This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain. The proposed method circumvents the curse-of-dimensionality that plagues traditional discretizations and the computational bottlenecks of gradient-based neural network solvers through a synergistic combination of techniques. First, a prescribed Gaussian envelope encodes the far-field decay of the wavefunction, enabling a space-time separation where the spatial approximation is handled by a frozen, single-hidden-layer neural network with data-driven sampled features. This yields a gradient-free formalism where spatial derivatives are analytically precomputed and time-dependence is evolved via reduced ODEs. Second, a stochastic-dimension sampler provides a conditionally unbiased estimate of the spatial operator by evaluating only a small subset of spatial dimensions at each time step, essentially reducing computational and memory costs. Discrete conservation laws are also enforced, ensuring long-term stability. Extensive numerical experiments on GPE in up to 1000 dimensions demonstrate that SD-FSNN achieves significantly higher accuracy and efficiency compared to state-of-the-art methods, including PINNs, randomized feature methods, and tensor-network approaches. The results confirm that SD-FSNN effectively mitigates the Kolmogorov $n$-width barrier for frozen-basis models on structured solution manifolds.

翻译：本文提出随机维度冻结采样神经网络（SD-FSNN），这是一种用于求解无界域上高维Gross-Pitaevskii方程（GPE）的新型计算框架。所提方法通过多项技术的协同组合，规避了传统离散化方法中普遍存在的维数灾难以及基于梯度的神经网络求解器中的计算瓶颈。首先，预设的高斯包络函数编码了波函数的远场衰减特性，实现了时空分离，其中空间近似由具有数据驱动采样特征的冻结单隐层神经网络处理。这产生了一种无梯度形式，空间导数被解析预计算，时间依赖性通过简化常微分方程演化。其次，随机维度采样器通过在每个时间步仅评估空间维度的一小部分子集，提供空间算子的条件无偏估计，从而大幅降低计算和内存成本。同时还施加了离散守恒律，确保了长期稳定性。在高达1000维的GPE上的大量数值实验表明，与PINN、随机特征方法和张量网络方法等现有技术相比，SD-FSNN实现了显著更高的精度和效率。结果证实，SD-FSNN有效缓解了冻结基模型在结构化解流形上的Kolmogorov $n$宽度障碍。