{"id":"13378c87-f215-41ae-8803-2ecfa8f9505f","arxiv_id":"2502.01938","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"K-HOrderDNN approximates univariate KST components with high-order networks, reducing basis count from (p+1)^d to about d(p+1) and showing strong accuracy on high-frequency, high-dimensional PDE tests.","lead":"This paper proposes a neural network for solving high-dimensional PDEs with rapidly oscillating solutions. It uses Kolmogorov's decomposition into one-dimensional pieces, cutting the number of basis functions from exponential to roughly linear in the dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"CoD theorem applies only to K_L with Lipschitz K-outer functions; no evidence the Section 4 PDE solutions lie in K_L, and the slow p^{-0.301} rate cannot explain the p=7 accuracies, so the practical CoD claim is not established.","rationale":"I read the paper in good faith: the theoretical framework is a plausible and structurally coherent extension of KST-based approximation, and the architectural change from (p+1)^d tensor-product basis functions to d(p+1) univariate basis functions is a genuine fix for the parameter explosion in HOrderDNN. Theorem 2's proof is essentially complete for the ReLU case, apart from minor constant looseness, and the parameter count formula is correct for the special class Kσ_{p,n}. The reader's weakest_assumption correctly identifies K_L membership as the main condition. I partially agree, and I sharpen the concern: the theorem's proved rate p^{-0.301} is so slow that it cannot account for the reported p=7 and p=9 errors in the numerical PDE tables. This suggests either the tested functions are far smoother in their KST representation than the worst case (unproven and unlikely for fixed pathological inner functions), or the implemented network is not operating within the analyzed class. Either way, the central claim that K-HOrderDNN 'breaks the curse of dimensionality' for the demonstrated high-dimensional PDE problems is not directly supported by the theory. I do not think this requires a change to the reader's conditional verdict: the theoretical result is still an interesting conditional statement, and the numerical results may be reproducible and practically valid. The missing Lemma E3 in Appendix E is a separate proof gap affecting the tanh extension, but I did not make it the primary concern because the K_L membership and rate mismatch bear more directly on the central claim. A concrete computational test with the special class Kσ_{p,n} would settle whether the theorem's mechanism is actually the one producing the reported accuracies.","tokens_in":28697,"tokens_out":11590,"duration_ms":125254,"concrete_test":"Implement the exact class Kσ_{p,n} from Section 3 for a 10D target such as Eq. (4.8), using a fixed constructive family of KST inner functions with Hölder exponent α=log_10 2, and measure the error as p and n increase. Check whether the error follows O(1/n + 1/p^α) and whether p=7,9 can reach the 10^{-3}-level errors reported in Table 10. If Kσ_{p,n} cannot reproduce those accuracies at these p values, then the practical CoD claim is not explained by Theorem 2, and the paper would need either to prove the PDE solutions lie in K_L or to analyze the actual learned-inner-function architecture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central guarantee is Theorem 2: for f in K_L, approximation by Kσ_{p,n} has error ≤ C_g d(2d+1)(1/n + 2KC_φ/p^α), α=log_10 2, with 2n+d+(p+1)(2d+1) parameters. The paper's abstract and conclusion phrase this as K-HOrderDNN 'breaking the curse of dimensionality,' but K_L is only dense in C([0,1]^d), and the paper does not show that any of the PDE solutions in Section 4 belongs to K_L. The tested solutions are products and sums of high-frequency sines (Eqs. 4.5–4.8); no argument or computation shows their K-outer functions are Lipschitz, and for the fixed Hölder KST inner functions there is no reason to expect this. Moreover, the theorem's rate in p is p^{-0.301}: to make the bound reach 4.4e-3 as in Table 10, one would need p of order 10^8, not p=7. The paper itself attributes the observed faster-than-p^{-α} convergence to the possibly higher smoothness of the learned inner functions (Section 4.2.5), i.e. to behavior outside the analyzed KST construction. Thus the CoD-free approximation result for K_L does not, by itself, establish tractability of the actual K-HOrderDNN for the high-dimensional high-frequency PDEs tested; the