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REVIEW 2 major objections 5 minor 58 references

Spectral connvergece of random feature method in one dimension

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that the random feature method converges super-exponentially for one-dimensional elliptic problems whose solutions are Gevrey-smooth with index below 1, and exponentially or algebraically for analytic and Sobolev solutions.

desk verdict A mostly solid analysis of RFM in 1D with a real error in the headline super-exponential rate; the proof gives a slower rate than claimed, but the paper is worth refereeing after the fix. read the letter →

arxiv 2507.07371 v1 pith:AK5ZLXDW submitted 2025-07-10 math.NA cs.NA

Pingbing Ming , Hao Yu This is my paper · ORCID
classification math.NAcs.NA MSC 65N1565N3068T2015A12
keywords RandomfeaturemethodSpectralaccuracyPartitionofunityGevreyclassSingularvaluesConditionnumberTwo-pointboundaryvalueproblemExponentialill-conditioning
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 proves that the Random Feature Method (RFM), a two-layer network with randomly sampled hidden frequencies trained by least squares, solves one-dimensional second-order elliptic boundary value problems with spectral accuracy. When the true solution lies in a Gevrey class of index $s<1$, the expected squared loss decays as $\Gamma(2N-1)^{-2/[3(1-s)]+\epsilon}$, which is faster than any exponential in the number of features $N$; analytic and Gevrey-$s>1$ solutions give exponential rates, and Sobolev solutions give algebraic rates. The result matters because it supplies the missing theory behind the numerically observed spectral accuracy of RFM and makes explicit how problem data, sampling range, and solution smoothness determine the required feature count. The paper also proves that the random feature matrix is exponentially ill-conditioned, with singular values decaying roughly as $\sqrt{nN}(SR)^{m-3}/(m-3)!$, and that the partition of unity method both accelerates convergence in patch size and mitigates the excessive singular-value decay.

What carries the argument

The argument is carried by a Taylor-plus-Vandermonde decomposition. The solution is split into even and odd parts, and the features $\cos(k_i x)$, $\sin(k_i x)$ are replaced by their Taylor polynomials, reducing the problem to two Vandermonde systems $V_N(-k_1^2,\dots,-k_N^2)\boldsymbol{X}=\boldsymbol{F}$ and $V_N\,\mathrm{diag}(k_i)\,\boldsymbol{Y}=\boldsymbol{G}$. The explicit inverse of a Vandermonde matrix in terms of elementary symmetric functions gives sharp bounds on the coefficients $\alpha_i$; a probabilistic moment estimate (Lemma 3.2) shows the event on which any $\alpha_i$ is exponentially large has probability at most $e^{-(c-1-\ln c)(N-1)}$. The Taylor remainders of the features, applied through the differential operator $L$, contribute the factor $(2ec\tau^2)^{2N-2}$ with $\tau=R\max(C_u,S)\,\Gamma(2N-1)^{(s-1)/(2N-2)}$, and choosing $c\sim \ln N$ balances the two error terms to produce the super-exponential rates. For the random feature matrix, the same idea constructs $2N-m+1$ independent coefficient vectors for which $\boldsymbol{\Phi}\boldsymbol{\alpha}$ is tiny, yielding the minimax upper bound on $\sigma_m(\boldsymbol{\Phi})$; the lower bound on the condition number follows from the Frobenius norm combined with a high-probability bound on a random quadratic form.

What would settle it

Take $u(x)=e^{kx}$ on $(-1,1)$ with $k=10$, so $u\in G^0$ with $C_u=k$; compute the minimal squared loss of RFM for increasing $N$ and plot it against $\Gamma(2N-1)^{-2/3}$. If the empirical error does not drop steeply once $N$ passes roughly $k$ and does not track a super-exponential slope, then Corollary 2.7 is not describing the observed behavior. A second check is to solve the same problem with a solution that has a boundary layer and verify whether the loss stagnates until $N$ reaches $(R\,C_u)^{1/(1-s)}$, as Remark 2.5 predicts.

