Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915858288214016 |
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| author | Liang, Zhangyong Mao, Zhiping Zhao, Xiaofei |
| author_facet | Liang, Zhangyong Mao, Zhiping Zhao, Xiaofei |
| contents | In this paper, we propose a neural multiscale decomposition method (NeuralMD) for solving the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in(0,1]$ from the relativistic regime to the nonrelativistic limit regime. The solution of the NKGE propagates waves with wavelength at $O(1)$ and $O(\varepsilon^2)$ in space and time, respectively, which brings the oscillation in time. Existing collocation-based methods for solving this equation lead to spectral bias and propagation failure. To mitigate the spectral bias induced by high-frequency time oscillation, we employ a multiscale time integrator (MTI) to absorb the time oscillation into the phase. This decomposes the NKGE into a nonlinear Schrödinger equation with wave operator (NLSW) with well-prepared initial data and a remainder equation with small initial data. As $\varepsilon \to 0$, the NKGE converges to the NLSW at rate $O(\varepsilon^{2})$, and the contribution of the remainder equation becomes negligible. Furthermore, to alleviate propagation failure caused by medium-frequency time oscillation, we propose a gated gradient correlation correction strategy to enforce temporal coherence in collocation-based methods. As a result, the approximation of the remainder term is no longer affected by propagation failure. Comparative experiments with existing collocation-based methods demonstrate the superior performance of our method for solving the NKGE with various regularities of initial data over the whole regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation Liang, Zhangyong Mao, Zhiping Zhao, Xiaofei Numerical Analysis In this paper, we propose a neural multiscale decomposition method (NeuralMD) for solving the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in(0,1]$ from the relativistic regime to the nonrelativistic limit regime. The solution of the NKGE propagates waves with wavelength at $O(1)$ and $O(\varepsilon^2)$ in space and time, respectively, which brings the oscillation in time. Existing collocation-based methods for solving this equation lead to spectral bias and propagation failure. To mitigate the spectral bias induced by high-frequency time oscillation, we employ a multiscale time integrator (MTI) to absorb the time oscillation into the phase. This decomposes the NKGE into a nonlinear Schrödinger equation with wave operator (NLSW) with well-prepared initial data and a remainder equation with small initial data. As $\varepsilon \to 0$, the NKGE converges to the NLSW at rate $O(\varepsilon^{2})$, and the contribution of the remainder equation becomes negligible. Furthermore, to alleviate propagation failure caused by medium-frequency time oscillation, we propose a gated gradient correlation correction strategy to enforce temporal coherence in collocation-based methods. As a result, the approximation of the remainder term is no longer affected by propagation failure. Comparative experiments with existing collocation-based methods demonstrate the superior performance of our method for solving the NKGE with various regularities of initial data over the whole regime. |
| title | Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.00266 |