unverified K_L membership and the extreme gap between the theoretical rate and the numerical p-values are the load-bearing weakness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes the Kolmogorov high-order deep neural network (K-HOrderDNN), a variant of HOrderDNN in which the tensor-product high-order layer, whose size is (p+1)^d, is replaced by a KST-inspired construction: a shared high-order univariate subnetwork h_p approximates the K-inner functions, and a fully connected subnetwork approximates the K-outer function. The paper proves approximation rates for a special class of ReLU K-HOrderDNNs (Theorem 2) and a tanh version (Theorem 4), giving parameter counts 2n+d+(p+1)(2d+1) and O(d^2(n^{-1}+p^{-alpha})) error for the Lipschitz-outer class K_L, where alpha=log_10 2; Theorem 3 extends the ReLU result to arbitrary continuous outer functions via the modulus of continuity. Numerical experiments fit high-frequency functions and solve 2D, 10D, 20D, and 50D Poisson and Helmholtz problems, reporting strong empirical gains for K-HOrderDNN(p>1) over PINN and HOrderDNN(p=1), with parameter counts avoiding the exponential (p+1)^d growth. The central limitations are that the CoD-free statement is conditional on membership in K_L, which is not verified for the PDE solutions tested, and the proof of Theorem 4 contains a missing lemma.","tokens_in":29089,"tokens_out":14746,"duration_ms":141752,"significance":"If the theorems were fully established and the K_L-membership gap were closed, the paper would provide a useful explicit mechanism: a dense subclass of C([0,1]^d) approximable with polynomial-in-d parameter count and an algebraic rate O(d^2(n^{-1}+p^{-alpha})). The paper is commendable for stating precise parameter counts, defining the approximation class cleanly, and running a broad numerical comparison against PINN, HOrderDNN, KAN, and TNN, including high-dimensional examples; the factor of roughly 225 improvement in Table 10 is striking. The main value is the architecture and its empirical behavior, while the theoretical CoD claim is narrower than the abstract and conclusion suggest. The paper is also honest in Section 4.2.5 that the faster-than-theory p-convergence is outside the analyzed KST regime. No code is provided, but the experimental setup is described in enough detail to be replicable.","major_comments":[{"comment":"The CoD-free guarantee is stated for f in K_L, the class whose KST outer function is Lipschitz. The paper does not verify that any of the PDE solutions in Section 4 belongs to K_L. The solutions in Eqs. (4.5)-(4.8) are sums and products of trigonometric polynomials; since the KST inner functions are fixed Holder functions with exponent alpha=log_10 2, the Lipschitz property of the outer function is not automatic and has not been checked. Consequently the polynomial-in-d error bound does not, as stated, apply to the target problems in Tables 6, 7, 9, and 10, and the strong high-dimensional results there remain empirical. Moreover, density of K_L in C([0,1]^d) does not transfer a single uniform CoD-free estimate, because the constant C_g in the bound can grow as the approximating member of K_L approaches the target.","section":"Section 3 (Theorem 2) and Section 4.3-4.4"},{"comment":"The proof of Theorem 4 asserts, after Eq. (E8), that |phi_q(x)-bar_L_q(x)| <= 3|phi_q(x)-L_q(x)| \"by applying the triangle inequality, the definition of bar_L_q, and Lemma E3.\" No Lemma E3 is stated or proved anywhere in the manuscript. The needed estimate can in fact be obtained from Lemma E2 with w large enough, which suggests that E3 is a typo, but as it stands the proof of the tanh result is incomplete. In addition, the proof of Lemma E2 contains a limit error: the quantity 1 - w tanh(1/w) tends to 0 as w tends to infinity, not as w tends to 0+.","section":"Appendix E, Theorem 4"},{"comment":"The parameter counts in Theorems 2 and 4 are for the special classes K^sigma_{p,n} (one-hidden-layer ReLU spline outer network) and K^{sigma1}_{p,n} (outer tanh network with layer widths at most N-1 and 6N), not for the implemented K-HOrderDNN, whose g-subnetwork has depth gd at least 2 and width gw. Section 4.2.5 identifies n with gw, but the theoretical n is the number of knots in a linear spline, not the width parameter of a deep fully connected GNN. For example, for the d=2, p=9, hw=45, gw=90 configuration reported in Table 4, Theorem 4's count is (2d+6N+14)(N-1)+13+(p+1)(2d+1), which equals 49,725 if N=90 and 1,575 if N=15; neither