Data-Driven Localized Wave Solutions and Parameters Discovery of the (1+1)-dimensional Ito Integro-Differential Equation

In this paper, we investigate the data-driven localized wave solutions and parameters discovery of the (1+1)-dimensional Ito integro-differential equation by using the ​Potential-Function-Transformation PINNs (PFT-PINNs) method. Firstly, through the introduction of a potential function transformation, we convert the original integro-differential equation into the differential form and simultaneously also reduce the order of the differential equation, which provide convenience in solving the (1+1)-dimensional Ito integro-differential equation by using the PINNs method. Secondly, based on the neural networks, the data-driven localized wave solutions including soliton, breather, rogue wave, fusion and fission solutions are obtained with the aid of the potential function transformation. The results show that the PFT-PINNs method possesses the good performance of PFT-PINNs method in solving the forward problem of the (1+1)-dimensional Ito integro-differential equation and can acquire more accurate localized wave solutions than the standard PINNs method. Finally the PFT-PINNs method is used to solve the inverse problem of the (1+1)-dimensional Ito integro-differential equation and the results demonstrate that the unknown coefficient parameters can be satisfactorily identified even with heavily noisy data.