SLeNN-ELM: A shifted Legendre neural network method for fractional delay differential equations based on extreme learning machine

In this paper, we introduce a shifted Legendre neural network method based on an extreme learning machine algorithm (SLeNN-ELM) to solve fractional differential equations with constant and proportional delays. Based on the properties of Caputo fractional derivatives and shifted Legendre polynomials, the fractional derivatives of SLeNN can be represented analytically without other numerical techniques. SLeNN, in terms of neural network architecture, uses a function expansion block to replace the hidden layer, and thus improving the computational efficiency by reducing parameters. In terms of solving technology of neural networks, the extreme learning machine algorithm is used to replace the traditional gradient-based training algorithm. It dramatically improves our solution efficiency. In addition, the proposed method does not require parameter initialization randomly, making the neural network solution stable. Finally, three examples with constant delays and three examples with proportional delays are given, and the effectiveness and superiority of the proposed method are verified by comparison with other numerical methods.