PERSON] et al. [8] presented a two-stage fractional dynamical system model for microbial batch processes. These researchers have established physical models that utilize fractional-order differentiation, leading to high-quality approximate solutions. In addition, many optimal control problems also involve fractional-order differentiation. For example, [PERSON], [PERSON], and [PERSON] et al. [9] focused on establishing necessary optimality conditions and exact penalization techniques for constrained fractional optimal control problems. [PERSON] et al. [10] solved fractional optimal control problems using a Chebyshev-Legendre operational technique. [PERSON] et al. [11] presented a control strategy for load frequencyusing a fractional-order controller combined with reduced-order modeling. [PERSON] et al. [12] used Caputo-Fabrizio fractional modeling for analyzing hearing loss due to the Mumps virus. The emergence of neural networks has greatly improved our ability to analyze and process huge volumes of data and deal with nonlinearity [13]. The concept of neural networks originated in the 1950s when people first used computers to simulate human thinking processes. The earliest neural network had a simple structure comprising neuron models and connection weights. With advancements in computer technology and data processing, neural network research experienced a breakthrough in the 1990s. During that period, the backpropagation algorithm and deep learning were introduced to enhance the performance of neural networks [14]. In recent years, fractional-order differentiation has found wide applications in neural networks with the rapid development of new technologies like big data, high-performance computing, and deep learning [15]. For example, [PERSON] et al. [16] analyzed the characteristic equations of a neural network model using two different delays as bifurcation parameters. They developed a new fractional-order neural network model with multiple delays and demonstrated the impact of fractional order on stability. [PERSON] et al. [17] employed optimization methods to adjust the weights of artificial neural networks to solve different fractional-order equations, including linear and nonlinear terms. This approach ensures that the approximate solutions meet the requirements of the fractional-order equations. Similarly, [PERSON] et al. [18] employed a fractional-order controller in training radial basis function neural networks, resulting in improved performance and robustness.