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REVIEW 4 major objections 5 minor 39 references

Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A fixed Shannon wavelet basis plus a single least-squares solve replaces gradient-trained PINNs, yielding higher accuracy and speed on the PDEs the paper tests.

desk verdict A straightforward wavelet least-squares method that works, but the paper overstates its PINN comparison and spectral-bias story. read the letter →

arxiv 2508.07546 v1 pith:WZWIQY3W submitted 2025-08-11 math.NA cs.NA

Feng Han , Jianguo Wang , Guoliang Peng , Xueting Shi This is my paper · ORCID
classification math.NAcs.NA MSC 65N3565M7042C40
keywords waveletneuralnetworksmultiresolutionanalysisShannonphysics-informedleast-squarescollocationspectralbiaspartialdifferentialequationsmesh-freemethods
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

The paper proposes a physics-informed multiresolution wavelet neural network (PIMWNN) for solving PDEs: the solution is represented by a single hidden layer of Shannon scaling and wavelet functions at several resolutions, and the unknown output weights are found by substituting this ansatz into the equation and boundary/initial conditions, then solving one linear least-squares problem. The central claim is that on a set of stationary and time-dependent linear and nonlinear test problems, PIMWNN is more accurate and substantially faster than physics-informed neural networks (PINNs), whose weights are trained by gradient descent. The authors further claim that raising the finest resolution $J$ extends the frequency content the network reproduces, so the method captures high-frequency solution components and counters the spectral bias that plagues gradient-trained networks. If these claims hold, PIMWNN offers a simple, interpretable alternative: no optimizer, no learning rate, and a direct accuracy-versus-scale tradeoff.

What carries the argument

The multiresolution wavelet neural network (MWNN): a single-hidden-layer network built from Shannon scaling nodes $\phi_{J_0,k}(x)=2^{J_0/2}\operatorname{sinc}(2^{J_0}x/L-k)$ and Shannon wavelet nodes $\psi_{j,k}(x)$ for $j=J_0,\dots,J-1$, so the network output is the truncated multiresolution expansion $f(x)=\sum_k c_{J_0,k}\phi_{J_0,k}(x)+\sum_{j=J_0}^{J-1}\sum_k d_{j,k}\psi_{j,k}(x)$ (Eq. 15). Because the basis is fixed and only the output weights are unknown, substituting the ansatz into the PDE and boundary conditions produces a rectangular linear system whose least-squares solution defines the trained network.

What would settle it

Compute the Gram matrix of the basis functions in Eq. (14) on $[0,1]$ for $J=7$ and $J=9$. If the off-diagonal entries do not approach zero or the matrix condition number grows steeply with $J$, the orthonormality assumption collapses; one would then expect the reported $L^2$ errors for a smooth function like $\exp(x)$ to stop improving as $J$ increases, which is a direct, checkable contradiction.

Watch

Extended reading notes

Core claim

The paper's central discovery is that a feed-forward network whose hidden layer consists of Shannon scaling functions at a coarse resolution plus Shannon wavelets at finer resolutions, Eq. (15), can be trained by assembling a linear least-squares system from the PDE, boundary, and initial conditions; no backpropagation or stochastic optimization is needed. Across the tested stationary and time-dependent linear and nonlinear equations, this PIMWNN achieves relative $L^2$ errors that decrease as the resolution $J$ increases, matches or exceeds the accuracy of PINNs while using a fraction of the training time, and reproduces high-frequency spectral content that PINNs lose. The authors argue thi

Load-bearing premise

The convergence and high-frequency claims rest on the assumption that the truncated Shannon scaling and wavelet functions on the finite interval $[0,L]$ form an orthonormal basis whose approximation error shrinks as $J$ grows — but Shannon orthogonality is only guaranteed on the entire real line, not on an interval with a finite set of translations.

