Lax-Pair-FIND: Discovering Lax pair from scarce data via deep learning

With the flourishing development of data science and machine learning, significant progress has been made in solving forward and inverse problems of partial differential equations (PDEs) and discovering mathematical equations that describe physical systems. Inspired by data-driven discovery of PDEs, we propose a method for discovering Lax pairs from data—the Lax-Pair-FIND algorithm. The algorithm is capable of identifying the linear evolution operator A based solely on sparse or even noisy data and a known spectral operator L, without prior knowledge of the equation form. For the identification of operator A, we first construct a library consisting of numerous candidate operator terms and then identify the key terms that constitute operator A through sparse optimization and other techniques. In contrast to discovering PDE, which directly leverages terms in the pre-constructed library for constraint computation, learning Lax pairs necessitates additional calculations of operator compositions to compute the Lax compatibility residual. Furthermore, test functions are designed and introduced to validate the operator equations. The proposed Lax-Pair-FIND algorithm has been applied to discover the Lax pairs of the advection, KdV, and mKdV equations, as well as the matrix-form Lax pairs of the Boussinesq–Burgers and nonlinear Schrödinger equations. Through numerical simulations and experimental validation, this method demonstrates excellent effectiveness and robustness when handling varying degrees of data sparsity and noise interference. The computational framework that integrates deep learning with a priori physical information demonstrates tremendous potential in discovering Lax pairs, showing promise in identifying both novel integrable systems and new Lax pair representations of existing systems.