Spectral coefficient learning physics informed neural network for time-dependent fractional parametric differential problems
S M Sivalingam, V Govindaraj, A. S. Hendy
TL;DR
This work addresses solving parametric time-fractional differential equations by marrying Legendre-Galerkin spectral discretization with physics-informed neural networks to learn the coefficient expansion in Legendre bases. The method replaces per-parameter PDE solves with a neural network that learns the Legendre coefficients \omega_k(\Upsilon) across the parametric domain, using a weak-form residual loss and Monte Carlo integration over parameters. Theoretical convergence is established under stability and Lipschitz assumptions, and the approach is extended to multidimensional FDEs; extensive numerical experiments on linear, heat-like, and nonlinear problems confirm accuracy improvements with larger networks and more samples. The results highlight the potential of spectral-PINN approaches for efficient, parametric solvers of time-fractional systems with memory effects, relevant to applications in weather, epidemiology, and beyond.
Abstract
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.