Data-driven solitons dynamics and parameters discovery in the generalized nonlinear dispersive mKdV-type equation via deep neural networks learning

In this paper, we study the dynamics of data-driven solutions and identify the unknown parameters of the nonlinear dispersive modified KdV-type (mKdV-type) equation based on physics-informed neural networks (PINNs). Specifically, we learn the soliton solution, the combination of a soliton and an anti-soliton solution, the combination of two solitons and one anti-soliton solution, and the combination of two solitons and two anti-solitons solution of the mKdV-type equation by two different transformations. Meanwhile, we learn the data-driven kink solution, peakon solution, and periodic solution using the PINNs method. By utilizing image simulations, we conduct a detailed analysis of the nonlinear dynamical behaviors of the aforementioned solutions in the spatial-temporal domain. Our findings indicate that the PINNs method solves the mKdV-type equation with relative errors of \(\mathcal {O}(10^{-3})\) $\mathcal{O}({10}^{-3})$ or \(\mathcal {O}(10^{-4})\) $\mathcal{O}({10}^{-4})$ for the multi-soliton and kink solutions, respectively, while relative errors for the peakon and periodic solutions reach \(\mathcal {O}(10^{-2})\) $\mathcal{O}({10}^{-2})$ . In addition, the tanh function has the best training effect by comparing eight common activation functions (e.g., \(\textrm{ReLU}(\textbf{x})\) $\text{ReLU}(\mathbf{x})$ , \(\textrm{ELU}(\textbf{x})\) $\text{ELU}(\mathbf{x})$ , \(\textrm{SiLU}(\textbf{x})\) $\text{SiLU}(\mathbf{x})$ , \(\textrm{sigmoid}(\textbf{x})\) $\text{sigmoid}(\mathbf{x})$ , \(\textrm{swish}(\textbf{x})\) $\text{swish}(\mathbf{x})$ , \(\textrm{sin}(\textbf{x})\) $\text{sin}(\mathbf{x})$ , \(\textrm{cos}(\textbf{x})\) $\text{cos}(\mathbf{x})$ , and \(\textrm{tanh}(\textbf{x})\) $\text{tanh}(\mathbf{x})$ ). For the inverse problem, we invert the soliton solution and identify the unknown parameters with relative errors reaching \(\mathcal {O}(10^{-2})\) $\mathcal{O}({10}^{-2})$ or \(\mathcal {O}(10^{-3})\) $\mathcal{O}({10}^{-3})$ . Furthermore, we discover that adding appropriate noise to the initial condition enhances the robustness of the model. Our research results are crucial for understanding phenomena such as interactions in travelling waves, aiding in the discovery of physical processes and dynamic features in nonlinear systems, which have significant implications in fields such as nonlinear optics and plasma physics.