Editorial extensions

If this is right

  • PDE solving reduces to assembling and solving a linear least-squares system, so training is deterministic, non-iterative for linear problems, and does not depend on optimizer hyperparameters.
  • Increasing the finest scale $J$ systematically raises the maximum representable frequency, giving an explicit knob for high-frequency content and a concrete route around spectral bias.
  • Time-dependent problems can be solved as a single space-time system (Eqs. 26-28), and nonlinear equations like Burgers can be handled by Picard iteration per time step.
  • Because the ansatz is a fixed wavelet basis, the network's approximation capability provably increases with the number of hidden nodes (scale), making the architecture interpretable.
  • The mesh-free collocation formulation accommodates irregular domains and Dirichlet boundary conditions without special treatment.

Reading between the lines

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

  • Mathematically, PIMWNN is a spectral collocation method with a Shannon wavelet basis; comparing it against conventional Fourier or Chebyshev collocation with equal degrees of freedom would isolate what the wavelet multiresolution, rather than the least-squares training, contributes to its accuracy.
  • The spectral-bias claim conflates ansatz and training algorithm: the fixed linear basis solves all frequencies at once, so a fair test is whether a gradient-descent-trained version of the same MWNN shows the frequency lag that the least-squares version avoids.
  • The Shannon basis's orthonormality on the finite interval $[0,L]$ is not automatic — computing the Gram matrix of the truncated basis would reveal whether the convergence theory in Section 2.2 carries over to the actual computational domain.
  • The space-time formulation means the method's cost grows with the product of spatial and temporal resolutions; for long-time or high-dimensional problems the authors' own discussion points to domain decomposition as the natural extension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a physics-informed multiresolution wavelet neural network (PIMWNN) method for solving PDEs. The solution is represented by a single-hidden-layer network whose activations are Shannon scaling and wavelet functions at multiple resolutions; the network is trained by substituting the ansatz into the PDE and boundary/initial conditions and solving the resulting linear least-squares problem. The method is applied to 1D and 2D stationary advection/diffusion equations, a high-frequency Helmholtz problem, a complex-domain diffusion problem, unsteady advection/diffusion equations, and Burgers' equation. The paper claims higher accuracy and faster speed than PINNs and asserts that the method addresses the spectral bias problem by increasing the wavelet scale.

Significance. If the numerical results are reproducible, PIMWNN is a simple and potentially useful least-squares spectral collocation method for moderate-dimensional, smooth PDE problems. Its strengths are the transparent trial basis, the absence of gradient-based optimization, the clean assembly of the least-squares system, and the quantitative relative L2 error reporting together with FFT-based spectral diagnostics. The paper also explicitly acknowledges the high-dimensional limitation of tensor-product wavelet bases. However, the theoretical justification on bounded domains is not established, the reported network-size formula is inconsistent with all tables, and the headline comparison against PINNs rests on a single under-trained baseline. These issues currently prevent the paper from supporting its broader claims.

