Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods

Chandler Haight, Alex D Rodriguez, Svetlana Roudenko

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) 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…