A semi-analytical physics-informed neural network framework with hessian analysis for singularly perturbed fractal convection-dominated anomalous diffusion problems.

The present study introduces the concept of physics-informed neural networks to develop a numerical approach for solving singularly perturbed convection-dominated anomalous diffusion problems. A key focus of this research is to analyse the boundary layers that arise in convection-dominated anomalous diffusion problems using an artificial neural network, namely a physics-informed neural network. We explore solutions to convection-dominated anomalous diffusion problems using neural-network corrector functions derived from underlying auxiliary conditions. The thin layers represent regions with rapid variations in the solution, exhibiting intricate behaviour and rendering existing numerical methods less effective. The proposed method is a semi-analytic method that captures the minute behaviour of the solution in the boundary layer and resolves the singularity that naturally arises in fractal boundary-value problems. An in-depth discussion will be provided, supported by numerical examples demonstrating promising, uncompromised results related to linear and nonlinear stationary anomalous diffusion equations.•PINN framework is proposed for singularly perturbed anomalous diffusion problems.•Corrector functions effectively capture thin boundary layers and singular behaviour.Hessian-based analysis is employed to investigate convergence and solution sensitivity.•The proposed method demonstrates strong performance on both linear and nonlinear problems.