Spectral coefficient learning physics informed neural network for time-dependent fractional parametric differential problems

The study of parametric differential equations plays a crucial role in weather forecasting and epidemiological modeling. These phenomena are better represented using fractional derivatives due to their inherent memory or hereditary effects. This paper introduces a novel scientific machine learning approach for solving parametric time-fractional differential equations by combining traditional spectral methods with neural networks. Instead of relying on automatic differentiation techniques, commonly used in traditional Physics-Informed Neural Networks (PINNs), we propose a more efficient global discretization method based on Legendre polynomials. This approach eliminates the need to simulate the parametric fractional differential equations across multiple parameter values. By applying the Legendre-Galerkin weak formulation to the differential equation, we construct a loss function for training the neural network. The trial solutions are represented as linear combinations of Legendre polynomials, with the coefficients learned by the neural network. The convergence of this method is theoretically established, and the theoretical results are validated through numerical experiments on several well-known differential equations.