Projected Sobolev Natural Gradient Descent for Neural Variational Monte Carlo Solution of the Gross-Pitaevskii Equation

Abstract: This paper proposes a neural variational Monte Carlo method based on deep neural networks to solve the Gross-Pitaevskii equation (GPE) via projected Sobolev natural gradient descent (NGD). Adopting an "optimize-then-discretize" strategy, we first apply a constraint-preserving continuous Riemannian gradient flow on an infinite-dimensional Riemannian manifold, which is subsequently mapped to the neural network parameter space via Galerkin projection. This process naturally induces a Sobolev energy metric that incorporates physical information, effectively mitigating stiffness during optimization. To address the explicit dependence on the normalization constant caused by the nonlinear interaction term in the GPE, we design a hybrid sampling strategy combining an integration stream and a MCMC stream to achieve precise estimation of the generalized Gram matrix and energy gradients. Numerical experiments on benchmark cases, including the harmonic oscillator potential in the strong interaction limit and multi-scale optical lattice potentials, demonstrate the high accuracy of the proposed method. Furthermore, it achieves an order-of-magnitude acceleration in convergence compared to standard optimizers like Adam, exhibiting superior robustness in handling strong nonlinearities and complex geometric constraints.