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.

翻译：神经算子在求解多种类型的偏微分方程（PDE）方面已展现出潜力。一旦利用一定量的观测数据进行训练后，其求解速度相较于传统数值求解器显著加快。然而，在求解时间依赖的偏微分方程，特别是动态系统的长时间预测方面，其数值性能仍有待提升。本文聚焦于通过神经算子求解非线性波动方程的长时间积分问题，其方法是以循环方式用预测值替代初始条件。在给定有限观测时间轨迹数据的情况下，我们利用这些非线性波动方程的一些固有特征，如守恒律和适定性，来改进算法设计并减少累积误差。我们的数值实验在非规则区域上的Korteweg-de Vries (KdV) 方程、sine-Gordon 方程以及 Klein-Gordon 波动方程中验证了这些改进。