标题： 保持结构的物理信息神经网络用于Korteweg--de Vries（KdV）方程 显示英文标题

标题： Structure-Preserving Physics-Informed Neural Network for the Korteweg--de Vries (KdV) Equation

摘要： 物理信息神经网络（PINNs）为求解非线性偏微分方程（PDEs）提供了一个灵活的框架，但传统的实现方法在长期积分过程中往往无法保持关键的物理不变量。 本文介绍了一个\emph{保持结构的PINN}框架，用于非线性Korteweg--de Vries（KdV）方程，这是一个非线性和色散波传播的典型模型。 所提出的方法将质量守恒和哈密顿能量直接嵌入损失函数中，确保在整个训练和预测过程中保持物理一致性与能量稳定性。 与标准\texttt{双曲正切函数}基于的 PINNs~\cite{raissi2019pinn,wang2022modifiedpinn}不同，我们的方法采用正弦激活函数，增强了谱表达能力，并准确捕捉了KdV孤子的振荡和色散特性。 通过代表性案例研究——包括单孤子传播（形状保持的平移）、双孤子相互作用（弹性碰撞伴随相位变化）以及余弦脉冲初始化（非线性色散破裂）——该模型成功再现了KdV动力学的标志性行为，同时保持了守恒不变量。 消融研究表明，将不变量约束优化与正弦特征映射相结合可以加速收敛，提高长期稳定性，并在无需多阶段预训练的情况下减轻漂移。 这些结果表明，计算效率高、具备不变量意识的正则化结合正弦表示，能够为如KdV方程这样的哈密顿偏微分方程生成稳健且能量一致的PINNs。 显示英文摘要

摘要： Physics-Informed Neural Networks (PINNs) offer a flexible framework for solving nonlinear partial differential equations (PDEs), yet conventional implementations often fail to preserve key physical invariants during long-term integration. This paper introduces a \emph{structure-preserving PINN} framework for the nonlinear Korteweg--de Vries (KdV) equation, a prototypical model for nonlinear and dispersive wave propagation. The proposed method embeds the conservation of mass and Hamiltonian energy directly into the loss function, ensuring physically consistent and energy-stable evolution throughout training and prediction. Unlike standard \texttt{tanh}-based PINNs~\cite{raissi2019pinn,wang2022modifiedpinn}, our approach employs sinusoidal activation functions that enhance spectral expressiveness and accurately capture the oscillatory and dispersive nature of KdV solitons. Through representative case studies -- including single-soliton propagation (shape-preserving translation), two-soliton interaction (elastic collision with phase shift), and cosine-pulse initialization (nonlinear dispersive breakup) -- the model successfully reproduces hallmark behaviors of KdV dynamics while maintaining conserved invariants. Ablation studies demonstrate that combining invariant-constrained optimization with sinusoidal feature mappings accelerates convergence, improves long-term stability, and mitigates drift without multi-stage pretraining. These results highlight that computationally efficient, invariant-aware regularization coupled with sinusoidal representations yields robust, energy-consistent PINNs for Hamiltonian partial differential equations such as the KdV equation.