Long-time Integration of Nonlinear Wave Equations with Neural Operators

• URL: http://arxiv.org/abs/2410.15617v1
• Date: Mon, 21 Oct 2024 03:36:34 GMT
• Title: Long-time Integration of Nonlinear Wave Equations with Neural Operators
• Authors: Guanhang Lei, Zhen Lei, Lei Shi,
• Abstract summary: We focus on solving the long-time integration of nonlinear wave equations via neural operators. We utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and a semilinear wave equation on the irregular domain.
• Score: 13.357441268268758
• License:
• Abstract: 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 a semilinear wave equation on the irregular domain.

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