This article proposes a novel numerical method, the Chebyshev Ensemble Extreme Learning Machine (CH-EN-ELM), to solve variable-order (VO) fractional partial differential equations (FPDEs). The study aims to provide accurate and robust solutions for important VO fractional models in fluid dynamics, including the Burger's, wave and diffusion equations. It seeks to improve solution accuracy and overcome the generalization and stability issues associated with conventional extreme learning machines (ELMs).
The study develops a feedforward neural network (FNN) scheme that integrates Chebyshev polynomials with an ensemble of ELMs, in which the network outputs of two or more independently trained ELM-based single-layer FNNs (SLFNN-ELMs) are aggregated to obtain a single output of the scheme. The method enhances network efficiency by using Chebyshev polynomials for input feature expansion and a radial basis function as the hidden-layer activation function of each SLFNN-ELM in the ensemble network.
The numerical results demonstrate that the proposed CH-EN-ELM method produces highly accurate solutions and exhibits a good convergence rate. The approach shows superior performance with strong error minimization capabilities when compared to other techniques in the literature. The method's effectiveness is confirmed across various examples, including the nonlinear time-fractional Burger's equation and diffusion-wave equations, outperforming methods presented in other studies.
The authors suggest future work could extend the approach to more complex models, such as VO fractal-fractional PDEs, coupled systems of VO FPDEs and higher-order problems. There is also potential to combine the CH-EN-ELM method with data-driven frameworks like physics-informed neural networks (PINNs) to enhance its application to real-world problems.
The article provides a robust, efficient, and easily implemented method for solving a wide range of VO FPDEs that model complex phenomena in physics, mechanics and fluid dynamics. The method is a promising alternative for tackling large-scale, multidimensional problems where traditional numerical methods often face difficulties.
This study is the first to propose an ensemble ELM technique that uses Chebyshev-augmented input patterns to solve VO FPDEs. The novel combination of ensemble learning with specific feature expansion provides a powerful and efficient framework for accurately solving complex fractional differential equations, demonstrating versatility across different types of fractional operators.
