AI and the World's Hard Problems · Physics · arXiv · 2026-10-21
Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
The authors develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles of the stationary equation underlying the nonlinear Schrödinger, generalized Korteweg-de Vries, and nonlinear Klein-Gordon equations, using explicit sech-type solutions as exact references for errors and hyperparameter selection. Beyond a standard Adam-trained PINN baseline, they add higher-order Gauss-Newton/Levenberg-Marquardt and L-BFGS refinements, Fourier-feature PINNs, and two new constrained Deep Ritz formulations implementing the classical Nehari constraint and the Weinstein quotient. The Deep Ritz methods perform best overall, with the Weinstein variant giving the best error-runtime balance, although the neural solvers remain less accurate and efficient than classical Petviashvili iteration in 1D, making the study a proof of concept for higher dimensions and more complicated nonlinearities.