Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs

F. Wang; S. Mishra; T. De Ryck; Y. Shang

REVIEW 3 major objections 6 minor 43 references

Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that randomized neural networks achieve dimension-independent Sobolev approximation rates, making them a practical tool for high-dimensional PDEs.

desk verdict Serious attempt at dimension-independent H^1/H^2 approximation rates for tanh RaNNs, but a false identity in Eq. (3.12) breaks the central proof; the numerics are plausible and the paper deserves major revision, not a full reject. read the letter →

arxiv 2501.12145 v1 pith:ECGJB44O submitted 2025-01-21 math.NA cs.NA

T. De Ryck , S. Mishra , Y. Shang , F. Wang This is my paper · ORCID

classification math.NAcs.NA MSC 68T0765C0541A2565M75

keywords randomizedneuralnetworksextremelearningmachinesphysics-informedcurseofdimensionalitySobolevapproximationhigh-dimensionalPDEsMonteCarlotanhactivation

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's central claim is that randomized neural networks — single-hidden-layer networks whose hidden weights are drawn at random and never trained, leaving only a linear least-squares solve for the output weights — can approximate smooth high-dimensional functions at error rates that do not deteriorate as the dimension grows. If this is right, it would mean that solving a linear system for the output weights, rather than running an expensive nonlinear optimization, is enough to approximate solutions of high-dimensional parabolic PDEs such as the heat equation, Black–Scholes, and Heston models. The paper gives dimension-independent convergence rates in the Sobolev norms $H^{1}$ and $H^{2}$, and supports them with experiments up to dimension 100.

What carries the argument

The load-bearing object is the smooth Heaviside approximation H_ε(x) = ½(1+tanh(x/ε)), introduced in equation (3.2). The proof starts from a representation of the target as an integral of a shifted Heaviside function, replaces the sharp step by H_ε, and then identifies the resulting expectation over random weights, biases, and shifts with a tanh randomized neural network. The approximation error is decomposed into the smoothing error from H vs H_ε and the Monte Carlo sampling error of a finite-width network, and both are bounded explicitly in the $H^{1}$ and $H^{2}$ norms.

What would settle it

Evaluate the unproved equality in equation (3.12) numerically for a smooth target such as u(x)=$e^{{-x^2}}$: if the difference between the two integrals does not decay polynomially in the smoothing parameter ε, the error decomposition in the proof is incomplete and the claimed rates do not follow.

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Extended reading notes

Core claim

The central discovery is that a randomized neural network of width N with tanh activations approximates Sobolev functions in the $H^{2}$ norm on a ball with an error bounded by C $M^{2}$ $d^{2}$ ||u||_{$H^{{4+s}}$} $N^{{-(γ/(γ+2))(s-d/2)/(2s+3)}}$ for any γ in (0,1/2), and a similar power law in the $H^{1}$ norm for functions in $H^{{3+s}}$. The exponents are independent of the dimension d whenever the regularity s exceeds d/2, and they approach $N^{{-1/10}}$ in $H^{2}$ and $N^{{-1/4}}$ in $H^{1}$ as the regularity grows. The proof represents the target function as a Fourier integral, rewrites it using a smooth tanh-based approximation to the Heaviside step function, and then shows that the resulting expectation over random weights, biases, and shifts is exactly a randomized neural network; the finite-width network is a Monte Carlo estimate of that expectation, and the error splits into a smoothing error and a sampling error.

Load-bearing premise

The proof takes as given that replacing the sharp step function by its smooth tanh approximation and shifting the integration limits leaves no unaccounted boundary error, and the size of that residual is never bounded.

Editorial extensions

If this is right

If the rates hold, a physics-informed extreme learning machine that solves a linear PDE by a single least-squares solve with N randomized tanh features has a total error governed by the approximation error, so the cost to reach a target accuracy in dimension d scales polynomially rather than exponentially in d.
The H^2 approximation bound directly controls the PDE residual for second-order operators, since ||L[u_θ]||_{L2} is bounded by a constant times ||u_θ - u||_{H^2}; a small Sobolev error therefore implies a small physics-informed loss.
For the heat equation, Black–Scholes, and Heston models in dimension up to 100, the numerical experiments report relative L2 errors of 1–3% with runtimes under six minutes, consistent with the claim that the curse of dimensionality is alleviated in practice.
The dimension-independent rates imply that increasing the width N improves accuracy without needing to add features per dimension; the hidden weights are drawn once and fixed, and only the output layer is fitted.

