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arxiv: 2604.09361 · v2 · submitted 2026-04-10 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

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

Zhangyong Liang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:53 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural networksGross-Pitaevskii equationshigh-dimensional PDEsstochastic samplingunbounded domainsrandom featuresstructure preservationphysics-informed networks
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The pith

A neural network with stochastic dimension selection and frozen random weights solves high-dimensional Gross-Pitaevskii equations at a cost independent of dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to solve high-dimensional Gross-Pitaevskii equations on unbounded domains without the exponential cost growth that plagues traditional discretizations such as Hermite bases. It achieves this by randomly sampling and freezing the hidden weights and biases of a neural network while selecting dimensions stochastically, so that total computational effort stays constant as dimension rises. A Gaussian-weighted ansatz enforces the required decay at infinity, a normalization projection layer maintains the physical mass, and an added energy constraint limits artificial dissipation during long-time evolution. Space-time separation with adaptive ODE solvers updates the coefficients while respecting causality. Tests show the resulting SD-FSNN trains faster and reaches higher accuracy than both gradient-based neural solvers and existing random-feature approaches across a range of dimensions and interaction parameters.

Core claim

The SD-FSNN approximates solutions to high-dimensional nonlinear Gross-Pitaevskii equations by freezing randomly sampled hidden-layer weights and biases, selecting dimensions stochastically, and embedding a Gaussian ansatz, normalization projection, and energy-conservation constraint; the construction is unbiased in dimension and reduces computational complexity from linear in dimension to dimension-independent while preserving mass and energy.

What carries the argument

The stochastic-dimension frozen sampled neural network (SD-FSNN), which freezes randomly sampled hidden weights and biases, selects dimensions stochastically, and augments the network with a Gaussian-weighted ansatz, normalization projection layer, and energy constraint.

If this is right

  • Computational cost remains constant rather than growing with spatial dimension.
  • Training requires far less time than iterative gradient-based optimization of all network parameters.
  • Mass is exactly preserved at every step through the normalization projection layer.
  • Energy dissipation is reduced over long integration intervals by the explicit conservation constraint.
  • The method applies uniformly across different interaction strengths without retuning the network architecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same frozen-sampling and stochastic-dimension strategy may transfer to other high-dimensional nonlinear Schrödinger-type equations on unbounded domains.
  • Because the method never optimizes the hidden parameters, it could serve as a fast surrogate for repeated solves in inverse problems or uncertainty quantification.
  • The structure-preserving layers suggest a template for embedding other conservation laws directly into random-feature models for physics.

Load-bearing premise

Random sampling of hidden weights, biases, and dimensions will produce a reliable approximation to the GPE solution operator without problem-specific tuning or unacceptable variance in high dimensions.

What would settle it

Numerical tests in which the SD-FSNN error or variance grows markedly with increasing dimension, or in which mass or energy drifts appreciably during long-time integration, when compared against known exact or reference solutions.

Figures

Figures reproduced from arXiv: 2604.09361 by Zhangyong Liang.

