Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation

TL;DR

NeuralMD tackles the NKGE across the full regime by explicitly separating fast time oscillations into phase with a WKB-based multiscale decomposition, yielding a slowly varying amplitude z governed by an NLSW-like equation and a small amplitude remainder r. It integrates this decomposition with a two-stage PINN training protocol to learn z first and then corrects with r, using a gated gradient correlation mechanism and multiscale time perturbations to preserve temporal coherence and mitigate propagation failure. A interpretability layer via Kolmogorov–Arnold networks provides transparent edge-based activations that reveal the learned operators. Extensive 1D/2D/3D experiments demonstrate superior accuracy and robustness over a wide ε∈(0,1], surpassing many baselines and maintaining efficiency, with reconstruction experiments showing effective dropping and reconstituting of time oscillations. The work offers both practical neural solvers for highly oscillatory PDEs and theoretical insights into generalization and convergence under multiscale training dynamics.

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 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.

Figures (21)

• Figure 1: The prediction solution of PINNs for $\varepsilon=0.1$.
• Figure 1: Error convergence curves under different $\varepsilon$. (a) The NKGE converges to the NLSW uniformly in time. (b) The NKGE converges to the NLSE with linear growth in time.
• Figure 1: The prediction solution of NeuralMD and baselines for $\varepsilon=0.8,T=5.0$.
• Figure 2: PINNs for a simple temporally oscillatory problem.
• Figure 2: Overall architecture of NeuralMD.The single input point is perturbed in time to form point sets across multiscale regions. The model gradients are propagated backward across multiple regions.

Theorems & Definitions (18)

• Proof 1
• Theorem 3.2: Gradient correlation in time
• Proof 2
• Remark 3.3: Consistency with multiscale decomposition
• Lemma A.2: Convex case
• Proof 3
• Lemma A.3: Non-convex case
• Proof 4
• Theorem A.4: Gated multiscale time region optimization
• Lemma A.5: Smoothness of gated multiscale time loss