Physics-informed neural networks (PINNs) combine deep learning with physical constraints to solve differential equations by embedding governing laws into the training process. This study examines the application of PINNs to the one-dimensional nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation, a prototypical reaction–diffusion model with applications in population dynamics and flame propagation. A standard PINN framework was employed, incorporating the governing equation, initial conditions, and boundary conditions into a unified loss function. Model predictions were validated against analytical solutions and numerical approximations obtained using the finite difference method. The results demonstrate that PINNs can accurately approximate the Fisher–Kolmogorov–Petrovsky–Piskunov solution, achieving strong agreement with both analytical and numerical benchmarks. An analysis of retraining strategies further elucidates the influence of optimizer state retention and network complexity on convergence behavior and solution accuracy. These findings highlight trade-offs between model expressiveness, training efficiency, and numerical fidelity, thereby confirming PINNs as a competitive and flexible alternative to classical numerical methods for nonlinear partial differential equations.