Soliton profiles in 1D: Classical numerical schemes vs. neural network-based solvers

Chandler Haight, Svetlana Roudenko and Zhongming Wang

Chaos, Solitons & Fractals, 2026, vol. 210, issue P1

Abstract: We present a comparative study of classical numerical solvers, such as Petviashvili’s method and finite-difference with Newton iterations, and neural-network-based methods for computing the solutions of a nonlinear elliptic equation, namely, ground states or solitary-wave profiles to the dispersive PDEs that include the nonlinear Schrödinger, Klein–Gordon and the generalized KdV equations. To benchmark we consider the 1D setting and confirm that classical approaches retain high-order accuracy and strong computational efficiency for single-instance problems. 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 the accuracy of operator-learning methods remains lower than that of classical methods or PINNs, in general.

Keywords: Solitary wave; Ground state; Finite-difference; Petviashvili’s method; Physics-informed neural networks; Deep operator network; Fourier neural operator (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p1:s0960077926007873

DOI: 10.1016/j.chaos.2026.118646

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