PE-RWP: A deep learning framework for polynomial feature extraction of rogue wave patterns and peregrine waves localization

Rogue wave (RW) patterns are nonlinear wave phenomena universally present in integrable systems. When one or more free internal parameters in the exact RW solutions of integrable systems like the nonlinear Schrödinger (NLS) equation are sufficiently large, the RW exhibit unique geometric patterns fully determined by the zero structure of the Yablonskii-Vorob'ev (Y-V) polynomial hierarchy. In this paper, we propose a deep learning-based “Polynomial Extractor for Rogue Wave Patterns” (PE-RWP). This approach replaces traditional asymptotic analysis of high-order RW solutions under large-parameter conditions, enabling automatic and precise identification of polynomial characteristics within RW patterns. We introduce a generalized polynomial hierarchy with parameters γ_k,j that encompasses all Y-V polynomials relevant to RW patterns, serving as the target objective for PE-RWP. The unique dual-branch network architecture (with a regression branch for determining parameter values and a classification branch for identifying corresponding polynomial types) enables PE-RWP to effectively output this generalized polynomial hierarchy and recognize RW patterns subjected to arbitrary scaling and rotational transformations. Furthermore, as an application based on the mathematical theory of RW patterns, we leverage the Y-V polynomials output by PE-RWP to achieve unsupervised localization of Peregrine waves through deep learning methods. Finally, through extensive experimental evaluation, both problems-polynomial extraction and Peregrine wave localization-are effectively solved with high accuracy.