Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation

TL;DR

This paper introduces NeuralMD, a neural multiscale decomposition method that effectively solves the nonlinear Klein-Gordon equation across different regimes by mitigating spectral bias and propagation failure through multiscale time integration and gradient correction.

Contribution

The paper presents a novel neural multiscale decomposition approach combining multiscale time integrator and gradient correction to improve solving the NKGE with high-frequency oscillations.

Findings

Outperforms existing collocation methods in accuracy and stability.

Effectively handles oscillations across all regimes of the NKGE.

Achieves convergence rate of O(ε^2) as ε approaches zero.

Abstract

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 Schr\"odinger 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.