Physics-informed fractional machine intelligence and space–time wavelet frameworks for non-local integro-partial differential equations involving weak singularities

Abstract

This paper presents a space–time multi-dimensional wavelet framework and a physics-informed fractional machine intelligence (PI-fMI) model to address the weak singularity involved in time-fractional integro-partial differential equations with mixed Volterra–Fredholm operators. Conventional machine learning approaches often struggle with weak initial singularities; however, our proposed approach overcomes this challenge through two complementary strategies in the context of fractional-order integro-differential equations. First, a wavelet-based numerical scheme is employed that utilizes the multi-resolution analysis with the collocation method to compute the wavelet coefficients, ensuring convergence for fractional-order integro-differential problems with sufficiently smooth solutions. Second, we introduce a PI-fMI model for problems that exhibit unbounded temporal derivatives at , which incorporates the discretization for fractional operators, a combination of the repeated quadrature rule, and automatic differentiation to handle integral operators that contain diffusion terms. Theoretical and numerical analyses demonstrate that the proposed approach successfully resolves the initial weak singularities where the traditional Haar wavelets fail to address such issues. Furthermore, the convergence of the PI-fMI model is analyzed for problems with nonlinear source terms, demonstrating its effectiveness under suitable hyperparameter choices. Theoretical findings are validated through extensive numerical experiments on several test problems exhibiting bounded and unbounded temporal derivatives at .