Watch

Extended reading notes

Core claim

For the two-point boundary value problem (2.1), the paper takes trial functions $u_N=\sum_{i=1}^N \alpha_i \cos(k_i x)+\alpha_{N+i}\sin(k_i x)$ with $k_i$ uniform on $(0,S)$ and analyzes $\mathbb{E}\,\min_{\boldsymbol{\alpha}} L(\boldsymbol{\alpha})$, the minimal squared loss for PDE and boundary residuals. The central quantitative result is Corollary 2.7: if $u\in G^s(\overline{\Omega})$ with $s<1$, then $\mathbb{E}\,\min L(\boldsymbol{\alpha}) \lesssim \Gamma(2N-1)^{-2/[3(1-s)]+\epsilon}$ for every $\epsilon>0$, a super-exponential rate. A Fourier-truncation argument (Corollary 2.8) extends this to $\exp(-\kappa N)$ for analytic solutions, $\exp(-\kappa N^{1/s})$ for Gevrey solutions with $s>1$, and $N^{-2s+4}$ for Sobolev solutions. With the partition of unity method, Theorem 2.11 proves convergence in the maximum patch size $r$ at rate $r^{4(N_p-4)/3}$ for Gevrey-$s<1$ solutions. The same Taylor-remainder machinery yields Theorem 2.14: the $m$-th singular value of the random feature matrix decays super-exponentially, so its condition number grows exponentially with $\min(n,2N)$; Theorem 2.17 shows how block diagonal structure ties the global singular values to those of local patch matrices, explaining why PUM slows the decay. Proposition 2.9 converts all loss bounds into $\|\cdot\|_{H^{1/2}}$ or $\|\cdot\|_{H^{3/2}}$ error estimates for the numerical solution.

Load-bearing premise

The load-bearing premise is that the true solution is extremely smooth across the whole closed interval, with a single growth constant $C_u$ controlling all derivatives; a steep layer or boundary singularity makes $C_u$ large, so the promised super-exponential speed-up only appears after an impractically large number of features.

Editorial extensions

If this is right

  • When the solution is Gevrey-smooth with index $s<1$, the expected squared loss decays faster than any exponential in the number of random features, so the spectral accuracy observed in RFM experiments is a genuine property of the method rather than a lucky-sample artifact.
  • The explicit error bound identifies the feature count $N\sim (R\max(C_u,S))^{1/(1-s)}$ needed to enter the fast regime, giving a practical predictor for when RFM will be accurate and when it will not.
  • Because the singular values of the raw random feature matrix decay exponentially, iterative solvers cannot reach high precision on the un-preconditioned system; direct or preconditioned solvers are needed, and the block estimates show that PUM reduces the singular-value decay rate by roughly the number of patches.
  • PUM provides a second, independent convergence lever: for Gevrey-$s<1$ solutions the error decays as $r^{4(N_p-4)/3}$ in the maximum patch size, so refining patches accelerates convergence even when the local feature count is fixed.
  • All loss bounds translate into $H^{1/2}$ or $H^{3/2}$ error bounds for the numerical solution $u_N$, so controlling the training loss controls the actual PDE approximation error.

Reading between the lines

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

  • If the same Taylor-remainder mechanism extends to higher dimensions as the paper conjectures, the effective convergence rate in $d$ dimensions should be obtained by replacing $N$ with $N^{1/d}$, which would make RFM attractive precisely in low-dimensional settings where direct linear algebra remains feasible.
  • The exponential ill-conditioning result suggests a practical diagnostic: the minimal singular value of a single patch's local matrix predicts the global conditioning, so one can choose patch size and sampling range $S$ to keep the local condition number bounded before assembling the global system.
  • One could test the sharpness of the theory by computing RFM errors on $u(x)=e^{kx}$ over $(-1,1)$, where $C_u=k$ is known exactly; the predicted transition at $N\sim k$ features and the $\Gamma(2N-1)^{-2/3}$ slope should be directly visible in a log-log plot.
  • The proof cleanly separates PDE coefficients from smoothness constants, which implies that coefficient variations should not alter convergence rates; this is a testable robustness property for variable-coefficient problems.
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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

2 major / 5 minor

Summary. The paper analyzes the convergence of the Random Feature Method (RFM) for one-dimensional second-order elliptic boundary value problems. The authors prove approximation error bounds for the continuous least-squares loss, establishing algebraic, exponential, or superexponential rates depending on whether the solution belongs to a Sobolev space, an analytic class, or a Gevrey class. They also prove convergence rates for the partition-of-unity (PUM) enhancement in terms of the patch size, and show that the random feature matrix has exponentially decaying singular values and an exponentially large condition number, with bounds indicating that PUM mitigates this decay. The proofs are self-contained and use Taylor expansions, Vandermonde systems, and probabilistic estimates on random frequencies.

