A novel singularity- and discontinuity-capturing PINN for time-fractional diffusion equations involving initial singularities and interfaces on complex curved surfaces

In this paper, we propose a novel singularity- and discontinuity-capturing physics-informed neural network (SDC-PINN) designed to tackle challenging time-fractional diffusion problems characterized by initial singularities and interfaces on complex curved surfaces. The SDC-PINN method incorporates a singularity-capturing feature in the temporal direction at the initial time, alongside a discontinuity-capturing feature in the spatial direction within the neural network input. This approach preserves the inherent properties of the solution, effectively and accurately capturing the solution’s sharpness as well as that of its derivatives at the interface, while adeptly resolving the initial singularity. To further address the initial singularity of the solution, we employ the classical nonuniform L2-1σ scheme in time to approximate the Caputo fractional derivative of the neural network output. Additionally, the proposed network accommodates the use of scattered training points, thereby facilitating the efficient handling of problems on more complex curved surfaces. Numerous numerical experiments are performed to assess the effectiveness and precision of the proposed SDC-PINN.