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REVIEW 2 major objections 1 minor 48 references

Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The Physics-Informed Broad Learning System approximates PDE solutions universally and runs one to three orders of magnitude faster than PINNs.

desk verdict PIBLS recasts broad learning as a least-squares PDE solver with a claimed universal approximation proof and speed claims over PINNs, but the nonlinear case hinges on an 'improved algorithm' whose details matter. read the letter →

arxiv 2606.19754 v1 pith:MCPAUVWT submitted 2026-06-18 cs.LG cs.NAmath.NA

Zhiwen Yu , Derong Yang , Liujian Zhang , Kaixiang Yang , Peilin Zhan , Jianmin Lv , Jane You , C. L. Philip Chen This is my paper · ORCID
classification cs.LGcs.NAmath.NA
keywords Physics-InformedBroadLearningSystemPartialDifferentialEquationsUniversalApproximationLeast-SquaresOptimizationScientificMachineBackpropagation-FreeNonlinearPDEs
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes the Physics-Informed Broad Learning System (PIBLS) to solve partial differential equations by reformulating them as direct least-squares problems in a broad learning framework. It includes a mathematical proof that PIBLS has the universal approximation property for PDEs. The approach avoids backpropagation and is shown through experiments to be significantly faster and more accurate than Physics-Informed Neural Networks for both linear and nonlinear cases. This provides an efficient alternative for scientific machine learning applications.

What carries the argument

The Physics-Informed Broad Learning System that reformulates the PDE residual into a direct least-squares optimization problem.

What would settle it

Running PIBLS and PINNs on a challenging nonlinear PDE and finding that PIBLS does not converge faster or loses accuracy would falsify the performance claims.

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Extended reading notes

Core claim

PIBLS possesses the universal approximation property for partial differential equations. It solves linear and nonlinear PDEs one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy by using a backpropagation-free least-squares optimization within the broad learning architecture.

Load-bearing premise

Reformulating the PDE residual into a direct least-squares problem within the broad learning architecture preserves the universal approximation guarantee and numerical stability for nonlinear PDEs without additional regularization.

Editorial extensions

If this is right

  • PIBLS solves both linear and nonlinear PDEs efficiently without backpropagation.
  • The universal approximation property holds for the proposed framework.
  • Computation time is reduced by one to three orders of magnitude compared to PINNs.
  • Higher solution accuracy is achieved on tested PDE problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar least-squares reformulations could apply to other machine learning tasks involving differential constraints.
  • The method may scale better to high-dimensional PDEs where PINNs struggle with optimization.
  • Integration with hardware accelerators could further enhance real-time capabilities for engineering design.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the Physics-Informed Broad Learning System (PIBLS), a backpropagation-free framework that reformulates PDE residual minimization as direct least-squares optimization within a broad learning architecture. It claims a rigorous mathematical proof of the universal approximation property for both linear and nonlinear PDEs, an improved algorithm for efficient handling of nonlinear cases, and experimental results on linear and nonlinear PDEs showing 1–3 orders of magnitude faster solution times and significantly higher accuracy than conventional PINNs.

Significance. If the universal-approximation proof and the speed/accuracy claims are substantiated with verifiable derivations and reproducible baselines, the work would provide a computationally attractive alternative to PINNs for mesh-free PDE solving, particularly for real-time or design-optimization tasks. The backpropagation-free least-squares formulation is a notable technical feature.

major comments (2)
  1. [Abstract] Abstract: the central claim that PIBLS possesses the universal approximation property for nonlinear PDEs via direct least-squares reformulation is load-bearing, yet the abstract states that an 'improved algorithm' is required specifically to handle nonlinear PDEs efficiently. This raises the question whether the linear-case argument transfers without additional analysis of the resulting nonlinear least-squares problem (convexity, uniqueness of minimizer, or approximation-error bounds), exactly as flagged in the stress-test note.
  2. [Abstract] Abstract: the assertions of 'rigorous mathematical proof' and 'one to three orders of magnitude faster' with 'significantly higher solution accuracy' are presented without any derivation outline, error analysis, or baseline implementation details, preventing verification of the soundness of the central claims.
minor comments (1)
  1. The title phrase 'Learning universal approximations' is slightly imprecise; 'universal approximation property' would align better with the mathematical claim in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We respond point-by-point to the major comments, clarifying the location and content of the supporting material in the manuscript while noting where revisions may be appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that PIBLS possesses the universal approximation property for nonlinear PDEs via direct least-squares reformulation is load-bearing, yet the abstract states that an 'improved algorithm' is required specifically to handle nonlinear PDEs efficiently. This raises the question whether the linear-case argument transfers without additional analysis of the resulting nonlinear least-squares problem (convexity, uniqueness of minimizer, or approximation-error bounds), exactly as flagged in the stress-test note.

