Solving localized wave solutions of the BBM equation using physics-informed neural networks

Abstract Focusing on the Benjamin-Bona-Mahony (BBM) equation, a nonlinear partial differential equation governing shallow water wave propagation, we explore the performance of physics-informed neural networks (PINNs) in predicting its solutions. A hybrid sampling strategy is firstly adopted to generate the training datasets for enhancing the accuracy of the PINNs and achieving effective spatiotemporal coverage. Then, by embedding the physical constraints into the loss function of the BBM equation and employing automatic differentiation, the PINNs successfully predicts multiple typical soliton solutions, including bright and dark solitons. We note that this method can also capture periodic solution of the BBM equation. We also compare this approach with the spectral method. The error of the predicted and numerical solutions with compared to the exact solution gradually increases and mainly concentrates on the flanking regions of peak and trough regions as evolutionary time grows. We also consider the impact of some main factors affecting the PINNs performance, including activation functions, hidden layers and neurons in each hidden layer. Extensive training reveals that selecting suitable activation functions is essential for accurately capturing the soliton morphology and dynamical behaviors. Furthermore, we numerically calculated the conservation of mass and energy for solutions of the BBM equation predicted by PINNs during long-term evolution. These findings have certain reference value for predicting localized waves in nonlinear partial differential equations using PINNs.