Data-driven forward–inverse problems for the variable coefficients Hirota equation using deep learning method

This paper investigates data-driven forward–inverse problems associated with the variable coefficients Hirota (VC-Hirota) equation using the physics-informed neural network (PINN) algorithm. First, we propose an improved PINN algorithm with a locally adaptive activation function to recover data-driven solitons and high-order solitons solutions for the VC-Hirota equation. Second, we demonstrate the effectiveness of the improved PINN algorithm in accurately predicting parameters under different noise intensities using a parameter regularization strategy and appropriate weight coefficients. Third, we introduce a PINNs approach that employs two neural networks to tackle the function discovery problem. The neural network with time and space coordinates in the input layer is used to train the prediction solution, and the neural network with only time coordinates in the input layer is trained to model the unknown function in the variable coefficient function. This work presents a successful attempt to use the PINN method to solve the function discovery problem of VC-Hirota equations.