标题： Lax 对的神经网络求解可积系统 显示英文标题

标题： Lax pairs informed neural networks solving integrable systems

摘要： Lax对是可积系统最重要的特征之一。 在本工作中，我们提出了针对具有Lax对的可积系统的Lax对信息神经网络（LPNNs），通过设计新颖的网络架构和损失函数，包括LPNN-v1和LPNN-v2。 LPNN-v1最显著的优势在于它可以将非线性可积系统的求解转化为Lax对谱问题的求解，并且不仅能高效求解数据驱动的局域波解，还能获得可积系统Lax对谱问题中的谱参数和相应的谱函数。 在LPNN-v1的基础上，我们还在LPNN-v2中额外引入了Lax对的相容性条件/零曲率方程，其主要优势是能够求解和探索具有Lax对的可积系统的高精度数据驱动局域波解及其相关的谱问题。 数值实验集中研究了具有Lax对谱问题的非常重要且具有代表性的可积系统的丰富局域波解，包括Korteweg-de Vries（KdV）方程和修正KdV方程的孤子解、非线性Schrödinger方程的 rogue 波解、sine-Gordon方程的扭结解、Camassa-Holm方程的非光滑尖峰解、短脉冲方程的脉冲解，以及Kadomtsev-Petviashvili方程的线孤子解和高维KdV方程的lump解。 本工作的创新之处在于首次将可积系统的Lax对信息整合到深度神经网络中，从而为研究数据驱动的局域波解和Lax对谱问题提供了一种新的方法和途径。 显示英文摘要

摘要： Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of nonlinear integrable systems into the solving of a linear Lax pairs spectral problems, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in Lax pairs spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs spectral problems, including the soliton solution of the Korteweg-de Vries (KdV) euqation and modified KdV equation, rogue wave solution of the nonlinear Schr\"odinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and Lax pairs spectral problems.