对比经典算法与神经网络求解孤波解，发现经典方法在单次计算中更准更快。
Soliton profiles: Classical Numerical Schemes vs. Neural Network - Based Solvers
- 用经典数值法和神经网络分别求解一维非线性偏微分方程的孤波解
- 经典方法精度高、效率好；神经网络训练慢且收敛难，低维下不如经典方法
- 算子学习法可复用，适合重复模拟或实时预测，但单次计算精度仍较低
我们比较了经典数值求解器（如Petviashvili方法或带牛顿迭代的有限差分法）与基于神经网络的方法，在求解包含非线性薛定谔方程、非线性克莱因-戈登方程及广义KdV方程的一维色散型偏微分方程的基态或孤波解方面的表现。结果表明，经典方法在单实例一维问题中保持高阶精度和强计算效率。物理信息神经网络（PINNs）虽能再现定性解，但在低维场景下通常精度和效率低于经典方法，主要受限于昂贵的训练过程和缓慢的收敛速度。我们还研究了算子学习方法，尽管训练阶段计算开销大，但预训练后可跨多个参数实例快速推理，适用于重复仿真或实时预测场景。然而，对于单实例计算，算子学习方法的精度总体仍低于经典方法或PINNs。
原文摘要 · Abstract (English)
We present a comparative study of classical numerical solvers, such as Petviashvili's method or finite difference with Newton iterations, and neural network-based methods for computing ground states or profiles of solitary-wave solutions to the one-dimensional dispersive PDEs that include the nonlinear Schrödinger, the nonlinear Klein-Gordon and the generalized KdV equations. We confirm that classical approaches retain high-order accuracy and strong computational efficiency for single-instance problems in the one-dimensional setting. Physics-informed neural networks (PINNs) are also able to reproduce qualitative solutions but are generally less accurate and less efficient in low dimensions than classical solvers due to expensive training and slow convergence. We also investigate the operator-learning methods, which, although computationally intensive during training, can be reused across many parameter instances, providing rapid inference after pretraining, making them attractive for applications involving repeated simulations or real-time predictions. For single-instance computations, however, the accuracy of operator-learning methods remains lower than that of classical methods or PINNs, in general.
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