Forecasting Key Global Factors using Hybrid Artificial Neural Networks and the Mackey-Glass Nonlinear Differential Equation

Document Type : Research Article

Authors

1 Biosystem department, Agricultural Research Institute, Iranian Research Organization for science and Technology

2 Department of Biotechnology Research Organization for Science and Technology, Tehran

3 General Office of Information Technology, Iranian Research Organization for Science and Technology

4 Department of Wood and Paper Science and Technology, Ka.C., Islamic Azad University, Karaj, Iran

Abstract

Accurate prediction of environmental and socio-economic indicators is of great importance for global development in all areas and for assessing risks related to climate change. In this study, artificial neural networks (ANNs) based on multilayer perceptron (MLP), long short-term memory (LSTM) and hybrid artificial neural networks, with and without the Mackey-Glass Nonlinear Differential Equation (MG), were used to predict world population, per capita gross domestic product (GDP), fossil fuel consumption and CO$_2$ emissions. Historical data were collected from official and reliable international sources for the years 1990 to 2022. To evaluate the performance of the proposed models, a set of reliable indicators including root mean square error (RMSE), mean absolute percentage error (MAPE) and coefficient of determination ($R^2$) were used. The results show that the hybrid neural network models that used the Mackey-Glass delayed differential equation significantly reduced the forecast error in all evaluation indices for all different variables. The Mackey-Glass equation improved the MAPE index by 12.5\% {}{}and increased the $R^2$ index by 8.7\%. In addition, the results of the sensitivity analysis show that the models are sensitive to the choice of input features, data preprocessing, and network architecture design. The differences between the model outputs highlight the need to pay close attention to the model complexity and how to represent the time series dynamics in long-term forecasts. Overall, the findings indicate that the hybrid neural models augmented with the nonlinear delayed differential equation provide a more accurate and reliable picture of future global trends. The results have important implications for climate policy design, global energy planning, and sustainable development strategies.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 10, Issue 2
December 2025
Pages 255-263

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History

  • Receive Date: 16 November 2025
  • Accept Date: 15 December 2025
  • Publish Date: 30 December 2025

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