Reading between the lines

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

The same representation-to-Monte-Carlo pipeline could yield explicit approximation bounds for other activation functions that are scaled antiderivatives of localized bumps, such as the sigmoid, as long as the smoothing error can be controlled in Sobolev norms.
The paper's rates come with constants that grow like d^2 and factors that depend on the Fourier integrability of the target; for functions whose Fourier transforms concentrate away from zero, these constants may grow exponentially in d, so the practical range of dimensions for a given accuracy may be smaller than the asymptotic result suggests.
Because the H^2 norm controls first and second derivatives, the same dimension-independent rates should apply to financial Greeks (derivatives of option prices) obtained from the randomized-network solution of Black–Scholes or Heston models.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know this paper is a serious attempt to prove dimension-independent H^1/H^2 approximation rates for tanh randomized neural networks, and the numerics show PIELM working in up to 100 dimensions. But the central proof has a false identity, and the stated theorems do not follow as written.

What is new: Gonon and others proved sup-norm rates for ReLU random features; this paper pushes toward Sobolev norms, which is the right norm for PDE residuals. The representation via a smooth Heaviside approximation H_epsilon and the Monte Carlo discretization is a natural extension. The experiments for the heat equation, Black-Scholes, and Heston model in d up to 100 are genuinely useful evidence that PIELM can be fast and accurate, although there are no error bars and no code.

The soft spot is load-bearing. In Proposition 3.4, Eq. (3.12) asserts that the integral of H_epsilon(y) from 0 to x·xi+u equals the integral of H_epsilon(x·xi+u-y) from 0 to B(xi). Substituting z = x·xi+u-y, the right side becomes the integral of H_epsilon(z) from x·xi+u-B(xi) to x·xi+u. For finite epsilon, H_epsilon(z) is strictly positive, so the difference from the left side is the integral from 0 to x·xi+u-B(xi), which is generally nonzero. No bound on this boundary-layer term is given anywhere. The proof bounds the first form and then applies that bound to the second form, which is what the Monte Carlo network actually represents. So Theorems 3.6 and 3.9 inherit the gap. This is not a cosmetic slip; it is the bridge between the smoothed integral and the network.

There is also a secondary defect: the constant I_ell in (3.44) uses F(1)-F(-1), and under Assumption 3.3, F(1) is negative while F(-1) is positive, making that factor negative. As written, I_ell can be undefined or imaginary. That is likely fixable, but it needs care.

The empirical section is closer to a methods paper than to rigorous support for the theorem. Code, repeated runs, and at least one comparison against another random-feature baseline would strengthen it. Still, the experiments are not the problem.

Bottom line: the paper is worth engaging, and a serious editor should send it to referees, but those referees should demand a repaired proof. The dimension-independent Sobolev claim is exactly the kind of result people in this area want, and much of the machinery is sound. If the boundary-layer term can be bounded, or the proof restructured, the paper could become an important contribution. As submitted, the central theorem is not proven.

Referee Report

3 major / 6 minor

Summary. The paper studies randomized neural networks (RaNNs) in the physics-informed extreme learning machine (PIELM) framework for high-dimensional PDEs. The theoretical part claims that tanh-activated RaNNs can approximate functions of oscillatory-integral form in H^1 and H^2 norms with dimension-independent rates (Theorem 3.6), and that Sobolev functions can be approximated by standard RaNNs with uniformly distributed hidden weights at rates that overcome the curse of dimensionality under sufficient regularity (Theorem 3.9). The numerical part reports PIELM experiments for the heat equation, the Black-Scholes model, and the Heston model in dimensions up to 100, with reported relative errors of a few percent and computation times of seconds to a few minutes.

Significance. If the main theorems were correct, the paper would make a useful contribution: dimension-independent H^1/H^2 approximation rates for randomized neural networks would supply a formal justification for PIELM-type methods in high-dimensional PDEs, complementing existing L^∞ results for random features. The proof strategy is constructive and the numerical experiments are extensive, which are strengths. However, the central proof contains a false equality in Eq. (3.12), and the constant I_ℓ in Theorem 3.6 is not well defined as written. These are load-bearing defects: Theorems 3.6 and 3.9 are not proven in the submitted manuscript, so the paper's main theoretical claim is currently unsupported.

major comments (3)

