Long-time Integration of Nonlinear Wave Equations with Neural Operators
This work addresses the challenge of long-time integration for dynamic systems in computational physics, offering incremental improvements for domain-specific applications.
The paper tackled the problem of improving neural operators for long-time prediction of nonlinear wave equations by incorporating intrinsic features like conservation laws to reduce accumulated error, achieving enhanced numerical performance in experiments on KdV, sine-Gordon, and Klein-Gordon equations.
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.