Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization

In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based Physics-Informed Neural Networks (PINNs) with the Cuckoo Search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite-difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising of two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations and three-dimensional fractional Bloch-Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Further, the extended use to problem areas with irregular geometries and difficult to define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations, in computing science and engineering.