major comments (4)
  1. [§2.2, Eqs. (14)–(15)] The multiresolution framework in §2.1 is stated for L2(R), but the construction in Eqs. (14) restricts the Shannon scaling and wavelet functions to [0,L] and to k=0,...,2^j. Shannon functions are orthonormal on R, not on a finite interval; truncating the integer shifts breaks both orthogonality of {φ_{j,k}} and the decomposition property V_j ⊕ W_j = V_{j+1} on [0,L]. Consequently Eq. (13) is not justified as an orthonormal multiresolution expansion, and the claim that approximation accuracy improves as J increases does not follow from the cited theory. The numerical least-squares method may still work with a non-orthogonal trial set, but the theoretical statement in §2.2 and the scale-convergence interpretation of Tables 1–4 need either a proof for the bounded truncated case or a weakened empirical statement.
  2. [Eq. (15) vs. Tables 1, 3, 4] The formula N = n_J0+1 + Σ_{j=J0}^{J-1}(n_j+1) = 2^J + J - J0 + 1 gives N=136 for J0=0, J=7, but Table 1 reports N=265; the same discrepancy appears throughout Tables 3 and 4 (e.g., J0=0, J=1 yields Table N=7 instead of 4). The tabulated values match 2^{J+1}+J+2, equivalently including the j=J wavelet level in the sum, not the formula in Eq. (15). Since N is the dimension of the least-squares system and the accuracy-versus-scale claim is tied to the network size, this inconsistency must be corrected before the experiments can be reproduced or the scaling behavior assessed.
  3. [§4.2.2, §5, abstract] The headline comparison with PINNs is supported by exactly one experiment: a 2-hidden-layer, 80-neuron tanh network trained with Adam for 10,000 steps at learning rate 0.001 using DeepXDE. No loss history, final loss, L-BFGS refinement, learning-rate schedule, hyperparameter search, or repeated-seed statistics are reported, and the baseline is not close to convergence for a high-frequency Helmholtz problem. The abstract's statement that PIMWNN achieves 'higher accuracy and faster speed than PINNs' (plural) is therefore not supported; the evidence supports only a comparison with one particular under-trained PINN implementation. I recommend adding at least one fully converged PINN baseline (e.g., Adam followed by L-BFGS, with training curves and relative L2 errors) and, ideally, comparisons with PIELM or randomized neural network methods on the same test problems.
  4. [§1 contributions, §4.1, §5] The claim that the method 'addresses the spectral bias problem in network training' is not established. The FFT diagnostics in Figs. 2 and 5 show that the Shannon-basis least-squares approximation reproduces the spectrum of the target function; they do not probe the training dynamics of a neural network, and PIMWNN determines its outer-layer weights by solving a linear system rather than by gradient descent. The contribution should be reformulated as 'captures high-frequency components with increasing scale' or 'avoids the gradient-descent spectral bias of PINNs by construction.' As written, the spectral-bias language overstates what the experiments measure.
minor comments (5)
  1. [§3.3, §4.6] The time-stepping loop and Picard iteration are described only in prose; Algorithm 1 covers only the stationary case. For reproducibility, please state the time loop, the Picard stopping criterion (or explain why fixed K=10 is always sufficient), and report the number of least-squares solves and total CPU time for the Burgers' equation experiment.
  2. [§4.5.1, §4.5.3] The comparisons against 'deep PINN' and 'PINN' in these subsections are taken from figures in Refs. [27] and [39], not from runs performed by the authors, and no quantitative errors are reported for those baselines. If these comparisons are retained, the source and the exact setup of each baseline should be stated; ideally, the baselines should be rerun with the same error metric.
  3. [Tables 1–4] Table 2 and Table 4 report N as products such as '7×7' and '152×46'. Please clarify whether these are total numbers of basis functions or per-dimension counts, and align the notation with the formula for N.
  4. [§4.2.1 and elsewhere] Typographical issues include 'dvection-diffusion' in the section heading, 'Fucntion' in Table 1, 'the the pointwise error' in §4.2.2, and inconsistent use of 'Burger' vs 'Burgers'. These are minor but should be corrected.
  5. [§2.2, Eq. (14)] The Shannon wavelet formula in Eq. (14b) should be checked against the standard Shannon wavelet definition; the factor 1/2 and the cosine argument affect the generated basis, and any typographical error would propagate into the numerical implementation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PIMWNN weights are solved from PDE residuals and boundary data, and accuracy is assessed against independent exact solutions.

full rationale

The paper's derivation chain is self-contained in the non-circular sense: the MWNN ansatz is inserted into the PDE and boundary conditions (Eqs. 19–21 and 26–28), producing a linear least-squares system whose unknowns are the network weights; the exact solution enters only through the source term, boundary data, and initial data, not through the fitted weights. Reported relative L2 errors are computed against those independent exact solutions, so the accuracy claims are not fitted inputs renamed as predictions. The spectral-bias discussion in Section 5 is a descriptive property of adding higher-scale Shannon wavelets to the trial space rather than a predicted quantity forced by the same data: increasing J by construction adds high-frequency basis functions, but the paper does not derive a quantitative spectral-bias result from the fitted solution itself. The one head-to-head PINN comparison in Section 4.2.2 uses a weak, undertrained baseline, and the bounded-domain orthonormality assumption in Eqs. (14)–(15) is a mathematical rigor gap; both are correctness or benchmark-design concerns, not circularity. The paper contains no load-bearing self-citations and does not invoke a uniqueness theorem from its own authors. Therefore no circular step can be exhibited from the paper's own equations or citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities or fitted physical constants. Its load-bearing assumptions are numerical: bounded-domain validity of the Shannon basis, unproved conditioning of the least-squares systems, and heuristic choices of scales, collocation counts, and time-stepping parameters. These are set by hand for each example rather than derived.

