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

• URL: http://arxiv.org/abs/2604.09361v1
• Date: Fri, 10 Apr 2026 14:31:36 GMT
• Title: Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains
• Abstract summary: We propose a frozen sampled neural network (SD-FSNN) for solving a class of high-dimensional Gross-Pitaevskii equations (GPEs) on unbounded domains.<n> SD-FSNN is unbiased across all dimensions, and its computational cost is independent of the dimension.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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