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

Area of Science:

• Computational Physics
• Data Science
• Machine Learning
• Applied Mathematics

Background:

• Significant advancements in data science and machine learning have enabled progress in solving partial differential equations (PDEs) and discovering physical system equations.
• Data-driven discovery methods are increasingly utilized for uncovering complex mathematical relationships in scientific data.

Purpose of the Study:

• To propose and validate a novel data-driven algorithm, Lax-Pair-FIND, for the discovery of Lax pairs.
• To identify the linear evolution operator (A) using sparse or noisy data and a known spectral operator (L), without prior knowledge of the equation's form.

Main Methods:

• Constructing a library of candidate operator terms for identifying operator A.
• Employing sparse optimization and other techniques to select key terms constituting operator A.
• Calculating Lax compatibility residuals through operator compositions and validating operator equations with designed test functions.

Main Results:

• Successfully discovered Lax pairs for advection, KdV, mKdV, Boussinesq-Burgers, and nonlinear Schrödinger equations.
• Demonstrated excellent effectiveness and robustness in handling data with varying sparsity and noise levels through numerical simulations.
• Validated the algorithm's capability to identify both novel integrable systems and new Lax pair representations for existing systems.

Conclusions:

• The Lax-Pair-FIND algorithm offers a powerful computational framework integrating deep learning with physical information for Lax pair discovery.
• This data-driven approach shows significant potential for advancing the field of discovering integrable systems and their associated Lax pairs.