用神经算子实现非线性波方程的长时间预测，误差更小。
Long-time Integration of Nonlinear Wave Equations with Neural Operators
- 通过循环替换初值，提升长期积分稳定性
- 在KdV、正弦-戈登等方程上显著降低累积误差
- 适合需要长时间模拟的物理系统建模
神经算子在求解偏微分方程方面展现潜力，训练后比传统数值求解器快得多。然而，在求解时变偏微分方程，尤其是动态系统的长时间预测方面，性能仍有待提升。本文针对非线性波方程的长时间积分问题，提出一种方法：以预测结果作为下一次迭代的初始条件，进行循环更新。在仅有有限时间轨迹数据的情况下，利用非线性波方程的内在特性，如守恒律和适定性，优化算法设计，减少累积误差。数值实验在不规则域上对Korteweg-de Vries（KdV）方程、正弦-戈登（sine-Gordon）方程及克莱因-戈登（Klein-Gordon）波方程进行了验证，结果表明该方法在长时间预测中具有更优表现。
原文摘要 · Abstract (English)
Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.
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