A physics-informed neural network method for an inverse source problem in a nonlinear time-fractional subdiffusion equation

We study a physics-informed neural network (PINN) method for an inverse source problem associated with a nonlinear time-fractional subdiffusion equation with Caputo derivative of order α ∈ ( 0 , 1 ) {α∈(0,1)} . The unknown source depends on both time and space and is identified from additional integral measurement data. Building on available well-posedness and stability results for the continuous inverse problem, we formulate a residual-based PINN approximation in which the state variable is represented by a neural network satisfying the homogeneous spatial boundary conditions, while the source is reconstructed in closed form through the analytical representation formula. Our main contribution is a rigorous error analysis of this computational framework. First, we derive a stability estimate showing that the error in the state and reconstructed source is controlled by the PDE and initial residuals, while the measurement residual is included in the PINN loss to enforce consistency with the observed integral data. Second, using Sobolev approximation results for shallow tanh networks, we quantify the approximation power of the neural ansatz in terms of the network size. Third, we relate the continuous generalization error to the discrete training loss by establishing explicit midpoint quadrature estimates. Combining these ingredients, we obtain computable a priori bounds for both the solution and source reconstruction errors, which reveal their dependence on the network width and quadrature resolution. Numerical experiments for representative examples support the theory and illustrate stable and accurate recovery across different fractional orders, discretization levels, and data perturbations. The results provide a mathematically justified scientific-computing framework for PINN-based inversion in nonlinear fractional diffusion problems.