Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
We develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles, which are solutions of $$ Q^{\prime\prime}=bQ-γQ^{p},\quad Q>0,\quad p>1, \quad b, γ>0, $$ 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) f...
| Authors | Chandler Haight, Alex D Rodriguez, Svetlana Roudenko |
|---|---|
| Primary category | math.NA |
| Published | 20 October 2026 |
| Last updated at source | 20 October 2026 |
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