A New Perspective for Scientific Modelling: Sparse Reconstruction-Based Approach for Learning Time-Space Fractional Differential Equations

Abstract

This paper studies a sparse reconstruction-based approach to learn time–space fractional differential equations (FDEs), i.e., to identify parameter values and particularly the order of the fractional derivatives. The approach uses a generalized Taylor series expansion to generate, in every iteration, a feature matrix, which is used to learn the fractional orders of both, temporal and spatial derivatives by minimizing the least absolute shrinkage and selection operator (LASSO) operator using differential evolution (DE) algorithm. To verify the robustness of the method, numerical results for time–space fractional diffusion equation, wave equation, and Burgers' equation at different noise levels in the data are presented. Finally, the methodology is applied to a realistic example where underlying fractional differential equation associated with published experimental data obtained from an in vitro cell culture assay is learned.