mlsanityData to 14 Oct 2026

Long-time Integration of Nonlinear Wave Equations with Neural Operators

arXiv 2410.15617 ↗First posted 21 Oct 2024math.NA cs.LG cs.NAcited by 1

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Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 15 of its 36 references that are in the map. First posted on arXiv 21 Oct 2024.

Builds on

15 of 36 references.

References · 15

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  7. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
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3 more
  1. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 89
  2. 2022Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equationscited by 28
  3. 2023Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networkscited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamicscited by 0