Optical bistable state exhibits its great potential applications in photonic integrated circuit and photonic neural network. However, the traditional optical bistable state will be influenced by the system disorders, which are not suitable for application. In this work, we investigate the topological bistable states in a layered structure with center inversion symmetry consisting of alternating layers of high index material TiO ₂and low index material SiO ₂. In the topological mode, the electric field is highly localized in the inversion center of the layered structure (also known as the interface) and exponentially decays into the bulk. Thus, when the nonlinear permittivity is strategically introduced into those layers, nonlinear phenomena such as the bistable state appears. The finite element numerical simulations reveal that the optimal bistable state appears when the layer period is 5 with a threshold power around 1.2 W/m. Benefiting from the topological characteristics, such a bistable state persists when random perturbations are introduced into the layer thickness and refractive index. Finally, we apply the bistable states to a photonic neural network. The bistable function shows prediction accuracy similar to the classic activation function ReLu and sigmoid in various learning tasks. These results provide a novel method for inserting highly robust optical bistable states from topological layered structure into photonic neural network.