Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

TL;DR

This paper introduces SD-FSNN, a neural network method for efficiently solving high-dimensional Gross-Pitaevskii equations on unbounded domains, outperforming existing methods in speed and accuracy.

Contribution

The paper presents a novel stochastic-dimension frozen sampled neural network that is unbiased, dimension-independent, and incorporates structure-preserving features for high-dimensional GPEs.

Findings

SD-FSNN outperforms iterative methods in training time and accuracy.

The method reduces complexity from linear to dimension-independent.

Experiments show superior performance across various dimensions and parameters.

Abstract

In this paper, we propose a stochastic-dimension frozen sampled neural network (SD-FSNN) for solving a class of high-dimensional Gross-Pitaevskii equations (GPEs) on unbounded domains. SD-FSNN is unbiased across all dimensions, and its computational cost is independent of the dimension, avoiding the exponential growth in computational and memory costs associated with Hermite-basis discretizations. Additionally, we randomly sample the hidden weights and biases of the neural network, significantly outperforming iterative, gradient-based optimization methods in terms of training time and accuracy. Furthermore, we employ a space-time separation strategy, using adaptive ordinary differential equation (ODE) solvers to update the evolution coefficients and incorporate temporal causality. To preserve the structure of the GPEs, we integrate a Gaussian-weighted ansatz into the neural network to enforce exponential decay at infinity, embed a normalization projection layer for mass normalization, and add an energy conservation constraint to mitigate long-time numerical dissipation. Comparative experiments with existing methods demonstrate the superior performance of SD-FSNN across a range of spatial dimensions and interaction parameters. Compared to existing random-feature methods, SD-FSNN reduces the complexity from linear to dimension-independent. Additionally, SD-FSNN achieves better accuracy and faster training compared to general high-dimensional solvers, while focusing specifically on high-dimensional GPEs on unbounded domains.