[Section 3.1, Eq. (3.12)] The equality asserted in Eq. (3.12) is false for finite ε. Writing F(a)=∫_0^a H_ε(y)dy, the second integral equals F(x·ξ+u)−F(x·ξ+u−B(ξ)), so the difference between the two expressions is −F(x·ξ+u−B(ξ)), which is nonzero because H_ε is strictly positive. The error estimates in (3.21)–(3.41) are derived for the first integral, while the Monte-Carlo network in (3.47) implements the second integral. The proof therefore misses an unquantified boundary-layer contribution to u−u_ε and, consequently, to the networks in Theorems 3.6 and 3.9. The authors need either to replace the false identity by a valid bound for the boundary-layer difference in H^1(H^2), or to modify the definition of u_ε and H_ε so that the two forms are genuinely equal.
[Theorem 3.6, Eq. (3.44)] The constant I_ℓ is not well defined as stated. Under Assumption 3.3, F(r)=2∫_0^{-r}πB(s)^{-1}ds is negative for r>0, and F(1)−F(−1) is always negative for any positive density πB. Hence the bracket [1+F(M‖ξ‖_1)+F(1)−F(−1)] can be negative, making the integrand in (3.44) negative and I_ℓ imaginary or undefined. This affects the validity of the error bound in Theorem 3.6. The definition should be corrected, for example by using a positive interval integral such as ∫_{-r}^0 πB(s)^{-1}ds, or by taking absolute values, and the proof of (3.50) should be checked with the corrected constant.
[Theorem 3.9, Eq. (3.59) and Eq. (3.76)] The H^1 convergence rate stated in Theorem 3.9 is inconsistent with the proof. The theorem statement gives an exponent (s−d/2)/(6s+9), while the proof in Eq. (3.76) and Remark 3.10 both give (s−d/2)/(4s+6). These differ by a factor of 3/2 in the denominator. Since the H^1 rate is one of the two main advertised results, this is not a cosmetic typo in a lemma; the theorem statement must be corrected to match the derivation, or the derivation must be redone if (3.59) is the intended claim.

minor comments (6)

[Eq. (3.22)] In the display following Eq. (3.22), the notation G(ε) appears in the integrand where G(ξ) is meant; the same confusion appears in the preceding line's bound.
[Section 4, Tables 1–5] All numerical results appear to be single runs with no standard deviations or multiple random seeds. Given that the hidden weights A and B are random, reporting statistics over independent trials would substantially strengthen the empirical claims.
[Section 4.1, Eq. (4.4)] The notation W·σ(A·(x,t)+b) is informal for the sum over features; the dimension of the bias vector b and the role of W₀ in Eq. (3.55) should be clarified.
[Section 4.2] The quantity denoted E_r^G in Tables 3 and 4 is not defined in the text; it appears to be the relative L² error, but this should be stated explicitly.
[Remark 3.10] The remark interprets the rates as tending to 1/4 and 1/10, but the abstract and introduction do not mention that these limits are obtained only as s→∞; a sentence clarifying the regularity requirement for dimension-independent rates would prevent overstatement.
[Introduction] The sentence 'there are no available results on the approximation error of using randomized neural networks in high-dimensional PDEs' is too strong in view of the cited work of Gonon [11] on Black-Scholes type PDEs; the novelty should be phrased as Sobolev-norm approximation for RaNNs specifically.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; theoretical rates rest on externally cited representation formulas and explicit error bounds, not on fitted parameters or self-citation chains.

full rationale

The derivation chain in Section 3 is not circular. The starting representation formula (3.8) is taken from Gonon (2023) and the Monte Carlo concentration lemma (Lemma 3.5) from Grohs et al. (2023), both external to the present authors; these are independent supports, and the paper's assumptions (Assumptions 3.1-3.3, 3.7) do not include the target approximation rates. The constants I_l and I*_l are upper bounds derived from the Fourier representation, not fitted to data, and the network weights W are constructed measurably rather than optimized against the target u. The Sobolev rates in Theorems 3.6 and 3.9 are proved by chaining the representation formula, the smooth-Heaviside approximation error, and a Monte Carlo estimate; no step redefines the target as the model. The numerical experiments in Section 4 are benchmarks against known exact or reference solutions and are not used to derive the theoretical rates. Self-citations (e.g., [4]-[6]) appear only as background on PINN training and error analysis and are not load-bearing. The contested equality (3.12) and the I_l sign issue are mathematical gaps or potential errors in the proof as written, not circular reductions: the second integral form is asserted, not shown to be the conclusion by construction. Even if the proof is incomplete, the argument is not equivalent to its inputs. Therefore no circularity is present.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The theoretical results rest on the Gonon representation formula, integrability assumptions on random weight densities, and Fourier inversion for Sobolev functions. The proof also depends on the asserted equality in Eq. (3.12), which is not justified and is in fact false; this is the main weakness. No new physical entities are introduced.

