Fractional Artificial Neural Networks for Growth Models

TL;DR

The paper addresses solving initial-value problems for fractional growth models using Caputo derivatives. It proposes Fractional Artificial Neural Networks (FANN) trained in R by minimizing $L$ based on the fractional derivative mismatch, and derives a backward-difference discretization for $D^{\alpha}$ to enable practical computation. The method is demonstrated on linear and nonlinear fractional growth cases—including periodic harvesting—showing alpha-dependent performance and validating against analytical solutions where available. This work provides a robust neural-network-based tool for fractional differential equations and suggests directions for architectural improvements and broader applications in fractional modeling, with validation on $u_0 E_{\alpha}(a t^{\alpha})$ and related dynamics.

Abstract

In this paper we present a method to solve initial value problems for fractional growth models, such as generalizations of the exponential and logistic with periodic harvesting models. Using a discretization of the Caputo derivative we propose a fractional artificial neural network, which is implemented in the statistical software R. Moreover, we show examples where the analytical solutions and the approximation of the artificial neural network are compared.

Figures (6)

• Figure 1: Fractional exponential growth.
• Figure 2: The loss functions for $\alpha= 1, 0.9, 0.8$ and $0.7$ respectively. The $x$-axis represents the training epochs while the $y$-axis represents the loss.
• Figure 3: Fractional logistics growth.
• Figure 4: The loss functions for $\alpha= 1, 0.9, 0.8$ and $0.7$ respectively.
• Figure 5: Fractional logistic model with periodic harvesting.

Theorems & Definitions (1)

• Definition 1