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arxiv: 2604.23999 · v1 · submitted 2026-04-27 · 🧮 math.NA · cs.LG· cs.NA

Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework

You Yang , Fei Wang This is my paper

Pith reviewed 2026-05-08 02:24 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA
keywords distributionrandomizedadaptiveneuralproblemad-rannleast-squareslow-dimensional
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The pith

AD-RaNN learns an effective low-dimensional sampling distribution for hidden parameters in randomized neural networks by optimizing a vector p via PDE-driven or data-driven adaptation and a two-stage least-squares procedure, improving accuracy on benchmark PDE problems.

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

Randomized neural networks solve partial differential equations by drawing random values for the hidden layer weights and biases, then solving a simple linear least-squares problem for the output weights. This avoids the heavy training of ordinary neural networks but works well only when the random values are sampled from a good distribution, which is usually chosen by hand for each new equation. The paper replaces that manual choice with an automatic step: it represents the sampling distribution by a short vector of numbers called p and tunes only those numbers. It does this in two stages. First it solves a regularized version of the problem to find a stable p. Then it solves the original unregularized problem once more with the chosen p to recover the final solution. Two concrete ways to choose the objective for tuning p are given: one that looks directly at the PDE residual and one that uses available data. The same idea is also extended to time-stepping schemes and to learning operators. Experiments on standard test problems show that the automatically chosen distributions give smaller errors than the usual hand-picked ones.

Core claim

AD-RaNN provides an effective distribution-level adaptation mechanism, reduces reliance on hand-crafted hidden-feature distributions, and achieves strong empirical accuracy.

Load-bearing premise

That a low-dimensional parameterization of the sampling distribution is expressive enough to capture near-optimal distributions for a broad class of PDEs, and that the two-stage ridge-regularized optimization produces a p that genuinely improves the final unregularized solution without introducing new instabilities.

Figures

Figures reproduced from arXiv: 2604.23999 by Fei Wang, You Yang.

Figure 1
Figure 1. Figure 1: In the architecture of Randomized Neural Networks, black solid lines denote parameters that have been view at source ↗
Figure 2
Figure 2. Figure 2: Two-layer RaNN with local basis functions. view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of RaNN-DeepONet. 4.4.2 AD-RaNN-DeepONet The performance of RaNN-DeepONet depends strongly on the random distributions used to generate the branch and trunk hidden features. This is precisely the same structural issue encountered earlier for randomized neural PDE solvers: once the hidden representation is randomized and frozen, the quality of the approximation is largely determined by how the … view at source ↗
Figure 4
Figure 4. Figure 4: Reference solution and numerical solutions for the sharp-layer problem view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of p = (rx, ry) for PDAD-DT (top) and DDAD-DT (bottom) with Nt = 100, 200, 400, 800 view at source ↗
Figure 6
Figure 6. Figure 6: Numerical solutions of (6.13) obtained by the PDAD method with (a1, a2) = (6, 6) (left) and (a1, a2) = (1, 20) (right). Here m1 = 3000 and mλ = 1000. 6.3 Burgers’ equation Burgers’ equation is an important partial differential equation that captures the interaction between nonlinear advection and viscous diffusion, and serves as a simplified model in many fluid-mechanical applications. To assess the perfor… view at source ↗
Figure 7
Figure 7. Figure 7: Results for the 1D Burgers’ equation (6.15) computed using PDAD-DT and DDAD-DT with Nt = 1000. From left to right, the columns show: the PDAD-DT solution, the optimized parameters p of PDAD-DT, the DDAD-DT solution, and the optimized parameters p of DDAD-DT. Case 2: Two-dimensional Burgers’ equation. We now extend Burgers’ equation to two dimensions and consider the problem    ut + u (ux + uy) − ε∆u… view at source ↗
Figure 8
Figure 8. Figure 8: Results for the 2D Burgers’ equation (6.16) with ε = 0.1 and Nt = 400 at t = 1. From left to right: the PDAD-DT solution, the optimized parameters p = (rx, ry) for PDAD-DT, the DDAD-DT solution, and the optimized parameters p = (rx, ry) for DDAD-DT. To further assess the performance of the discrete-time AD-RaNN, we reduce ε to 0.01, which produces a much sharper shock layer and significantly increases the … view at source ↗
Figure 9
Figure 9. Figure 9: Results for 2D Burgers’ equation (6.16) obtained by PDAD-DT (top) and DDAD-DT (bottom) with ε = 0.01 and Nt = 400. From left to right, the columns show: the numerical solution at t = 1, the residual points selected for the second layer at t = 1, the optimized first-layer parameters p, and the second-layer parameter r2. 6.4 Allen-Cahn equation Case 1: One-dimensional Allen-Cahn equation The Allen-Cahn equat… view at source ↗
Figure 10
Figure 10. Figure 10: Results for the Allen-Cahn equation (6.18) computed using PDAD-DT and DDAD-DT with Nt = 1000. From left to right, the columns show: the PDAD-DT solution, the optimized parameters p of PDAD-DT, the DDAD-DT solution, and the optimized parameters p of DDAD-DT. Case 2: Two-dimensional Allen-Cahn equation We now extend the Allen-Cahn equation to two dimensions and consider the problem with Dirichlet boundary c… view at source ↗
Figure 11
Figure 11. Figure 11: Results for the Allen-Cahn equation (6.19) obtained by PDAD-DT (top) and DDAD-DT (bottom) with Nt = 400 at t = 1. From left to right, the columns show: the numerical solution, the residual points selected for the second layer, the optimized first-layer parameters p, and the second-layer parameter r2. formulation of mean curvature flow: ∂tx = ∂ρρx |∂ρx| 2 , (6.20) where ρ ∈ (0, 1). A first-order discrete-t… view at source ↗
Figure 12
Figure 12. Figure 12: Schematic of the neural network architecture with a hard-constrained periodic boundary layer. view at source ↗
Figure 13
Figure 13. Figure 13: Comparison between the reference solution (blue dashed) and the DDAD-DT solution (red solid) for the view at source ↗
Figure 14
Figure 14. Figure 14: DDAD-DT solutions for the initial condition view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of exact solutions (left), predictions (middle), and absolute errors (right) for equation view at source ↗
read the original abstract

Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance, however, depends strongly on the sampling distribution of the hidden-layer parameters, which is usually chosen heuristically and problem by problem. This distribution sensitivity is a central bottleneck in randomized neural PDE solvers. In this work, we propose Adaptive-Distribution Randomized Neural Networks (AD-RaNN), a framework that promotes randomized feature generation from a fixed heuristic choice to a low-dimensional adaptive optimization problem. Instead of training all hidden weights and biases, AD-RaNN parameterizes the hidden-feature sampling distribution by a low-dimensional vector p and optimizes only p, thereby preserving the least-squares structure of RaNNs while reducing manual distribution tuning. The method uses a two-stage strategy: ridge-regularized reduced training for stable distribution-parameter optimization, followed by an unregularized least-squares refit for final solution recovery. We develop two adaptive mechanisms, PDE-Driven Adaptive Distribution (PDAD) and Data-Driven Adaptive Distribution (DDAD), and deploy them in space-time solvers, discrete-time solvers, and operator-learning models. We also incorporate an adaptive layer-growth enhancement for localized structures. For the reduced optimization problem, we establish well-posedness of the reduced objectives, consistency of ridge-regularized minimizers, an efficient gradient formula, and a practical lower-bound estimate for the ridge parameter. Numerical experiments on benchmark problems show that AD-RaNN provides an effective distribution-level adaptation mechanism, reduces reliance on hand-crafted hidden-feature distributions, and achieves strong empirical accuracy.

Editorial analysis

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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.

Circularity Check

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full rationale

The paper presents AD-RaNN as an algorithmic framework that optimizes a low-dimensional distribution parameter p via a two-stage ridge-regularized procedure before an unregularized refit. This optimization is explicitly part of the proposed method rather than a claimed first-principles derivation whose output reduces to its inputs by construction. Well-posedness, consistency, and gradient formulas are derived for the reduced problem itself; numerical results are presented as separate empirical evidence. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the provided text, and the central performance claims rest on the method's design and experiments rather than tautological renaming or fitting.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on the assumption that the sampling distribution can be usefully parameterized by a low-dimensional vector p whose optimization yields better features than fixed heuristics, plus standard well-posedness results for the reduced ridge-regularized least-squares problems.

free parameters (1)
  • distribution parameter vector p
    Low-dimensional vector that defines the adaptive sampling distribution and is optimized in the first stage.
axioms (1)
  • domain assumption Well-posedness of the reduced ridge-regularized objectives
    Invoked to guarantee existence and consistency of the minimizer for p.
invented entities (1)
  • AD-RaNN framework with PDAD and DDAD mechanisms no independent evidence
    purpose: Adaptive low-dimensional distribution learning for randomized neural PDE solvers
    New method introduced to replace heuristic distribution choice.

pith-pipeline@v0.9.0 · 5601 in / 1555 out tokens · 38076 ms · 2026-05-08T02:24:26.897521+00:00 · methodology

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