Robust Deep Learning Framework Using Hermite Interpolation for Time‐Dependent Caputo Fractional Differential Equations

ABSTRACT In this manuscript, we propose two innovative approaches for approximating the time‐dependent Fokker‐Planck and diffusion equations, the Hermite interpolation neural network (HINN) and the robust Hermite interpolation neural network (R‐HINN). In HINN, we use a basic feedforward neural network with Hermite interpolation, while R‐HINN adds extra connections (lateral connections) within the network to make it more robust, along with Hermite interpolation. By leveraging the infinitely differentiable properties of deep neural networks, Hermite interpolation is utilized to approximate the Caputo derivative. Our results demonstrate that R‐HINN achieves superior performance compared to HINN, L1, L1‐2, and L1‐2‐3 schemes, offering significant improvements in error levels as well as computational efficiency. The HINN & R‐HINN both utilize the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) algorithm, a quasi‐Newton optimizer that accelerates convergence and enhances accuracy. Rigorous comparisons with analytical solutions validate the effectiveness of the R‐HINN, showcasing its potential as a powerful tool for solving fractional differential equations.