标题： 非线性波动方程的摄动分析与神经网络初值估计 显示英文标题

标题： Perturbation Analysis and Neural Network-Based Initial Condition Estimation for the Sine-Gordon Equation

摘要： sine-Gordon 方程是一种基本的非线性偏微分方程，描述了多种物理系统中孤立子动力学和相演化，包括约瑟夫森结和超导电路。 在这项研究中，我们分析了外部扰动（如阻尼力和驱动力）对孤立子解稳定性的影响。 利用严格的 Hilbert 空间框架，我们建立了扰动方程的适定性，并推导出正则性结果。 特别是，我们提供了扰动函数有界的充分条件，该条件在确定孤立子结构的持久性方面起着关键作用。 此外，我们提出了一种基于神经网络的方法来解决估计未知初始条件的反问题。 通过在模拟的偏微分方程解上训练数据驱动模型，我们证明网络可以从有限或噪声观测中准确恢复初始状态。 数值模拟验证了理论结果，并突显了结合数学分析与机器学习技术研究非线性波现象的潜力。 这种方法为孤立子行为提供了有价值的见解，并且在设计基于约瑟夫森结的量子计算系统方面具有潜在应用。 显示英文摘要

摘要： The sine-Gordon equation is a fundamental nonlinear partial differential equation that governs soliton dynamics and phase evolution in a variety of physical systems, including Josephson junctions and superconducting circuits. In this study, we analyze the effects of external perturbations such as damping and driving forces on the stability of soliton solutions. Using a rigorous Hilbert space framework, we establish well-posedness and derive regularity results for the perturbed equation. In particular, we provide sufficient conditions for the boundedness of the perturbation function, which plays a crucial role in determining the persistence of the soliton structure. Furthermore, we propose a neural network-based approach for solving the inverse problem of estimating unknown initial conditions. By training a data-driven model on simulated PDE solutions, we demonstrate that the network can accurately recover the initial states from limited or noisy observations. Numerical simulations validate the theoretical results and highlight the potential of combining mathematical analysis with machine learning techniques to study nonlinear wave phenomena. This approach offers valuable insights into soliton behavior and has potential applications in the design of quantum computing systems based on Josephson junctions.