Discovery of Governing Equations and Dynamic Reconstruction of Nonlinear Waves Driven by Customized Library

ABSTRACT Nonlinear waves driven by multiple physical mechanisms are commonly described by different forms of the nonlinear Schrödinger (NLS) equation, yet conventional physics‐based modeling relies heavily on strong prior assumptions and expert knowledge, limiting its efficiency and applicability to complex systems. Here, we propose a physics learning strategy driven by a customized library (PhyCuLi) to automatically discover governing NLS equations from sparse and complex wave data while reconstructing nonlinear wave dynamics. The framework employs a tailored library that explicitly incorporates both the real and imaginary components of the field variables, enabling unified identification of NLS equations governed by multi‐physical‐field interactions. By further introducing time‐ and space‐dependent modulation functions, the method consistently models both homogeneous systems with constant coefficients and inhomogeneous systems with variable coefficients within a single framework. Using sparse wave data generated by five representative physical mechanisms, we demonstrate PhyCuLi robustly identifies the underlying NLS equations and accurately reconstructs complex nonlinear wave evolutions. Moreover, by regulating nonlinear terms in the customized library through physical prior knowledge, multiple distinct yet physically consistent NLS formulations are discovered from the same dataset, revealing the intrinsic non‐uniqueness of data‐driven equation discovery and broadening its applicability to complex nonlinear systems.