Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains
2026-04-10 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce a new neural network method called SD-FSNN to solve high-dimensional Gross-Pitaevskii equations, which are mathematical models used in quantum physics. Their approach avoids the usual increase in computing resources with more dimensions and trains faster by randomly fixing parts of the network rather than using slow optimization. They also add special features to respect the physics of the problem, like ensuring the solution behaves well at infinity and conserves important quantities over time. Tests show their method works better and faster than existing techniques, especially for problems with many variables.
Gross-Pitaevskii equationhigh-dimensional PDEsneural networksrandom feature methodsstochastic dimensionGaussian ansatzspace-time separationODE solversmass normalizationenergy conservation
Authors
Zhangyong Liang
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.