Abstract:

With the help of the physics-informed neural networks (PINNs), the forward and inverse problems of extended fifth-order mKdV(emKdV) equation are tackled, and the dynamic behaviors of solitons are also analyzed and simulated in this paper. The hyperbolic tangent function $\tanh$ is selected as the activation function to solve the one, two and three-soliton solutions of the equation. Moreover, the data-driven solutions obtained by PINNs method are compared with the exact solution given by the simplified Hirota method. Specifically, the accuracy of one-soliton solution is $\mathcal{O}(10^{-4})$, and the accuracy of the two-soliton and three-soliton solutions is $\mathcal{O}(10^{-3})$. For the inverse problem, the coefficients of the equation are discovered by the data of one, two and three-soliton solutions, respectively. Meanwhile, the robustness of the PINNs algorithm is explored under different noises. The accuracy of the data-driven coefficients can reach $\mathcal{O}(10^{-3})$ or $\mathcal{O}(10^{-2})$ respectively, when 1% initial noise or observation noise is added to the training data. And the prediction accuracy can still reach $\mathcal{O}(10^{-2})$ even if 3% initial noise or observation noise is added. According to the analysis of experimental data, the impact of observation noise on PINNs model is slightly greater than the initial noise.

Key words: Physics-informed neural networks, Fifth-order emKdV equations, Data-driven solutions, Nonlinear dynamics