Significance. If the results are correct, the paper gives the first rigorous explanation of the spectral accuracy observed numerically for RFM in low dimensions, quantifies the benefit of PUM, and characterizes the severe ill-conditioning of the random feature matrix. The authors provide detailed, mostly coherent proofs with explicit constants and parameter dependencies, and the argument is for the most part self-contained rather than fitted to data. However, the flagship superexponential rate in Corollary 2.7 does not follow from the proof as written, and the analysis is limited to the continuous loss rather than the discrete collocation loss actually used in practice. These issues are significant but appear repairable within the manuscript's scope.

major comments (2)
  1. [§3.1, proof of Corollary 2.7] The claimed rate in Corollary 2.7 does not follow from the proof. With c = 2(1-s)/(3(N-1)) ln Γ(2N-1), the first term of (2.4) is N e^{-(c-1-ln c)(N-1)} = Γ(2N-1)^{-2(1-s)/3 + o(1)}. The second term equals (2ec)^{2N-2} [R max(C_u,S)]^{4N-4} Γ(2N-1)^{2(s-1)} = Γ(2N-1)^{-2(1-s)+o(1)}, because c^{2N-2} is Γ^{o(1)} for this choice of c. Hence the proof establishes only E min L ≲ Γ(2N-1)^{-2(1-s)/3+ε}, not the stated Γ(2N-1)^{-2/[3(1-s)]+ε}. The displayed equality in the proof, (2ecτ^2)^{2N-2} = [2R^2 max^2]^{2N-2} Γ(2N-1)^{-ε} Γ(2N-1)^{-2/3(1-s)+ε}, is algebraically inconsistent with the definition τ = R max(C_u,S) Γ(2N-1)^{(s-1)/(2N-2)}; it would require (ec)^{2N-2} = Γ(2N-1)^{4(1-s)/3}, which the chosen c does not satisfy. For example, at s = 1/2 the proof gives Γ(2N-1)^{-1/3+ε}, while the corollary claims Γ(2N-1)^{-4/3+ε}. This affects the headline claims in the abstract, Table 1, and Remark 2.4. Since a weaker superexponential bound does follow from the first term, the error is repairable, but Theorem 2.4 and all statements depending on this exponent must be corrected.
  2. [§2.1, Theorem 2.4 and §2.3] The error estimates in Section 2.1 bound E min_{α∈R^{2N}} L(α) for the continuous loss (2.3), whereas the actual RFM algorithm and the numerical experiments cited in [10] solve a discrete least-squares problem with collocation points, as defined in Section 2.3. The paper does not prove any bound for the discrete loss, nor does it quantify the relation between the minimizers of the continuous and discrete losses. As a result, the headline 'spectral convergence of RFM' is established only for the idealized continuous-loss version of the method. The authors should either extend the analysis to the discrete collocation loss or explicitly frame the results as bounds for the continuous-loss idealization.
minor comments (5)
  1. [Title and Abstract] The title contains the typo 'CONNVERGECE' and should read 'CONVERGENCE'; the abstract has a similar typographical error in the first sentence.
  2. [§1.2] In the heading 'Exponential convergnece in Gevrey and analytic class', 'convergnece' should be 'convergence'.
  3. [Throughout] The phrase 'exponential ill conditionality' should be 'exponential ill-conditioning' for consistency with standard terminology.
  4. [§3.1, proof of Corollary 2.7] The expression '− 2/3(1−s)' is ambiguous between Γ^{-2/[3(1-s)]} and Γ^{-2(1-s)/3}; the displayed formulas would be clearer with parentheses, e.g., Γ(2N-1)^{-2(1-s)/3}.
  5. [Corollary 2.8(b)] The quantity C_{φu} in the possible choice of κ is not defined; it should be the Gevrey constant of the product φu, and a brief explanation of the notation would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral rates are derived from stated Gevrey/analytic regularity via Taylor remainder and random-weight estimates, with no fitted parameter or self-citation chain as a load-bearing premise.

