Data-driven soliton solutions and parameters discovery of the coupled nonlinear wave equations via a deep learning method

In this paper, we extend the physics-informed neural networks (PINNs) to learn data-driven solutions and discover the coefficients of multi-component and high-dimensional coupled nonlinear partial differential equations (cNPDEs). In the forward problems, the data-driven solutions include "bell"-type (dark and bright) and "anti-kink"-type solitons of the generalized Hirota–Satsuma coupled KdV equation are investigated and reconstructed. Moreover, some special shape one-soliton and two-soliton solutions such as "Kink"-shaped, "U"-shaped, "Y"-shaped, and "X"-shaped of a (2+1)-dimensional cNPDE are learned and simulated as well. Particularly, some factors affecting the performance of PINNs are investigated, such as the structure of neural networks, the kinds of activation functions and optimizers, as well as the number of collocation points and iterations. For the inverse problems, the coefficients of the equations are successfully identified and driven by various kinds of solution data. The impact of the number and moment of observations on the model is discussed while the network's robustness is validated by introducing mensurable initial noises to the training data. Numerical experiments not only show that the propagation direction and dynamical behaviors of the cNPDE can be well uncovered by utilizing the deep learning method, but also exhibit the advantage of finding equations with multiple parameters.