Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
Abstract
We develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles, which are solutions of Q′′=bQ−γQp, Q>0, p>1, b, γ>0, the stationary profile equation for the nonlinear Schrödinger, generalized Korteweg-de Vries, and nonlinear Klein-Gordon equations. Its explicit sech-type solutions serve as exact reference for error computation and benchmark hyperparameter selection. We improve on the standard Adam-trained physics-informed neural network (PINN) baseline by introducing higher-order optimization via Gauss-Newton/Levenberg-Marquardt and L-BFGS refinements, as well as Fourier-feature PINNs, and constrained Deep Ritz (DR) formulations. In DR approaches we introduce two new neural-network formulations, which implement two classical variational characterizations: a Nehari-constrained Deep Ritz method, in which the height is learned by projecting onto the Nehari/Pokhozhaev manifold, and a Weinstein Deep Ritz method, in which a normalized profile minimizes the Weinstein quotient. The Deep Ritz formulations perform best overall, in particular, the Weinstein variant gives the best error-runtime balance. The neural-network based solvers remain less accurate and efficient than classical Petviashvili iteration in 1D, however, the current study provides a proof of concept: the mesh-free formulation, non-reliance on exact profiles or the related features make them promising in higher dimensions and for more complicated nonlinearities and dispersive operators.
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BibTeX
@misc{haight2026soliton,
title = {{Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods}},
author = {Chandler Haight and Alex D Rodriguez and Svetlana Roudenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27500},
eprint = {2610.27500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27500},
}
RIS
TY - UNPB
TI - Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
AU - Chandler Haight
AU - Alex D Rodriguez
AU - Svetlana Roudenko
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27500
UR - https://arxiv.org/abs/2610.27500
AB - We develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles, which are solutions of Q^′′=bQ−γQ^p, Q>0, p>1, b, γ>0, the stationary profile equation for the nonlinear Schrödinger, generalized Korteweg-de Vries, and nonlinear Klein-Gordon equations. Its explicit sech-type solutions serve as exact reference for error computation and benchmark hyperparameter selection. We improve on the standard Adam-trained physics-informed neural network (PINN) baseline by introducing higher-order optimization via Gauss-Newton/Levenberg-Marquardt and L-BFGS refinements, as well as Fourier-feature PINNs, and constrained Deep Ritz (DR) formulations. In DR approaches we introduce two new neural-network formulations, which implement two classical variational characterizations: a Nehari-constrained Deep Ritz method, in which the height is learned by projecting onto the Nehari/Pokhozhaev manifold, and a Weinstein Deep Ritz method, in which a normalized profile minimizes the Weinstein quotient. The Deep Ritz formulations perform best overall, in particular, the Weinstein variant gives the best error-runtime balance. The neural-network based solvers remain less accurate and efficient than classical Petviashvili iteration in 1D, however, the current study provides a proof of concept: the mesh-free formulation, non-reliance on exact profiles or the related features make them promising in higher dimensions and for more complicated nonlinearities and dispersive operators.
ER -