A Data-Driven Adaptive Deep Learning Method for Nonlinear Schrödinger Type Equation

Abstract

Abstract This paper develops a residual-based adaptive refinement physics-informed neural networks (RAR-PINNs) method for solving the Gross–Pitaevskii (GP) equation and Hirota equation, two paradigmatic nonlinear partial differential equations (PDEs) governing quantum condensates and optical rogue waves, respectively. The key innovation lies in the adaptive sampling strategy that dynamically allocates computational resources to regions with large PDE residuals, addressing critical limitations of conventional PINNs in handling: (1) Strong nonlinearities (| u | 2 u terms) in the GP equation; (2) High-order derivatives ( u xxx ) in the Hirota equation; (3) Multi-scale solution structures. Through rigorous numerical experiments, we demonstrate that RAR-PINNs achieve superior accuracy [relative L 2 errors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> ] and computational efficiency (faster than standard PINNs) for both equations. The method successfully captures: (1) Bright solitons in the GP equation; (2) First- and second-order rogue waves in the Hirota equation. The RAR adaptive sampling method demonstrates particularly remarkable effectiveness in solving steep gradient problems. Compared with uniform sampling methods, the errors of simulation results are reduced by two orders of magnitude. This study establishes a general framework for data-driven solutions of high-order nonlinear PDEs with complex solution structures.