full rationale

After walking the derivation chain, I find no circular step. Theorem 2.4 is proven from the Gevrey/Taylor remainder estimates (3.5)-(3.6), the Vandermonde coefficient bounds in Lemma 3.1, and the probabilistic estimate Lemma 3.2; no term in the final bound is set equal to the quantity being predicted. Corollary 2.7 merely chooses c to balance the two terms of (2.4); even if the quoted choice of c yields a weaker exponent than claimed, that is a possible correctness issue with the displayed rate, not a circularity, because the target rate is not assumed as an input. Corollary 2.8 uses Fourier truncation and external Paley-Wiener / Gevrey-Paley-Wiener theorems to reduce the problem to Theorem 2.4. The PUM results reduce global error to local patches via Proposition 2.10 and then apply Theorem 2.4 on each patch, again without fitting any parameter to data. The singular-value bounds in Theorem 2.14 are proved by constructing explicit null-space vectors from Taylor remainders, and Theorem 2.17 is a matrix interlacing/minimax statement. Numerical comparisons to [10] and [12] are confirmatory illustrations, not premises of the proofs, and those references are not by the present authors, so no self-citation chain is load-bearing. No fitted parameter is renamed a prediction, and no uniqueness theorem from the authors' prior work is invoked. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no invented entities and fits no free parameters. It relies on the stated smoothness assumptions on the solution and the standard properties of Gevrey classes, Vandermonde matrices, and Fourier analysis.

assumptions (6)
  • domain assumption The true solution u belongs to a Gevrey class G^s(\bar{\Omega}) (Definition 2.1) or a Sobolev space H^s(\Omega) with s >= 2, with finite constants M_u and C_u.
    This smoothness assumption drives the spectral rates; it is stated in Theorem 2.4 and Corollary 2.8.
  • domain assumption The hidden weights k_i are i.i.d. uniform on (0,S), and collocation points lie in [-R,R].
    The algorithm and the probability estimates rely on this sampling distribution.
  • domain assumption The differential operator L and boundary operator B have L^\infty coefficients with bounds \Lambda_i, and the problem is well-posed.
    These bounds enter the constants in the error estimates.
  • standard math For the loss-to-error conversion (Proposition 2.9), L is properly elliptic and coefficients are C^\infty, relying on the regularity estimate from Schechter [50].
    The paper invokes an external theorem to translate loss bounds into H^{1/2} or H^{3/2} error bounds.
  • standard math For the high-frequency truncation in Corollary 2.8, the Paley-Wiener theorem and the Gevrey Paley-Wiener-Schwartz theorem from [47] are used.
    These classical results control the Fourier tails.
  • standard math For the Sobolev case, a bounded linear extension operator from H^s(\Omega) to H^s(\mathbb{R}) exists (from [6]).
    Used to obtain Fourier decay of the extension.

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Cite this review

Pith. "Pith review of Spectral connvergece of random feature method in one dimension." pith.science (2026). https://pith.science/paper/AK5ZLXDW

@misc{pith2026250707371,
  author       = {Pith},
  title        = {Pith review of: Spectral connvergece of random feature method in one dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AK5ZLXDW}},
  note         = {Machine review of arXiv:2507.07371}
}
read the original abstract

Among the various machine learning methods solving partial differential equations, the Random Feature Method (RFM) stands out due to its accuracy and efficiency. In this paper, we demonstrate that the approximation error of RFM exhibits spectral convergence when it is applied to the second-order elliptic equations in one dimension, provided that the solution belongs to Gevrey classes or Sobolev spaces. We highlight the significant impact of incorporating the Partition of Unity Method (PUM) to enhance the convergence of RFM by establishing the convergence rate in terms of the maximum patch size. Furthermore, we reveal that the singular values of the random feature matrix (RFMtx) decay exponentially, while its condition number increases exponentially as the number of the features grows. We also theoretically illustrate that PUM may mitigate the excessive decay of the singular values of RFMtx.

Figures

Figures reproduced from arXiv: 2507.07371 by the authors.