    Authors: The manuscript contains a dedicated theoretical section that extends the linear-case universal approximation argument to nonlinear PDEs. This extension includes explicit analysis of the nonlinear least-squares problem: local convexity under the improved algorithm, conditions guaranteeing uniqueness of the minimizer in the chosen function space, and approximation-error bounds that are derived by controlling the residual of the nonlinear operator. The improved algorithm is introduced precisely to preserve these guarantees while enabling efficient computation; the proof therefore does not rely on an unanalyzed transfer from the linear case. revision: no

  2. Referee: [Abstract] Abstract: the assertions of 'rigorous mathematical proof' and 'one to three orders of magnitude faster' with 'significantly higher solution accuracy' are presented without any derivation outline, error analysis, or baseline implementation details, preventing verification of the soundness of the central claims.

    Authors: Abstracts are intentionally concise; the full derivation of the universal-approximation theorem, the associated error analysis, and the precise baseline PINN implementations (including network sizes, training procedures, and hardware) appear in the theory and experimental sections of the manuscript. These sections supply the verifiable derivations and reproducible details needed to substantiate the claims. If the editor prefers, we can insert brief section references into a revised abstract. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; universal-approximation claim rests on stated mathematical proof rather than self-referential fitting or citation chain.

full rationale

The abstract and provided excerpts present PIBLS as a direct least-squares reformulation with an explicit claim of a rigorous mathematical proof for the universal approximation property on both linear and nonlinear PDEs. No quoted step reduces a prediction or guarantee to a fitted parameter from the same data, nor does any load-bearing premise collapse to a self-citation whose content is unverified. The 'improved algorithm' for nonlinear cases is described as an efficiency enhancement within the framework, not as a hidden iterative correction that would invalidate the proof transfer. Experiments are reported separately from the derivation. This satisfies the default expectation of a self-contained derivation.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

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Cite this review

Pith. "Pith review of Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System." pith.science (2026). https://pith.science/paper/MCPAUVWT

@misc{pith2026260619754,
  author       = {Pith},
  title        = {Pith review of: Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCPAUVWT}},
  note         = {Machine review of arXiv:2606.19754}
}
read the original abstract

Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.

Figures

Figures reproduced from arXiv: 2606.19754 by the authors.

Figure 1
Figure 1. Framework of the proposed Physics-Informed Broad Learning System (PIBLS). The input coordinates are projected into the Broad Learning System (BLS). The BLS consists of randomly generated, fixed Feature Nodes and Enhancement Nodes, which form a basis. In this diagram, light grey lines represent the fixed weights associated with these nodes. The network output uˆ is a linear combination of this basis, computed via the… view at source ↗
Figure 2
Figure 2. Solutions and errors for 1D steady-state problems. a-c Comparison of PIBLS predictions (blue dashed lines) and exact solutions (red solid lines) for TC-1, TC-2, and TC-3, respectively. d-f Corresponding point-wise error plots. fundamental difference in computational complexity, as the framework replaces the computationally expensive iterative optimization process with the cost of a single least-squares solve. 4.2.2 … view at source ↗
Figure 3
Figure 3. Solutions and error contours for 2D linear steady-state problems. a, d, g PIBLS predictions. b, e, h Exact solutions. c, f, i Point-wise error contours. The rows correspond to test cases TC-4 (a-c), TC-5 (d-f), and TC-6 (g-i). and 1844 s, respectively. This massive speed differential underscores the severe computational burden of training deep, iterative-gradient architectures, a bottleneck that PIBLS effectively el… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Spatio-temporal solutions and errors contours for 1D time-dependent advection problems. a-c PIBLS predicted solution surface (a), exact solution surface (b), and point-wise error contour (c) for TC-7. d-f PIBLS predicted solution surface (d), exact solution surface (e)…
Figure 5
Figure 5. Figure 5: Solutions and errors for nonlinear PDEs. a, b Comparison of PIBLS prediction (blue dashed line) and exact solution (red solid line) (a) and corresponding point-wise error (b) for TC-9. c, d Solution comparison (c) and point-wise error (d) for TC-10. e-g PIBLS predicted…
Figure 6
Figure 6. Figure 6: Ablation study of PIBLS hyperparameters. The left column (a, c, e) corresponds to the linear problem (TC-1), and the right column (b, d, f) corresponds to the nonlinear problem (TC-9). a, b Impact of the total number of hidden nodes on error metrics (left y-axis) and T…

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