{"id":"1cf9fadb-4e75-4702-804f-c60ed00069c5","arxiv_id":"2501.12145","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Randomized neural networks are claimed to approximate Sobolev functions in H^1 and H^2 at dimension-independent rates, but the key proof step (Eq. 3.12) is invalid as written.","lead":"This paper proves approximation rates for randomized neural networks in Sobolev norms and tests the method on high-dimensional heat, Black-Scholes, and Heston equations. The main proof contains an incorrect equality that undermines the theoretical claim, though the numerical experiments show the approach can be fast and reasonably accurate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.4's Eq. (3.12) is a false identity, and the un-bounded boundary-layer difference propagates into Theorems 3.6 and 3.9, so the claimed H^2 rates are not proven as submitted.","rationale":"I find the reader's weakest-assumption correct and load-bearing. The central claim is a proof of dimension-independent H^2 approximation rates; that proof relies on identifying the function for which Proposition 3.4 proves approximation with the function the Monte-Carlo estimator represents. Equation (3.12) is the only bridge, and it is false for smooth positive Hε. The difference term has a second derivative concentrated near x·ξ+u=B(ξ), and no estimate for it appears. This is not a matter of disagreement with consensus; it is an internal inconsistency in the proof. I note, however, that the gap is plausibly repairable: for all (ξ,u) in the support of α one has u≤1 and hence x·ξ+u−B(ξ)≤−M‖ξ‖_1, so the missing term is exponentially small and might be bounded by O(ε^{d+1}). A revised version could likely fix the proof. But as submitted, the theorem statements are not supported, and the empirical sections do not compensate because they lack error bars, code, and do not validate the rates. Therefore the reader's REJECT verdict is unchanged.","tokens_in":23608,"tokens_out":28727,"duration_ms":291060,"concrete_test":"Re-derive Proposition 3.4 using only the second line of (3.12) as the definition of uε and add the exact correction D(x)=∫∫F(x·ξ+u−B(ξ))α(ξ,u)dudξ. Prove an explicit bound on ‖D‖_{H^2_μ(B_M^d)}, e.g. by using the support of α to show x·ξ+u−B(ξ)≤−M‖ξ‖_1 and hence |Hε′|≤(2/ε)e^{−2M‖ξ‖_1/ε}, giving ‖D‖=O(ε^{d+1}) (or at least O(ε^γ) for some γ>0). If such a bound holds, the gap closes and Theorems 3.6/3.9 survive; if only O(1) or divergent bounds are obtained, the claimed H^2 rates are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 3.4 defines uε by the first integral in (3.12), ∫_0^{x·ξ+u} Hε(y)dy, and then asserts equality with the second integral ∫_0^{B(ξ)} Hε(x·ξ+u−y)dy. Since Hε is strictly positive, this identity is false. Writing F(a)=∫_0^a Hε(y)dy, the second form equals F(x·ξ+u)−F(x·ξ+u−B(ξ)), so the difference between the two forms is −F(x·ξ+u−B(ξ)); its second derivative contains the boundary-layer term −ξ_jξ_l Hε′(x·ξ+u−B(ξ)). The estimates (3.21)–(3.41) apply only to the first form, while the Monte-Carlo network in (3.47) represents the second form. Therefore the error bound in Theorem 3.6, and hence the Sobolev-function result in Theorem 3.9, is missing this unquantified component. No bound on this difference is supplied anywhere in the paper. A secondary defect is that the constant Iℓ in (3.44) uses F(1)−F(−1) with Assumption 3.3's F, which is negative for any density πB, making Iℓ undefined as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23929,"tokens_out":14146,"duration_ms":146474,"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":[{"comment":"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.","section":"Section 3.1, Eq. (3.12)"},{"comment":"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.","section":"Theorem 3.6, Eq. (3.44)"},{"comment":"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.","section":"Theorem 3.9, Eq. (3.59) and Eq. (3.76)"}],"minor_comments":[{"comment":"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":"Eq. (3.22)"},{"comment":"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":"Section 4, Tables 1–5"},{"comment":"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":"Section 4.1, Eq. (4.4)"},{"comment":"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.","section":"Section 4.2"},{"comment":"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.","section":"Remark 3.10"},{"comment":"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.