A Legendre-Based Modified Extreme Learning Machine for Convection-Diffusion Equations: Steady-State, Continuous-Time, and Discrete-Time Formulations
Abstract
Convection-diffusion equations are fundamental models for transport phenomena, but their numerical solution becomes challenging in convection-dominated regimes due to steep gradients and sharp boundary layers. This motivates the development of accurate approaches capable of effectively resolving such features while avoiding costly iterative optimization. In this study, a modified ELM framework based on shifted Legendre polynomials is developed for convection-diffusion equations in steady-state, continuous-time, and discrete-time formulations. The proposed approach is evaluated on steady and unsteady benchmark problems with relatively high Peclet numbers, including continuous and discontinuous boundary conditions. Finite element method solutions are used as independent solution references. By exploiting the linearity of the governing equations, MELM determines the output weights through a single least-squares solve, avoiding the iterative gradient-based training required by PINNs. The shifted Legendre polynomial basis provides an effective spectral representation for resolving steep gradients and sharp boundary layers. For the DT-MELM formulation, the coefficient matrix remains unchanged across successive time levels, allowing its LU or QR factorization to be computed once and reused. A systematic comparison with PINNs that incorporate advanced training strategies demonstrates that MELM provides higher accuracy and shorter solution times for steady-state and DT-MELM formulations across the benchmark problems considered. In contrast, CT-MELM exhibits scalability limitations as the size of the space--time basis increases. Overall, the results demonstrate the effectiveness of MELM for steady-state and discrete-time convection--diffusion problems, while also showing that PINNs provide better scalability for large space--time formulations where global CT-MELM becomes computationally demanding.
Related
- A Fully Discrete Local Discontinuous Galerkin Method for Quasilinear Stochastic Convection-Diffusion-Type Equations
- Optimization-based Cell Average Limiters as Metric Projections: Constrained Evolution and Convergence Analysis for Convection-diffusion Equations
- Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
- High-Moment Stability and Error Analysis of a Fully Discrete LDG-IMEX Method for High Dimensional Nonlinear Stochastic Convection-Diffusion Equations
- Sign-switched antithetic coupling in gradient random-walk methods for convection-diffusion equations
BibTeX
@misc{ankush2026legendrebased,
title = {{A Legendre-Based Modified Extreme Learning Machine for Convection-Diffusion Equations: Steady-State, Continuous-Time, and Discrete-Time Formulations}},
author = {Gadade Navnath Ankush and Ram Shiromani and Tomas Gandara and Ramon Codina and Sivaram Ambikasaran},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02800},
eprint = {2611.02800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02800},
}
RIS
TY - UNPB
TI - A Legendre-Based Modified Extreme Learning Machine for Convection-Diffusion Equations: Steady-State, Continuous-Time, and Discrete-Time Formulations
AU - Gadade Navnath Ankush
AU - Ram Shiromani
AU - Tomas Gandara
AU - Ramon Codina
AU - Sivaram Ambikasaran
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02800
UR - https://arxiv.org/abs/2611.02800
AB - Convection-diffusion equations are fundamental models for transport phenomena, but their numerical solution becomes challenging in convection-dominated regimes due to steep gradients and sharp boundary layers. This motivates the development of accurate approaches capable of effectively resolving such features while avoiding costly iterative optimization. In this study, a modified ELM framework based on shifted Legendre polynomials is developed for convection-diffusion equations in steady-state, continuous-time, and discrete-time formulations. The proposed approach is evaluated on steady and unsteady benchmark problems with relatively high Peclet numbers, including continuous and discontinuous boundary conditions. Finite element method solutions are used as independent solution references. By exploiting the linearity of the governing equations, MELM determines the output weights through a single least-squares solve, avoiding the iterative gradient-based training required by PINNs. The shifted Legendre polynomial basis provides an effective spectral representation for resolving steep gradients and sharp boundary layers. For the DT-MELM formulation, the coefficient matrix remains unchanged across successive time levels, allowing its LU or QR factorization to be computed once and reused. A systematic comparison with PINNs that incorporate advanced training strategies demonstrates that MELM provides higher accuracy and shorter solution times for steady-state and DT-MELM formulations across the benchmark problems considered. In contrast, CT-MELM exhibits scalability limitations as the size of the space--time basis increases. Overall, the results demonstrate the effectiveness of MELM for steady-state and discrete-time convection--diffusion problems, while also showing that PINNs provide better scalability for large space--time formulations where global CT-MELM becomes computationally demanding.
ER -