free parameters (3)
  • Resolution levels J0, J (and Jx, Jy in 2D) = e.g., J0=0, J=7..11 in Table 1; Jx*Jy up to 4x4
    Chosen by hand per example; the paper shows error decreases as J increases. These are numerical discretization parameters, not fitted theory constants, but they are user-selected and would need to be set for each new problem.
  • Collocation point counts Nf, Nb (and Nt) = Nf from 100 to 20,000; Nb from 2 to 400 depending on the example
    Selected per experiment; no adaptive strategy or sensitivity study is reported.
  • Time step dt and Picard iteration count K = dt=0.001, K=10 for Burgers' equation
    Chosen by hand in Section 4.6; no convergence study with respect to dt or K is given.
assumptions (5)
  • ad hoc to paper The truncated Shannon scaling and wavelet functions in Eq. (14) form an orthonormal multiresolution basis on the bounded domain [0,L].
    Eqs. (14)-(15) use these functions with finite k. Shannon bases are orthonormal on R, not on a bounded interval; truncation and restriction break orthogonality. No proof for the bounded-domain case is given.
  • standard math Multiresolution analysis approximation property: adding wavelets at finer scales guarantees L2 convergence of the network expansion.
    Cited standard MRA theory in Section 2.1, but its transfer to the collocation and least-squares setting and to bounded domains is assumed.
  • domain assumption The PDE problems considered have classical solutions smooth enough for pointwise collocation of the strong form and for boundary data to be enforced pointwise.
    The method substitutes the network into the strong form and evaluates at points; no regularity discussion or variational formulation is provided.
  • domain assumption The least-squares problems (21) and (28) are well-conditioned enough that solving them yields a reliable approximation.
    No condition number estimates or regularization strategy are reported; ill-conditioning of wavelet collocation matrices is a known concern.
  • ad hoc to paper Picard iteration for the Burgers' equation converges within the fixed K=10 iterations.
    Section 3.3 linearizes with Picard iteration, but no convergence criterion or proof is given; K=10 is fixed in Section 4.6.

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Cite this review

Pith. "Pith review of Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations." pith.science (2026). https://pith.science/paper/WZWIQY3W

@misc{pith2026250807546,
  author       = {Pith},
  title        = {Pith review of: Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZWIQY3W}},
  note         = {Machine review of arXiv:2508.07546}
}
read the original abstract

In this paper, a physics-informed multiresolution wavelet neural network (PIMWNN) method is proposed for solving partial differential equations (PDEs). This method uses the multiresolution wavelet neural network (MWNN) to approximate unknown functions, then substituting the MWNN into PDEs and training the MWNN by least-squares algorithm. We apply the proposed method to various problems, including stationary/nonstationary advection, diffusion and advection-diffusion problems, and linear/nonlinear time-dependent problems. Numerical experiments show that the PIMWNN method can achieve higher accuracy and faster speed than Physics Informed Neural Networks (PINNs). Moreover, the PIMWNN method, being mesh-free, can handle different boundary conditions easily and solve the time-dependent problems efficiently. The proposed method is expected to solve the spectral bias problem in network training. These characteristics show the great potential of the PIMWNN method used in the field of numerical solving methods for PDEs.

Figures

Figures reproduced from arXiv: 2508.07546 by the authors.

