Physics-informed neural networks with Riccati-type pseudopotential constraints for solving nonlinear evolution equations.

Abstract

Physics-informed neural networks (PINNs) provide a powerful framework for solving nonlinear evolution equations, but their accuracy and stability often deteriorate for systems with high-frequency structures, strong nonlinear interactions, or high-order derivatives. Although structure-informed extensions can improve physical consistency by incorporating geometric or integrable constraints, many such formulations rely on high-dimensional auxiliary variables, which may introduce latent gauge redundancies and enlarge the optimization space. To circumvent these inherent limitations, we propose RP-PINN, a physics-informed architecture based on Riccati-type pseudopotentials. By combining the Wahlquist-Estabrook prolongation structure with Nucci's direct method, the proposed framework replaces high-dimensional auxiliary representations with Riccati-type equations, which are embedded into the loss function as low-order rigid constraints. Numerical experiments on the Korteweg-de Vries, nonlinear Schrödinger, sine-Gordon, and Harry Dym equations demonstrate that RP-PINN can accurately reproduce soliton, rogue wave, and kink solutions while mitigating the adverse effects of high-order automatic differentiation. These results demonstrate the effectiveness and applicability of RP-PINN to integrable systems admitting Riccati-type pseudopotentials.