P. Deepika
Department of Mathematics, Vellore Institute of Technology, 632014, Vellore, Tamil Nadu, India.

L. Mohana Sundari
Department of Software Systems, Vellore Institute of Technology, 632014, Vellore, Tamil Nadu, India.

DOI https://doi.org/10.33889/IJMEMS.2026.11.4.071

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Abstract

Parabolic convection diffusion equations with small diffusion effects play a central role in modeling transport dominated processes arising in heat transfer, fluid flow, and mass transport. When diffusion is very small, these models exhibit sharp boundary layers near the spatial boundaries, which significantly complicates numerical approximation and often leads to loss of accuracy for conventional discretization techniques. Motivated by these challenges, this work focuses on a singularly perturbed parabolic convection diffusion model with Robin type boundary conditions, which naturally arise in problems involving convective heat exchange and surface interactions. To address the multiscale nature of the solution, a weighted ensemble neural network approach is proposed. The method combines several feedforward neural networks, each trained to approximate different localized features of the solution, and blends their outputs using smoothly varying spatial weight functions. A purely supervised learning framework is adopted, where training data are obtained from known exact solutions. Initial and boundary conditions are enforced through data based loss terms, while spatial derivatives are used only to ensure consistency at the boundaries. Unlike physics informed neural network approaches, the governing equation is not imposed as a residual constraint during training. By embedding the expected boundary layer behavior into the network structure and employing ensemble weighting, the proposed method accurately captures sharp solution gradients without the need for mesh refinement or domain decomposition. Numerical experiments for a range of perturbation parameters demonstrate that the method delivers stable and accurate approximations and generally outperforms single network models in resolving boundary layer effects.

Keywords- Weighted ensemble neural network, Singular perturbation problems, Parabolic convection–diffusion equations, Robin boundary conditions, Supervised learning.

Citation

Deepika, P., & Sundari, L. M. (2026). A Weighted Ensemble Neural Network Method for Singularly Perturbed Parabolic Convection-Diffusion Equations with Robin Boundary Conditions. International Journal of Mathematical, Engineering and Management Sciences, 11(4), 1743-1769. https://doi.org/10.33889/IJMEMS.2026.11.4.071.