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"I agree with the reader's core assessment: Eq. (3.12) is false and the missing boundary-layer bound invalidates the proof of the main theorems as submitted. I have recommended major revision rather than rejection because the error appears repairable in principle: a compactly supported smooth approximation of the Heaviside function, or an explicit exponential bound for the boundary-layer term, might restore Theorem 3.6, and the I_ℓ and exponent problems are local corrections. However, the burden is on the authors to supply these estimates; if the boundary-layer difference cannot be controlled with the stated rates, the paper should be rejected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou 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.\n\nWhat 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.\n\nThe 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.\n\nThere 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":24470,"tokens_out":2998,"would_cite":false,"duration_ms":30771,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65C05","41A25","65M75"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that randomized neural networks achieve dimension-independent Sobolev approximation rates, making them a practical tool for high-dimensional PDEs.","keywords":["randomized neural networks","extreme learning machines","physics-informed learning","curse of dimensionality","Sobolev approximation","high-dimensional PDEs","Monte Carlo approximation","tanh activation"],"falsifier":"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.","tokens_in":23359,"feed_emoji":"🎲","tokens_out":11977,"duration_ms":112696,"temperature":0.7,"pith_summary":"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.","feed_headline":"Randomized nets get dimension-free error rates","feed_subtitle":"Proof plus 100-dim tests for heat, Black–Scholes, and Heston equations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the representation formula u(x)=∫∫(x·ξ+u)_+ α(ξ,u)dudξ used as the starting point of the approximation proof.","marker":"[11]"},{"why":"Provides the refined representation and approximation bounds for random neural networks that the paper adapts to the smooth tanh setting.","marker":"[12]"},{"why":"Gives the concentration inequality (Corollary 2.5) used in Lemma 3.5 to bound the Monte Carlo sampling error between the expectation and the finite-width network.","marker":"[13]"},{"why":"Introduces the physics-informed extreme learning machine framework that the numerical experiments use to solve the PDEs.","marker":"[8]"}],"fun_headline_variants":["RaNNs break curse of dimensionality for high-dim PDEs","Dimension-independent error rates proven for randomized nets","Randomized neural nets tame high-dimensional PDE problems","RaNNs achieve dimension-free accuracy for PDE solving","Dimension-free approximation bounds for random neural nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["RaNNs break curse of dimensionality for high-dim PDEs","Dimension-independent error rates proven for randomized nets","Randomized neural nets tame high-dimensional PDE problems","RaNNs achieve dimension-free accuracy for PDE solving","Dimension-free approximation bounds for random neural nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001087,"raw_usage":{"total_tokens":4492,"prompt_tokens":845,"completion_tokens":3647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":3574}},"tokens_in":461,"tokens_out":3647,"duration_ms":27071,"temperature":1.0,"reasoning_tokens":3574,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:31:55.043908+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the representation formula u(x)=∫∫(x·ξ+u)_+ α(ξ,u)dudξ used as the starting point of the approximation proof."},{"cited_title":"Gonon, L","cited_arxiv_id":null,"evidence_quote":"Provides the refined representation and approximation bounds for random neural networks that the paper adapts to the smooth tanh setting."},{"cited_title":"Grohs, F","cited_arxiv_id":null,"evidence_quote":"Gives the concentration inequality (Corollary 2.5) used in Lemma 3.5 to bound the Monte Carlo sampling error between the expectation and the finite-width network."},{"cited_title":"Dwivedi and B","cited_arxiv_id":null,"evidence_quote":"Introduces the physics-informed extreme learning machine framework that the numerical experiments use to solve the PDEs."}],"review_version":1}