Projected Sobolev Natural Gradient Descent for Efficient Neural Network Solution of the Gross--Pitaevskii Equation

Abstract

This paper introduces a projected Sobolev natural gradient descent (NGD) method for computing ground states of the Gross--Pitaevskii equation. By projecting a continuous Riemannian Sobolev gradient flow onto the normalized neural network tangent space, we derive a discrete NGD algorithm that preserves the normalization constraint. The numerical implementation employs variational Monte Carlo with a hybrid sampling strategy to accurately account for the normalization constant. To enhance computational efficiency, a matrix-free Nyström-preconditioned conjugate gradient solver is adopted to approximate the NGD operator without explicit matrix assembly. Numerical experiments demonstrate that the proposed method converges significantly faster than physics-informed neural network approaches and exhibits near-constant wall-clock time and memory over the tested high-dimensional range when the hidden architecture is fixed and the implicit Gram operator is used. Moreover, the resulting neural-network solutions provide high-quality initial guesses that substantially accelerate subsequent refinement by traditional high-precision solvers.