A shifted Chebyshev neural network approach for nonlinear variable-order fractional differential equations

This paper presents Shifted Chebyshev Neural Network (SChNN), a functional link artificial neural network framework to solve variable-order fractional differential equations (VOFDEs). The framework employs shifted Chebyshev orthogonal polynomials as orthogonal basis functions for input feature expansion, significantly enhancing computational efficiency through reduced structure complexity to solve linear and nonlinear VOFDEs. To further optimize performance, we integrate a Taylor-series approximation of the smooth Mish activation function, which allows more flexibility when dealing with variable-order (VO) derivatives. The training process uses Broyden, Fletcher, Goldfarb, and Shanno (BFGS) optimization to minimize a mean square error (MSE) loss function, ensuring robust convergence properties. Comprehensive numerical experiments demonstrate that the proposed SChNN achieve a high accuracy, with a further validation process using the exact solution.

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