Solving inverse Gardner–Kawahara problems with physics-informed neural networks: a data-driven approach

Abstract Inverse problems involving nonlinear partial differential equations (PDEs) pose significant challenges due to their ill-posed nature and reliance on sparse or noisy observations. Traditional approaches often require complete knowledge of initial and boundary conditions, which may not be available in practical scenarios. Physics-informed neural networks (PINNs) have recently emerged as a powerful approach for addressing such problems by embedding physical laws into the structure of deep neural networks. In this work, we employ PINNs to solve the inverse problem for the Gardner-Kawahara equation, a high-order nonlinear dispersive PDE that modeling wave propagation in fluids and plasmas. The proposed PINNs framework accurately reconstructs unknown parameters and solution fields from limited data, even in the presence of noise. Numerical experiments conducted under varying parameter settings and noise levels demonstrate strong agreement with exact solutions, thereby highlighting the method’s accuracy, robustness, and computational efficiency. These results confirm the potential of PINNs for inverse modeling of complex nonlinear systems, even in the absence of complete initial or boundary information, and demonstrate that they outperform traditional methods in handling sparse and noisy data.