free parameters (4)

gamma (Theorem 3.6 and 3.9 rate parameter) = gamma in (0,1/2), e.g., tending to 1/2
The H^2 convergence exponent contains gamma/(2(2+gamma)) or gamma/(gamma+2) * (s-d/2)/(2s+3). The constant C_gamma grows as gamma approaches 1/2, trading rate against constant size.
epsilon and delta (smoothing and boundary-layer parameters) = powers of N and R chosen in the proofs
These are chosen by hand in the proofs to balance approximation and Monte Carlo errors. They are not fitted to data but are free choices that determine the final rate.
beta_1, beta_2 (boundary scaling hyperparameters) = 5, 10, 50, 100 depending on dimension and problem
In the Black-Scholes and Heston experiments, these hyperparameters balance the interior PDE residual against boundary conditions. They are chosen manually per dimension and affect the numerical error.
Hidden weight sampling range = U(-0.01, 0.01) for heat, U(-0.1, 0.1) for finance
The uniform range for random hidden weights is chosen by hand. The scale strongly affects the condition of the least-squares system and the expressivity of the network.

assumptions (4)

standard math Representation formula (3.8): u(x) = ∫∫ (x·ξ+u)_+ α(ξ,u) du dξ for Fourier-type functions (Gonon 2023, eq. 10).
Borrowed from [11] as the starting point for constructing H_ε approximations and Monte Carlo random features.
domain assumption Assumptions 3.1, 3.2, 3.3: A and B have strictly positive Lebesgue densities, Y uniform on [0,2M||A||_1+1], and F(r) finite.
These conditions define the class of random weight distributions under which the expectation representation and the bounds on I_ℓ^* are derived.
standard math Fourier inversion for u ∈ H^{4+s}(R^d): u(x)=∫ e^{ix·ξ} G(ξ)dξ with G related to the Fourier transform of u.
Used in Lemma 3.8 and Theorem 3.9 to move from the abstract Fourier-integral class to concrete Sobolev functions.
ad hoc to paper Equation (3.12): ∫_0^{x·ξ+u} H_ε(y)dy = ∫_0^{B(ξ)} H_ε(x·ξ+u−y)dy for finite ε.
This asserted equality is load-bearing for Theorems 3.6 and 3.9, but it is false because H_ε is strictly positive. The proof provides no bound for the difference, so the theorems are not established.

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Pith. "Pith review of Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs." pith.science (2026). https://pith.science/paper/ECGJB44O

@misc{pith2026250112145,
  author       = {Pith},
  title        = {Pith review of: Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECGJB44O}},
  note         = {Machine review of arXiv:2501.12145}
}

abstract

We present approximation results and numerical experiments for the use of randomized neural networks within physics-informed extreme learning machines to efficiently solve high-dimensional PDEs, demonstrating both high accuracy and low computational cost. Specifically, we prove that RaNNs can approximate certain classes of functions, including Sobolev functions, in the $H^2$-norm at dimension-independent convergence rates, thereby alleviating the curse of dimensionality. Numerical experiments are provided for the high-dimensional heat equation, the Black-Scholes model, and the Heston model, demonstrating the accuracy and efficiency of randomized neural networks.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A. R. Barron. Universal approximation bounds for superp ositions of a sigmoidal function. IEEE Transactions on Information theory , 39(3):930–945, 1993

work page 1993

[2] [2]

J. Chen, X. Chi, Z. Yang, et al. Bridging traditional and m achine learning-based algorithms for solving PDEs: the random feature method. J Mach Learn , 1:268–98, 2022

work page 2022

[3] [3]

Dang and F

H. Dang and F. W ang. Local randomized neural networks wit h hybridized discontinuous Petrov–Galerkin methods for Stokes–Darcy ﬂows. Physics of Fluids , 36(8), 2024

work page 2024

[4] [4]

De Ryck, F

T. De Ryck, F. Bonnet, S. Mishra, and E. de Bézenac. An oper ator preconditioning perspective on training in physics-informed machine learning. International Conference on Learning Representations , 2024

work page 2024

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