Soliton profiles: Classical Numerical Schemes vs. Neural Network - Based Solvers

TL;DR

This paper compares classical numerical methods and neural network-based approaches for computing solitary wave profiles in one-dimensional dispersive PDEs, highlighting their respective strengths and limitations in accuracy and efficiency.

Contribution

It provides a comprehensive comparison of classical and neural network methods, including PINNs and operator-learning techniques, for solitary wave profile computation in 1D dispersive PDEs.

Findings

Classical methods are highly accurate and efficient for single problems.

PINNs can reproduce qualitative solutions but are less accurate and slower.

Operator-learning methods are promising for repeated simulations after training.

Abstract

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 Schr\"odinger, 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.