Figure 1
Figure 1. Singular values and sparsity pattern of RFMtx. Left: The singular values of RFMtx without PUM (18 features, 𝑆 = 1, 𝑅 = 1/2) and the upper bound provided by (2.6). Middle: Sparsity pattern of RFMtx with 501 equidistant collocation points, 6 patches (𝑃 = 5) and 20 local features on each patch (𝑁𝑝 = 10). Right: The singular values of RFMtx with/without PUM, and the upper and lower bounds given by (2.7), where 𝑆 = 1, 𝑅 … view at source ↗
Figure 2
Figure 2. Bounds for singular values of RFMtx and bound for J(𝛼). Left: The singular values of RFMtx with/without PUM, and the upper and lower bounds given by (2.7). e𝐿 is taken as the identity operator. Right: Numerical integration of J(𝛼) and its upper bound 0.72. References [1] Alaoui, A., Mahoney, M.W.: Fast randomized kernel ridge regression with statistical guarantees. In: Advances in Neural Information Processing Syste… view at source ↗

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Works this paper leans on

58 extracted references · 55 canonical work pages

  1. [10]

    Chen, J., Chi, X., E, W., Yang, Z.: Bridging traditional and machine learning-based algorithms for solving PDEs: The random feature method. J. Mach. Learn. 1(3), 268–298 (2022)

    work page 2022

  2. [1]

    In: Advances in Neural Information Processing Systems, vol

    Alaoui, A., Mahoney, M.W.: Fast randomized kernel ridge regression with statistical guarantees. In: Advances in Neural Information Processing Systems, vol. 28. Curran Associates, Inc. (2015)

    work page 2015

  3. [2]

    Babu ˇska, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Meth. Engng.40(4), 727–758 (1997)

    work page 1997

  4. [3]

    In: Conference on Learning Theory, vol

    Bach, F.: Sharp analysis of low-rank kernel matrix approximations. In: Conference on Learning Theory, vol. 30, pp. 185–209. PMLR (2013) SPECTRAL CONNVERGECE OF RANDOM FEATURE METHOD IN ONE DIMENSION 33

    work page 2013

  5. [4]

    Bach, F.: On the equivalence between kernel quadrature rules and random feature expansions. J. Mach. Learn. Res. 18(21), 1–38 (2017)

    work page 2017

  6. [5]

    Barnett, A.H.: How exponentially ill-conditioned are contiguous submatrices of the Fourier matrix? SIAM Review 64(1), 105–131 (2022)

    work page 2022

  7. [6]

    Universitext

    Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)

    work page 2011

  8. [7]

    arXiv:2410.06308

    Chen, C., Zhou, Q., Yang, Y., Xiang, Y., Luo, T.: Quantifying training difficulty and accelerating convergence in neural network-based pde solvers (2024). arXiv:2410.06308

    arXiv 2024

Show all 58 references
  1. [8]

    arXiv:2305.05642

    Chen, H., Long, J., Wu, L.: A duality framework for analyzing random feature and two-layer neural networks (2024). arXiv:2305.05642

    2024 arXiv

  2. [9]

    In: General theory of partial differential equations and microlocal analysis, Pitman Res

    Chen, H., Rodino, L.: General theory of PDE and Gevrey classes. In: General theory of partial differential equations and microlocal analysis, Pitman Res. Notes Math. Ser., vol. 349, pp. 6–81 (1996)

    1996

  3. [11]

    East Asian J

    Chen, J., E, W., Luo, Y.: The random feature method for time-dependent problems. East Asian J. Appl. Math. 13(3), 435–463 (2023)

    2023

  4. [12]

    Chen, J., E, W., Sun, Y.: Optimization of random feature method in the high-precision regime. Commun. Appl. Math. Comput. 6(2), 1490–1517 (2024)

    2024

  5. [13]

    arXiv:2409.15818

    Chen, J., Tan, L.: High-precision randomized iterative methods for the random feature method (2024). arXiv:2409.15818

    2024 arXiv

  6. [14]

    Chen, Z., Schaeffer, H.: Conditioning of random Fourier feature matrices: double descent and generalization error. Inf. Inference 13(2), iaad054 (2024)

    2024

  7. [15]

