Title: Enhanced PINNs for data-driven solitons and parameter discovery for (2+ 1)-dimensional coupled nonlinear Schrödinger systems Show Chinese title

Title: 增强的PINNs用于数据驱动的孤立波和(2+1)维耦合非线性薛定谔系统的参数发现

Abstract: This paper investigates data-driven solutions and parameter discovery to (2+1)-dimensional coupled nonlinear Schr\"odinger equations with variable coefficients (VC-CNLSEs), which describe transverse effects in optical fiber systems under perturbed dispersion and nonlinearity. By setting different forms of perturbation coefficients, we aim to recover the dark and anti-dark one- and two-soliton structures by employing an enhanced physics-based deep neural network algorithm, namely a physics-informed neural network (PINN). The enhanced PINN algorithm leverages the locally adaptive activation function mechanism to improve convergence speed and accuracy. In the lack of data acquisition, the PINN algorithms will enhance the capability of the neural networks by incorporating physical information into the training phase. We demonstrate that applying PINN algorithms to (2+1)-dimensional VC-CNLSEs requires distinct distributions of physical information. To address this, we propose a region-specific weighted loss function with the help of residual-based adaptive refinement strategy. In the meantime, we perform data-driven parameter discovery for the model equation, classified into two categories: constant coefficient discovery and variable coefficient discovery. For the former, we aim to predict the cross-phase modulation constant coefficient under varying noise intensities using enhanced PINN with a single neural network. For the latter, we employ a dual-network strategy to predict the dynamic behavior of the dispersion and nonlinearity perturbation functions. Our study demonstrates that the proposed framework holds significant potential for studying high-dimensional and complex solitonic dynamics in optical fiber systems. Show Chinese abstract

Abstract: 本文研究了具有可变系数的(2+1)维耦合非线性薛定谔方程的基于数据的解决方案和参数发现，这些方程描述了在扰动色散和非线性下光纤系统中的横向效应。通过设置不同的扰动系数形式，我们旨在利用增强的基于物理的深度神经网络算法，即物理信息神经网络（PINN），恢复暗孤子和反暗孤子的一阶和二阶结构。增强的PINN算法利用局部自适应激活函数机制来提高收敛速度和准确性。在缺乏数据采集的情况下，PINN算法将在训练阶段结合物理信息以增强神经网络的能力。我们证明将PINN算法应用于(2+1)维VC-CNLSEs需要物理信息的不同分布。为了解决这个问题，我们在基于残差的自适应细化策略的帮助下提出了一种区域特定的加权损失函数。同时，我们对模型方程进行了数据驱动的参数发现，分为两类：常数系数发现和可变系数发现。对于前者，我们旨在使用单个神经网络的增强PINN预测在不同噪声强度下的交叉相位调制常数系数。对于后者，我们采用双网络策略来预测色散和非线性扰动函数的动力学行为。我们的研究表明，所提出的框架在研究光纤系统中的高维和复杂孤子动力学方面具有重要的潜力。