Modeling the Transmission Dynamics of Soluble and Insoluble Water Pollutants: A Comparative Study of Random Projection and Deep Neural Networks
4 affiliations
- 1School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China
- 2Department of Statistics and Operations Research, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
- 3School of Mechanical and Electrical Engineering, Chengdu University of Technology, Chengdu 610059, China
- 4Livestock and Dairy Development Department, Khyber Pakhtunkhwa, Peshawar 25000, Pakistan
Abstract
Water pollution remains a major environmental and public-health problem, especially in developing countries with limited treatment infrastructure. We formulate a four-compartment model of ordinary differential equations (ODEs) for the transmission of soluble and insoluble water pollutants that extends an earlier model with a natural dilution rate. We prove positivity and boundedness of solutions, derive the basic reproduction number R0 and the equilibria, and show that the pollutant-free equilibrium is locally asymptotically stable when R0<1. The model is solved with two complementary neural-network strategies: an adaptive physics-informed random projection neural network (RPNN), in which Gaussian hidden-layer parameters are drawn at random and fixed and only the output weights are computed by Newton iterations without any training data, and a deep neural network (DNN) surrogate with tanh hidden layers trained by the Levenberg–Marquardt algorithm on reference trajectories. Both are compared with the MATLAB solvers ode15s, ode23t, ode23s and ode45. For four case studies at a relative tolerance of 10−6, the RPNN attained L2 errors of 3.35×10−8–2.44×10−7, compared with 7.23×10−5–1.57×10−4 (ode15s), 6.12×10−4–6.50×10−4 (ode23t), 6.28×10−4–9.64×10−4 (ode23s), 8.64×10−6–1.22×10−5 (ode45) and 3.03×10−3–1.04×10−2 (DNN). This accuracy comes at a higher cost per run: the RPNN required 5.2–10.1 times the CPU time of ode15s and 4.0–21.3 times that of ode45, and a work–precision analysis shows that the explicit solver ode45 remains the most efficient solver for this small system. The accuracy of the RPNN was maintained over 200 parameter sets sampled within ±25% of the baseline, and the DNN recovered the dynamics from training data with up to 10% noise. Sensitivity analysis identifies the natural removal rate, the treatment rate of soluble pollutants and the pollutant inflow as the parameters that control R0. The RPNN is an accurate, data-free and tolerance-controlled solver for stiff environmental ODE models, whereas the DNN is a fast but less accurate surrogate.
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BibTeX
@article{waseem2026modeling,
title = {{Modeling the Transmission Dynamics of Soluble and Insoluble Water Pollutants: A Comparative Study of Random Projection and Deep Neural Networks}},
author = {Waseem Waseem and Ziyad A. Alhussain and Anwarud Din and Khurshaid Anwar},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214009},
url = {https://www.mdpi.com/2227-7390/14/21/4009},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Modeling the Transmission Dynamics of Soluble and Insoluble Water Pollutants: A Comparative Study of Random Projection and Deep Neural Networks
AU - Waseem Waseem
AU - Ziyad A. Alhussain
AU - Anwarud Din
AU - Khurshaid Anwar
PY - 2026
DA - 2026/11/03
JO - Mathematics
DO - 10.3390/math14214009
UR - https://www.mdpi.com/2227-7390/14/21/4009
AB - Water pollution remains a major environmental and public-health problem, especially in developing countries with limited treatment infrastructure. We formulate a four-compartment model of ordinary differential equations (ODEs) for the transmission of soluble and insoluble water pollutants that extends an earlier model with a natural dilution rate. We prove positivity and boundedness of solutions, derive the basic reproduction number R0 and the equilibria, and show that the pollutant-free equilibrium is locally asymptotically stable when R0<1. The model is solved with two complementary neural-network strategies: an adaptive physics-informed random projection neural network (RPNN), in which Gaussian hidden-layer parameters are drawn at random and fixed and only the output weights are computed by Newton iterations without any training data, and a deep neural network (DNN) surrogate with tanh hidden layers trained by the Levenberg–Marquardt algorithm on reference trajectories. Both are compared with the MATLAB solvers ode15s, ode23t, ode23s and ode45. For four case studies at a relative tolerance of 10−6, the RPNN attained L2 errors of 3.35×10−8–2.44×10−7, compared with 7.23×10−5–1.57×10−4 (ode15s), 6.12×10−4–6.50×10−4 (ode23t), 6.28×10−4–9.64×10−4 (ode23s), 8.64×10−6–1.22×10−5 (ode45) and 3.03×10−3–1.04×10−2 (DNN). This accuracy comes at a higher cost per run: the RPNN required 5.2–10.1 times the CPU time of ode15s and 4.0–21.3 times that of ode45, and a work–precision analysis shows that the explicit solver ode45 remains the most efficient solver for this small system. The accuracy of the RPNN was maintained over 200 parameter sets sampled within ±25% of the baseline, and the DNN recovered the dynamics from training data with up to 10% noise. Sensitivity analysis identifies the natural removal rate, the treatment rate of soluble pollutants and the pollutant inflow as the parameters that control R0. The RPNN is an accurate, data-free and tolerance-controlled solver for stiff environmental ODE models, whereas the DNN is a fast but less accurate surrogate.
SN - 2227-7390
ER -