Using PINN Machine learning Method with DGM Network for Solving Time-Fractional Black–Scholes Models

The classical Black-Scholes model fails to capture crucial market features such as memory effects and non-Markovian behavior. The time-fractional Black-Scholes equation (TFBSM) has emerged as a powerful alternative that incorporates these phenomena through fractional calculus. However, accurate numerical solution of TFBSMs remains challenging due to the non-local nature of fractional derivatives, complex boundary conditions, and the singularity in fractional operators. Existing numerical methods often suffer from discretization limitations and computational ineffciency when handling these complexities. This paper introduces a novel hybrid framework that synergistically combines Physics-Informed Neural Networks (PINNs) with the Deep Galerkin Method (DGM) architecture. Our approach leverages the PINN methodology to directly embed the TFBSM governing equation, initial conditions, and boundary conditions into the optimization objective through a carefully designed loss function. The DGM network, with its specialized architecture resembling Long Short-Term Memory (LSTM) networks, provides enhanced capability for capturing long-term temporal dependencies essential for fractional calculus. The Caputo time-fractional derivative is accurately implemented using Gauss-Legendre quadrature with 100-300 nodes, ensuring precise computation of the fractional operator. Comprehensive numerical experiments on two benchmark problems with analytical solutions demonstrate the effectiveness of our approach. For Example 1 (fractional ODE), the method achieves relative L2 errors as low as 3.06 × 10−3 at t = 1.0 with α = 0.5. For Example 2 (TFBSM with nonhomogeneous boundary conditions), relative L2 errors remain below 2.40 × 10−2 across the entire time domain with α = 0.7. The method exhibits excellent stability, with smooth loss decay curves indicating robust convergence. This work presents the first successful integration of PINNs with DGM for solving time-fractional Black-Scholes equations. The key innovations include a novel PINN-DGM hybrid architecture specifically tailored for fractional PDEs, accurate implementation of Caputo derivatives using Gauss-Legendre quadrature within the neural network framework, comprehensive numerical validation demonstrating superior performance compared to conventional methods, and establishment of a robust, mesh-free paradigm for financial PDEs with memory effects. The numerical results demonstrate that the proposed method accurately approximates the solution of the time-fractional Black–Scholes equation, with improved stability and convergence.