Figure 1
Figure 1. Figure 1: Core ideas of SD-FSNN. SD-FSNN avoids back-propagation through space￾time separation and random spatial sampling. Spatial basis functions with exponential decay are constructed and frozen. Differential operators are then evaluated on a small dimensional subset for efficiency. Finally, time-dependent coefficients are dynamically updated by solving an ODE system. 3.1. Neural network ansatz. To capture the ph… view at source ↗
Figure 2
Figure 2. Figure 2: Sampling strategy in SD-FSNN. (a) Data-agnostic sampling. (b) Data-driven [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: presents the space-time evolution of the real part of the wave function Re(ψ) for the 1D GPE with β = 10. The left panel shows the reference solution ob￾tained from a refined Hermite spectral method, the middle panel displays the SD-FSNN prediction, and the right panel depicts the pointwise absolute error. The SD-FSNN prediction is visually indistinguishable from the reference solution, accurately repro￾du… view at source ↗
Figure 4
Figure 4. Figure 4: Performance comparisons between SD-FSNN and baseline methods on the [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SD-FSNN results for the 2D GPE with β = 10 at t ∈ {0, 0.3, 0.6, 1}. The panels display the reference solution (left), prediction (middle), and absolute error (right) for |ψ| 2 on [−1, 1]2 . Example 4.3 (3D GPE). First, we assess the 3D GPE, a regime where tradi￾tional Hermite spectral methods face severe memory and computational bottlenecks due to the exponential scaling of basis functions. By employing th… view at source ↗
Figure 6
Figure 6. Figure 6: SD-FSNN results for the 3D GPE with β = 10 at t ∈ {0, 0.3, 0.6, 1}. The panels display the reference solution (left), prediction (middle), and absolute error (right) for |ψ| 2 on subsampled points in [−4, 4]3 . To further highlight the computational advantage of SD-FSNN, we compare its running time against the fourth-order time-splitting spectral (TSSP) method across dimensions d = 1, . . . , 8 for two tim… view at source ↗
Figure 7
Figure 7. Figure 7: Elapsed time comparison between TSSP (with various step sizes [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Long-time integration on t ∈ [0, 100] and x ∈ [−6, 6]. The panels display the reference solution, SD-FSNN prediction, and pointwise absolute error. 0 25 50 75 100 Time T 10 10 10 9 10 8 10 7 10 6 R elativ e 2 error [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relative L2 error of the solution predicted by SD-FSNN versus time T. 4.3. Combination coefficients across β. After space-time separation the wave function admits the discrete expansion (4.1) ψ(x, t) = nX−1 i=0 ci(t) ϕi(x), where {ϕi} n−1 i=0 are n fixed spatial features and {ci(t)} n−1 i=0 are the corresponding time￾dependent combination coefficients. For the 1D GPE, we use n = 80 basis functions. The ful… view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution of temporal basis coefficients under various [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ablation study on the exponential decay strategy for unbounded domains. [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Ablation study on the stochastic dimension sampling strategy. (a) Relative [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ablation study on the structure-preserving projections for mass normalization [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Analysis of the Kolmogorov n-width barrier. (a) Relative L2 error versus the number of basis functions M for a fixed dimension d = 10. (b) Relative L2 error versus the dimension d, with the number of features scaled as M = 150d. 6. Conclusions. We introduce SD-FSNN, an unbiased randomized estimation method for solving high-dimensional GPEs on unbounded domains. The computa￾tional cost of SD-FSNN is indepe… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a stochastic-dimension frozen sampled neural network (SD-FSNN) for solving high-dimensional Gross-Pitaevskii equations (GPEs) on unbounded domains. It claims SD-FSNN is unbiased across all dimensions with computational cost independent of dimension (avoiding exponential costs of Hermite bases), randomly samples hidden weights/biases to outperform iterative gradient-based optimization in training time and accuracy, uses space-time separation with adaptive ODE solvers for evolution coefficients and temporal causality, and incorporates a Gaussian-weighted ansatz, normalization projection layer, and energy conservation constraint to preserve mass and energy. Comparative experiments are said to demonstrate superior performance over existing methods, with complexity reduced from linear to dimension-independent relative to other random-feature approaches.

Significance. If the claims of dimension-independent cost/accuracy and controlled variance hold with supporting evidence, this could advance efficient numerical solution of high-dimensional nonlinear Schrödinger equations relevant to quantum many-body systems and Bose-Einstein condensates. The integration of random-feature sampling with structure-preserving elements (Gaussian decay, mass normalization, energy constraint) addresses the curse of dimensionality in a targeted way for unbounded-domain GPEs.

major comments (3)
  1. [SD-FSNN architecture and sampling description] The central claim that SD-FSNN is unbiased across dimensions with cost and accuracy independent of d requires that the Monte Carlo-style estimator from random hidden parameters and stochastic dimension selection has error/variance that remains controlled as d grows. For the cubic nonlinearity, random-feature approximations are unbiased only in expectation for linear operators; the nonlinear term couples coordinates, so variance typically scales with d unless neuron count or sampling distribution is explicitly scaled. No analysis or bounds are provided showing the number of samples can stay fixed while error stays O(1) in d.
  2. [Comparative experiments and results] The abstract states that comparative experiments demonstrate superior performance across dimensions and interaction parameters, but no quantitative error tables, convergence rates, or details on how the stochastic sampling variance is controlled appear in the results. This leaves the accuracy and dimension-independence claims resting on unshown empirical evidence.
  3. [Structure-preserving components] The Gaussian ansatz and normalization layer enforce decay and mass but do not bound sampling variance in the high-d feature space. No explicit scaling of the number of random features or sampled dimensions with d is given to support the dimension-independent cost claim.
minor comments (2)
  1. [Abstract] The abstract could more explicitly separate theoretical claims (unbiasedness, dimension independence) from empirical observations.
  2. [Method notation] Notation for the stochastic dimension selection process and the 'frozen' sampling of weights/biases could be clarified for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below, indicating planned revisions where appropriate to strengthen the presentation of the SD-FSNN approach.

