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.