Deep learning approaches for $ \beta- $conformable fractional differential equations: A PINN, NRPINN-s, and NRPINN-un based solutions

In this paper, we present a numerical approach for solving $ \beta- $conformable fractional differential equations (FDEs) using physics-informed neural networks (PINNs) and their enhanced versions, NRPINN-s and NRPINN-un. The proposed method combines the flexibility of artificial neural networks with the inherent structure of fractional calculus, allowing the solution of complex initial and boundary value problems without domain discretization. The loss function in the model includes initial and boundary conditions and is constructed using modifiable parameters (weights and biases). The efficiency of the proposed methods is demonstrated by numerical experiments on heat and wave equations. The obtained results show that NR-PINNs are superior to classical PINNs in improving convergence, reducing local errors and showing good performance. Visual and tabular comparisons of the solutions obtained from the method with analytical solutions confirm the effectiveness of the approach.