read point-by-point responses

  1. Referee: The central claim that SD-FSNN is unbiased across dimensions with cost and accuracy independent of d requires that the Monte Carlo-style estimator from random hidden parameters and stochastic dimension selection has error/variance that remains controlled as d grows. For the cubic nonlinearity, random-feature approximations are unbiased only in expectation for linear operators; the nonlinear term couples coordinates, so variance typically scales with d unless neuron count or sampling distribution is explicitly scaled. No analysis or bounds are provided showing the number of samples can stay fixed while error stays O(1) in d.

    Authors: We acknowledge that the manuscript does not include a formal variance bound for the nonlinear term. The stochastic dimension sampling draws a fixed number of coordinates independently of d, and the frozen random features are drawn from a Gaussian distribution chosen to match the expectation of the integral operator. The Gaussian-weighted ansatz and normalization projection further localize the approximation. While unbiasedness holds in expectation, we agree that explicit control of variance for the cubic nonlinearity merits additional discussion. We will add a subsection on sampling variance with supporting empirical plots of error versus d at fixed sample size. revision: yes

  2. Referee: The abstract states that comparative experiments demonstrate superior performance across dimensions and interaction parameters, but no quantitative error tables, convergence rates, or details on how the stochastic sampling variance is controlled appear in the results. This leaves the accuracy and dimension-independence claims resting on unshown empirical evidence.

    Authors: The current results section relies primarily on figures; we agree that tabulated quantitative metrics would make the claims more transparent. We will expand the experiments section to include tables reporting L2 errors, relative energy drift, and standard deviations over repeated runs for dimensions d = 2 to d = 10 and several interaction strengths, together with a short paragraph quantifying observed variance stability. revision: yes

  3. Referee: The Gaussian ansatz and normalization layer enforce decay and mass but do not bound sampling variance in the high-d feature space. No explicit scaling of the number of random features or sampled dimensions with d is given to support the dimension-independent cost claim.

    Authors: The dimension independence follows from keeping both the number of random features and the number of stochastically sampled dimensions fixed (independent of ambient d), with the expectation taken over the random selection. The Gaussian ansatz aids localization but is not claimed to bound variance by itself. We will revise the architecture description to state the fixed sample sizes explicitly and add a brief remark explaining why no d-dependent scaling is required, supported by the new variance plots mentioned above. revision: partial

Circularity Check

0 steps flagged

No circularity: SD-FSNN architecture properties and performance claims are independent of fitted inputs or self-referential definitions

full rationale

The paper introduces SD-FSNN via stochastic dimension selection, frozen random sampling of hidden weights/biases, Gaussian ansatz, normalization projection, and energy constraint. These are architectural choices whose dimension-independence and unbiasedness are asserted as direct consequences of the sampling and projection design rather than derived from any fitted parameter or prior result that loops back to the target GPE solution. No equations reduce a claimed prediction to a fitted quantity by construction, no uniqueness theorem is imported from self-citations, and no ansatz is smuggled via prior work. Comparative experiments against random-feature methods and high-dimensional solvers provide external validation instead of internal renaming or self-definition. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that a randomly initialized and frozen neural network plus stochastic dimension sampling can represent the solution operator of the nonlinear GPE, plus standard assumptions from neural network approximation theory and ODE solvers.

free parameters (2)
  • number of sampled dimensions per step
    Chosen to balance bias and variance; value not stated in abstract.
  • number of random features / hidden units
    Determines network capacity; value not stated.
axioms (2)
  • domain assumption The solution of the GPE decays exponentially at infinity, justifying the Gaussian-weighted ansatz.
    Invoked to enforce boundary behavior on unbounded domains.
  • domain assumption Randomly sampled weights and biases provide a sufficiently rich function space for the evolution coefficients.
    Core justification for freezing the hidden layers instead of training.

pith-pipeline@v0.9.0 · 5539 in / 1551 out tokens · 46355 ms · 2026-05-10T16:53:47.419855+00:00 · methodology

discussion (0)

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