Enhanced neural networks solve one-dimensional soliton profile equations
Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
Machine Learning
Summary
This work looks at solving a specific math problem that appears in physics involving wave shapes called solitons. The authors improve neural-network methods to find accurate solutions for these wave shapes in one dimension. They compare different training and optimization techniques as well as new ways to impose constraints on the solutions. Their best approach balances accuracy with runtime better than standard neural-network methods. While classic iterative methods still work better in this simple case, these neural methods show promise for more complex problems where traditional methods struggle.
What this means in practice
- •For computational physics teams: Model nonlinear wave phenomena using mesh-free neural solvers that handle complex equations beyond simple one-dimensional cases.
- •For machine learning engineers in scientific computing: Implement enhanced training and constraint techniques for neural PDE solvers to improve solution quality without relying on exact analytic forms.
Authors
Chandler Haight, Alex D Rodriguez, Svetlana Roudenko
Abstract
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 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.