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arxiv: 2604.13830 · v1 · submitted 2026-04-15 · 🧮 math.NA · cs.LG· cs.NA

Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport

Haoning Dang , Fei Wang , Yifan Chen , Zhouyu Liu , Dong Liu , Hongchun Wu This is my paper

Pith reviewed 2026-05-10 12:32 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA
keywords randomized neural networksintegro-differential equationsneutron transport equationcollocation methodmesh-free methodsconvex least squaresnonlocal operatorsrandom features
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The pith

Randomized neural networks approximate solutions to linear integro-differential equations by solving a convex least-squares problem for the output weights.

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

The paper develops randomized neural networks for linear integro-differential equations that arise in transport and kinetic problems. Hidden layer parameters are chosen randomly and fixed, leaving only the output layer coefficients to be determined by a linear least-squares fit at chosen collocation points. This converts the training into a convex problem that avoids the difficulties of nonconvex optimization in standard physics-informed networks. When applied to the steady neutron transport equation, the resulting approximations achieve accuracy on par with other neural and deterministic solvers but with substantially reduced training time in the experiments shown. This matters because many such equations involve dense integral couplings that make traditional discretizations expensive, and the random feature approach handles them without losing efficiency.

Core claim

The authors show that a randomized neural network, constructed by randomly fixing the parameters of the hidden layers and determining the output weights via least squares, serves as an effective mesh-free collocation scheme for the steady neutron transport equation. The global support of these random basis functions accommodates the nonlocal scattering integrals without additional computational penalty, and numerical tests indicate competitive accuracy at lower training cost than the compared methods.

What carries the argument

The randomized neural network (RaNN), defined by randomly selected and fixed hidden-layer parameters with trainable output weights solved by linear least squares, which provides a dense approximation basis suitable for nonlocal operators.

If this is right

  • The integral operators in the equations can be incorporated directly into the least-squares system without creating dense matrices that dominate memory and time.
  • Training remains a convex problem, leading to stable and fast optimization independent of the nonlocality.
  • The number of trainable parameters stays small while the approximation retains global support across phase space.
  • Boundary conditions of various types can be enforced naturally within the collocation framework.
  • Performance advantages appear in the tested steady-state neutron transport settings compared to both neural and deterministic baselines.

Where Pith is reading between the lines

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

  • The same random-basis idea could be extended to other nonlocal problems in radiative transfer or kinetic theory by reusing the collocation setup.
  • Potential exists for combining RaNN with adaptive selection of random features to further improve efficiency in high dimensions.
  • Since training cost is low, the method might support inverse problems or uncertainty quantification in neutron transport more readily than slower alternatives.
  • Limitations may arise if the random features require very large numbers to achieve high accuracy in certain regimes.

Load-bearing premise

The randomly fixed hidden-layer parameters must form a basis that is rich enough to represent the solution when the output weights are chosen to minimize the residual in the least-squares sense.

What would settle it

Observing that the approximation error does not decrease as the number of random hidden units increases, or that the linear system becomes severely ill-conditioned for realistic neutron transport parameters, would indicate the random basis is insufficient.

Figures

Figures reproduced from arXiv: 2604.13830 by Dong Liu, Fei Wang, Haoning Dang, Hongchun Wu, Yifan Chen, Zhouyu Liu.

Figure 1
Figure 1. Figure 1: A schematic diagram of a randomized neural network (RaNN). [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scalar-flux profiles (left) and pointwise absolute differences from the MCX reference (right) for Example [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scalar-flux profiles (left) and pointwise absolute differences from the MCX reference (right) for Example [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustrations of the pin cells in Example [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scalar flux for Example 3 (Case 1): RaNN prediction (left), MCX reference (middle), and absolute error (right) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scalar flux for Example 3 (Case 2): RaNN prediction (left), MCX reference (middle), and absolute error (right) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scalar flux for Example 3 (Case 3): RaNN prediction (left), MCX reference (middle), and absolute error (right). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scalar flux for Example 4 (Case 1): RaNN prediction (left), MCX reference (middle), and absolute error (right) [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Scalar flux for Example 4 (Case 2): RaNN prediction (left), MCX reference (middle), and absolute error (right) [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scalar flux for Example 4 (Case 3): RaNN prediction (left), MCX reference (middle), and absolute error (right). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Absolute errors for FV as (Nφ, Nµ) = (16, 16) in Example 4, left to right: Case 1 - Case 3. In the reported test cases, the reflecting boundary problems lead to smaller relative errors than the corresponding vacuum cases. The sketching variant again provides a significant runtime reduction with only a modest degradation in accuracy, illustrating a practical trade-off between computational cost and accurac… view at source ↗
Figure 12
Figure 12. Figure 12: For each energy group (left to right: Groups 1-7), we show the RaNN scalar flux (top), the MCX [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
read the original abstract