Figure 1
Figure 1. Structure of multiresolution wavelet neural networks and schematic of PIMWNN. In the subfigures (a) and (b) , the yellow nodes represent the scale transformation. The pink nodes represent the scaling function node and the orange nodes represent the wavelet function node. In the subfigure (c), the pink nodes represent the subnetwork of scaling function and the orange nodes represent subnetwork of wavelet function. Th… view at source ↗
Figure 2
Figure 2. Representation of function 𝑓1 (𝑥) and FFT of PIMWNN solutions. we incrementally increased the scale 𝐽, and find that spectra of PIMWNN solution and exact solution can match very well with each other before 𝑘=1000 with 𝐽=10 ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Representation of function 𝑓2 (𝑥, 𝑦) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Solution and error plots of equations (35a)-(35c). error 𝐿2 = 6.887 × 10−2. The FFT of solution are shown in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Solution and error plots for 1D Helmholtz equation Eq.(37a). 0.0 0.2 0.4 0.6 0.8 1.0 x 0.5 1.0 1.5 2.0 2.5 u(x) uex upred (a) Solution. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 1.2 error ×10−1 |uex − upred| (b) Pointwise error. 101 102 103 k 0.0 0.5 1.0 1.5 2.…
Figure 6
Figure 6. Figure 6: PINN solution and error plots for 1D Helmholtz equation Eq.(37a). 4.3. 2D steady case Next, we solve 2D stationary advection and diffusion equations, given by 𝑎𝑢𝑥 + 𝑏𝑢𝑦 = 𝑓, (𝑥, 𝑦) ∈ [−1, 1] × [−1, 1], (39a) (𝑢𝑥𝑥 + 𝑢𝑦𝑦) = 𝑓, (𝑥, 𝑦) ∈ [−1, 1] × [−1, 1]. (39b) The Dirich…
Figure 7
Figure 7. Figure 7: Solution and error plots for 2D advection equation Eq.(39a). −1.0 −0.5 0.0 0.5 1.0 x −1.0 −0.5 0.0 0.5 1.0 y uext 0.48 0.60 0.72 0.84 0.96 1.08 1.20 1.32 1.44 (a) Exact solution. −1.0 −0.5 0.0 0.5 1.0 x −1.0 −0.5 0.0 0.5 1.0 y upred 0.48 0.60 0.72 0.84 0.96 1.08 1.20 1…
Figure 8
Figure 8. Figure 8: Solution and error plots for 2D diffusion equation Eq.(39b). 𝑢(𝑥, 𝑦) = 1 2 + exp(−(2𝑥 2 + 4𝑦 2 )), (40b) respectively. The results for these two cases are given in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Solution and error plots for 2D diffusion equation Eq.(41) on a complex domain. where Ω = {(𝑥, 𝑦)|𝑥 = 0.5 + 𝑟(𝜃) cos(𝜃), 𝑦 = 0.5 + 𝑟(𝜃) sin(𝜃), 𝑟(𝜃) = 0.2 + 0.15 sin(5𝜃), 𝜃 ∈ [0, 2𝜋)}, which has a complex boundary, something like a flower. The function 𝑓 is chosen such…
Figure 10
Figure 10. Figure 10: Solution and error plots for 1D unsteady advection Eq.(44). The exact solution is given by 𝑢(𝑥, 𝑡) = exp(−50(𝑥 − 𝑡) 2 ). (47) This example is taken from [27]. The authors solved this equation with the PIELM method. The exact and PIELM solutions are shown in [PITH_FUL…
Figure 11
Figure 11. Figure 11: Solution and error plots for 1D unsteady advection Eq.(46). 4.5.3. Linear diffusion equation with the amplitude of solution grows exponentially over time To evaluate the performance of PIMWNN in a situation where the solution’s amplitude grows exponentially over time,…
Figure 12
Figure 12. Figure 12: Solution and error plots for diffusion equation Eq.(48). The exact solution and the PIMWNN solution of the above problem are shown in [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Solution and error plots for Burger’s equation Eq.(50). 5. DISCUSSION This paper aims to develop a physics-informed multiresolution wavelet neural network for solving partial differential equations. Based on multiresolution wavelet theory, we construct the multiresolu…
Figure 14
Figure 14. Figure 14: Solution and error plots for Burger’s equation at different time. We also plan to apply the PIMWNN method to a broader range of problems to explore its potential applications, including electromagnetic field problems, plasma physics problems, and fluid mechanics probl…

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.