    Chi, X., Chen, J., Yang, Z.: The random feature method for solving interface problems. Comput. Methods Appl. Mech. Engrg. 420, 116719 (2024)

    2024

  8. [16]

    arXiv:2408.17225

    Dang, H., Wang, F., Jiang, S.: Adaptive growing randomized neural networks for solving partial differential equations (2024). arXiv:2408.17225

    2024

  9. [17]

    Dong, S., Li, Z.: Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations. Comput. Methods Appl. Mech. Engrg. 387, 114129 (2021)

    2021

  10. [18]

    CSIAM Transactions on Applied Mathematics 1(4), 561–615 (2020)

    E, W., Ma, C., Wu, L., Wojtowytsch, S.: Towards a mathematical understanding of neural network-based machine learning: What we know and what we don’t. CSIAM Transactions on Applied Mathematics 1(4), 561–615 (2020)

    2020

  11. [19]

    E, W., Wang, Q.: Exponential convergence of the deep neural network approximation for analytic functions. Sci. China Math. 61(10), 1733–1740 (2018)

    2018

  12. [20]

    E, W., Yu, B.: The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1–12 (2018)

    2018

  13. [21]

    Fabiani, G., Calabr `o, F., Russo, L., Siettos, C.: Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines. J. Sci. Comput. 89(2), 44 (2021)

    2021

  14. [22]

    Feischl, M., Schwab, C.: Exponential convergence in 𝐻1 of hp-FEM for Gevrey regularity with isotropic singularities. Numer. Math. 144(2), 323–346 (2020)

    2020

  15. [23]

    Gautschi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math.4, 117–123 (1962)

    1962

  16. [24]

    Gonon, L.: Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality. J. Mach. Learn. Res. 24(189), 1–51 (2023)

    2023

  17. [25]

    Gonon, L., Grigoryeva, L., Ortega, J.P.: Approximation bounds for random neural networks and reservoir systems. Ann. Appl. Probab. 33(1), 28–69 (2023)

    2023

  18. [26]

    SIAM (1977)

    Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: theory and applications. SIAM (1977)

    1977

  19. [27]

    Gui, W., Babu ˇska, I.: Theℎ, 𝑝 andℎ-𝑝 versions of the finite element method in 1 dimension. Part II. The error analysis of theℎ- andℎ-𝑝 versions. Numer. Math. 49(6), 613–657 (1986) 34 P. B. MING AND H. YU

    1986

  20. [28]

    Part 1: The basic approximation results

    Guo, B., Babu ˇska, I.: Theℎ-𝑝 version of the finite element method. Part 1: The basic approximation results. Comput. Mech. 1(1), 21–41 (1986)

    1986

  21. [29]

    Hashemi, A., Schaeffer, H., Shi, R., Topcu, U., Tran, G., Ward, R.: Generalization bounds for sparse random feature expansions. Appl. Comput. Harmon. Anal. 62, 310–330 (2023)

    2023

  22. [30]

    Springer (2003-2009)

    H ¨ormander, L.: The analysis of linear partial differential operators I-IV. Springer (2003-2009)

    2003

  23. [31]

    IEEE Trans

    Huang, G., Chen, L., Siew, C.K.: Universal approximation using incremental constructive feedforward networks with random hidden nodes. IEEE Trans. Neural Networks17(4), 879–892 (2006)

    2006

  24. [32]

    Cambridge University Press (2004)

    Katznelson, Y.: An introduction to harmonic analysis. Cambridge University Press (2004)

    2004

  25. [33]

    Li, Z., Ton, J.F., Oglic, D., Sejdinovic, D.: Towards a unified analysis of random Fourier features. J. Mach. Learn. Res. 22(108), 1–51 (2021)

    2021

  26. [34]

    In: Mathematical and Scientific Machine Learning Conference, vol

    Ma, C., Wu, L., E, W.: The slow deterioration of the generalization error of the random feature model. In: Mathematical and Scientific Machine Learning Conference, vol. 107, pp. 373–389. PMLR (2020)

    2020

  27. [35]

    Macon, N., Spitzbart, A.: Inverses of Vandermonde matrices. Amer. Math. Monthly 65(2), 95–100 (1958)

    1958

  28. [36]