Integro-differential equations arise in a wide range of applications, including transport, kinetic theory, radiative transfer, and multiphysics modeling, where nonlocal integral operators couple the solution across phase space. Such nonlocality often introduces dense coupling blocks in deterministic discretizations, leading to increased computational cost and memory usage, while physics-informed neural networks may suffer from expensive nonconvex training and sensitivity to hyperparameter choices. In this work, we present randomized neural networks (RaNNs) as a mesh-free collocation framework for linear integro-differential equations. Because the RaNN approximation is intrinsically dense through globally supported random features, the nonlocal integral operator does not introduce an additional loss of sparsity, while the approximate solution can still be represented with relatively few trainable degrees of freedom. By randomly fixing the hidden-layer parameters and solving only for the linear output weights, the training procedure reduces to a convex least-squares problem in the output coefficients, enabling stable and efficient optimization. As a representative application, we apply the proposed framework to the steady neutron transport equation, a high-dimensional linear integro-differential model featuring scattering integrals and diverse boundary conditions. Extensive numerical experiments demonstrate that, in the reported test settings, the RaNN approach achieves competitive accuracy while incurring substantially lower training cost than the selected neural and deterministic baselines, highlighting RaNNs as a robust and efficient alternative for the numerical simulation of nonlocal linear operators.

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

2 major / 1 minor

Summary. The paper proposes randomized neural networks (RaNNs) as a mesh-free collocation framework for linear integro-differential equations. Hidden-layer parameters are fixed randomly while only output weights are optimized via convex least squares; the approach is applied to the steady neutron transport equation and is claimed to deliver competitive accuracy at substantially lower training cost than selected neural and deterministic baselines.

Significance. If the empirical performance claims are shown to be robust, the method would provide a practical, stable alternative for nonlocal linear operators in high-dimensional phase space, with the convex least-squares training offering a clear computational advantage over nonconvex PINN optimization.

major comments (2)
  1. Numerical Experiments section: the reported competitive accuracy and lower training cost for the neutron transport test cases are based on single realizations of the random hidden weights; no statistics over multiple seeds, no sensitivity plots versus number of features or activation choice, and no error bars are provided, so it is unclear whether the results generalize or depend on favorable random draws.
  2. Method and Approximation sections: the claim that randomly fixed features yield a sufficiently rich basis for the nonlocal scattering integral rests on the numerical tests alone; no a priori error estimates, density arguments, or robustness analysis with respect to the random-feature distribution are given, which is load-bearing for the central assertion that RaNNs are a reliable general framework.
minor comments (1)
  1. Abstract: the phrase 'extensive numerical experiments' is used without naming the specific test problems, boundary conditions, or baseline implementations, making the performance claims difficult to contextualize.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have made revisions to improve the clarity and robustness of the presented results.

read point-by-point responses

  1. Referee: Numerical Experiments section: the reported competitive accuracy and lower training cost for the neutron transport test cases are based on single realizations of the random hidden weights; no statistics over multiple seeds, no sensitivity plots versus number of features or activation choice, and no error bars are provided, so it is unclear whether the results generalize or depend on favorable random draws.

    Authors: We agree that presenting results from single realizations limits the ability to assess variability. In the revised manuscript we now include statistics computed over 20 independent random seeds for the hidden-layer parameters in all neutron transport test cases. Tables report mean relative errors together with standard deviations, and the corresponding figures include error bars. We have also added sensitivity plots that vary the number of random features (from 50 to 500) and compare three activation functions (tanh, ReLU, and sigmoid), confirming that the reported accuracy advantage is consistent across these choices and does not rely on particularly favorable draws. revision: yes

  2. Referee: Method and Approximation sections: the claim that randomly fixed features yield a sufficiently rich basis for the nonlocal scattering integral rests on the numerical tests alone; no a priori error estimates, density arguments, or robustness analysis with respect to the random-feature distribution are given, which is load-bearing for the central assertion that RaNNs are a reliable general framework.

    Authors: We acknowledge that the manuscript does not derive new a priori error bounds tailored to the integro-differential setting. The Approximation section motivates the random-feature basis by referencing established density results for random features in reproducing-kernel Hilbert spaces and universal approximation theorems for random Fourier features. In the revision we have expanded this discussion with additional citations to theoretical work on random-feature approximations of integral operators and have included a short paragraph on robustness with respect to the random-feature distribution, drawing on concentration inequalities for random projections. A complete rigorous error analysis for the scattering term in the neutron-transport equation is left as future work; the current contribution focuses on the practical performance of the convex training procedure, which is supported by the extensive numerical evidence across multiple regimes. revision: partial

Circularity Check

0 steps flagged

RaNN framework for integro-differential equations shows no circularity

full rationale

The paper introduces randomized neural networks as a collocation scheme for linear integro-differential equations by randomly fixing hidden-layer parameters and solving only output weights via least squares. All performance claims, including competitive accuracy and lower training cost for the neutron transport equation, are supported exclusively by separate numerical experiments rather than any derivation, prediction, or uniqueness result that reduces to the method's own inputs or fitted quantities by construction. No self-citations, ansatzes, or renamings are load-bearing in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that random hidden-layer features suffice for collocation of nonlocal linear operators; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Randomly fixed hidden-layer parameters yield an effective approximation basis for solutions of linear integro-differential equations when only output weights are trained.
    This is the foundational premise of the RaNN collocation method as stated in the abstract.

pith-pipeline@v0.9.0 · 5565 in / 1365 out tokens · 45460 ms · 2026-05-10T12:32:44.075282+00:00 · methodology

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Reference graph

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