    Mei, S., Misiakiewicz, T., Montanari, A.: Generalization error of random feature and kernel methods: hypercontractivity and kernel matrix concentration. Appl. Comput. Harmon. Anal. 59, 3–84 (2022)

    2022

  29. [37]

    Mei, S., Montanari, A.: The generalization error of random features regression: Precise asymptotics and the double descent curve. Comm. Pure Appl. Math. 75(4), 667–766 (2022)

    2022

  30. [38]

    Melenk, J.M.: ℎ𝑝 -finite element methods for singular perturbations, Lecture Notes in Mathematics , vol

  31. [39]

    Neural Comput

    Mhaskar, H.N.: Neural networks for optimal approximation of smooth and analytic functions. Neural Comput. 8(1), 164–177 (1996)

    1996

  32. [40]

    Montanelli, H., Yang, H., Du, Q.: Deep ReLU networks overcome the curse of dimensionality for generalized bandlimited functions. J. Comput. Math. 39(6), 801–815 (2021)

    2021

  33. [41]

    iii (1960)

    Muir, T.: History of the theory of determinants, vol. iii (1960)

    1960

  34. [42]

    Paley, R.E.A.C., Wiener, N.: Fourier transforms in the complex domain, vol. 19. American Mathematical Society, Providence, RI (1934)

    1934

  35. [43]

    Advances in neural information processing systems 20 (2007)

    Rahimi, A., Recht, B.: Random features for large-scale kernel machines. Advances in neural information processing systems 20 (2007)

    2007

  36. [44]

    In: 2008 46th Annual Allerton Conference on Communication, Control, and Computing, pp

    Rahimi, A., Recht, B.: Uniform approximation of functions with random bases. In: 2008 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 555–561 (2008)

    2008

  37. [45]

    Advances in neural information processing systems 21 (2008)

    Rahimi, A., Recht, B.: Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning. Advances in neural information processing systems 21 (2008)

    2008

  38. [46]

    Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019)

    2019

  39. [47]

    World Scientific (1993)

    Rodino, L.: Linear partial differential operators in Gevrey spaces. World Scientific (1993)

    1993

  40. [48]

    Advances in neural information processing systems 28 (2015)

    Rudi, A., Camoriano, R., Rosasco, L.: Less is more: Nystr ¨om computational regularization. Advances in neural information processing systems 28 (2015)

    2015

  41. [49]

    Advances in neural information processing systems 30 (2017)

    Rudi, A., Rosasco, L.: Generalization properties of learning with random features. Advances in neural information processing systems 30 (2017)

    2017

  42. [50]

    Schechter, M.: On 𝐿𝑝 estimates and regularity II. Math. Scand. 13(1), 47–69 (1963)

    1963

  43. [51]

    arXiv:2412.19207

    Shang, Y., Heinlein, A., Mishra, S., Wang, F.: Overlapping schwarz preconditioners for randomized neural networks with domain decomposition (2024). arXiv:2412.19207

    2024 arXiv

  44. [52]

    Shang, Y., Wang, F., Sun, J.: Randomized neural network with Petrov–Galerkin methods for solving linear and nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 127, 107518 (2023)

    2023

  45. [53]

    Sirignano, J., Spiliopoulos, K.: DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018) SPECTRAL CONNVERGECE OF RANDOM FEATURE METHOD IN ONE DIMENSION 35

    2018

  46. [54]

    In: Advances in Neural Information Processing Systems, vol

    Williams, C., Seeger, M.: Using the Nystr ¨om method to speed up kernel machines. In: Advances in Neural Information Processing Systems, vol. 13 (2000)

    2000

  47. [55]

    Xu, J.: Finite neuron method and convergence analysis. Commun. Comput. Phys. 28(5), 1707–1745 (2020)

    2020

  48. [56]

    Zang, Y., Bao, G., Ye, X., Zhou, H.: Weak adversarial networks for high-dimensional partial differential equations. J. Comput. Phys. 411, 109409 (2020)

    2020

  49. [57]

    Zhang, Z., Bao, F., Ju, L., Zhang, G.: Transferable neural networks for partial differential equations. J. Sci. Comput. 99(1), 2 (2024) SKLMS, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing 100190, China ...

    2024

  50. [1796]

    Springer-Verlag, Berlin (2002)

    2002

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.