matches the reported 14,136 parameters. Thus the proved parameter-efficiency rates are not directly tied to the network used in the experiments.","section":"Section 3 and Section 4.2.5"},{"comment":"The experiments converge in p much faster than the theoretical rate p^{-alpha} with alpha=log_10 2 approx 0.301. In Table 10, K-HOrderDNN(p=7) reaches 4.40E-03, whereas the worst-case p-term in the bound at p=7 is only about 0.55 times its value at p=1, so the theoretical rate cannot explain even one order of magnitude of the observed improvement. Section 4.2.5 explicitly attributes the faster decay to the potentially higher smoothness of the K-inner function, i.e., to behavior outside the analyzed KST construction. Statements such as \"This observation aligns well with Theorem 2\" in Section 4.2.1 therefore overstate the theory's explanatory power for the reported accuracies; either add a theorem for smoother inner functions or confine the theory claim to the n-rate.","section":"Section 4.2.5, Tables 8 and 10"}],"minor_comments":[{"comment":"The abstract says K-HOrderDNN reduces the number of introduced basis functions to d(p+1), while the definition and Theorem 2 use a shared set of p+1 univariate basis functions with (p+1)(2d+1) trainable coefficients. These counts should be reconciled to avoid confusion.","section":"Abstract and Section 2.3"},{"comment":"The tanh class is denoted K^{sigma1}_{p,n}, but its definition and Theorem 4 use N for the outer network size and never define n; the notation should be unified, for example K^{sigma1}_{p,N}.","section":"Section 3, definition of K^{sigma1}_{p,n}"},{"comment":"The sentence \"the depths of PINN and HOrderDNN match the depth of K-HOrderDNN\" is ambiguous because K-HOrderDNN has two subnetworks h_p and g with separate depths hd and gd; the rule L = hd + gd + 1 should be stated explicitly before it is used in the experiments.","section":"Section 4.1"},{"comment":"Several entries in Table 3 are backslashes without an explanatory note; please add a note that HOrderDNN(p>1) was not executed for d=20 and d=50 because the parameter count is prohibitive.","section":"Table 3"},{"comment":"The roles of G1 and G2 in Eq. (2.9) are not specified in terms of input and output dimensions; clarifying these layers would help connect the definition to the theoretical class K^sigma_{p,n} in Section 3.","section":"Section 2.3, Eq. (2.9)"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically competent and the numerical study is broad. The missing Lemma E3 is very likely a typo, since Lemma E2 supplies the needed estimate, so the proof of Theorem 4 is repairable locally. The more serious issue is the scope gap between the theoretical CoD claim (for K_L) and the numerical targets (whose K_L membership is not checked); this needs either a membership criterion for the test problems, an additional theorem for smoother inner functions, or a more cautious wording of the abstract and conclusion. The parameter-count mismatch between the analyzed special class and the deployed network should also be addressed. I do not recommend rejection: the architecture is plausible and the empirical evidence is useful, but the load-bearing claims need revision. The manuscript fits the journal's scope in numerical analysis and scientific computing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read of arXiv:2502.01938. The thing to know: K-HOrderDNN replaces HOrderDNN's (p+1)^d tensor-product basis with d(p+1) univariate high-order basis functions assembled through KST. That is a genuine fix for the parameter explosion, and the numerical study is extensive: function fitting, Poisson and Helmholtz, d=2 to d=50, comparisons against PINN, HOrderDNN, KAN, and TNN, plus a frequency analysis and a study of how errors scale with n and p. The paper earns credit for that breadth, and for reporting honestly that the observed p-convergence is faster than the proved rate, which they attribute to the learned inner functions being smoother than the worst-case KST bound.\n\nThe main soft spot is exactly where the stress-test note lands. Theorem 2 establishes a polynomial-in-d parameter count with error O(d^2(n^{-1} + p^{-alpha})), alpha = log_10 2, but only for f in K_L, the class of continuous functions whose KST outer function is Lipschitz. The paper never shows the high-dimensional PDE solutions in Section 4 belong to K_L. The tested solutions are sums and products of high-frequency sines; nothing there gives a Lipschitz outer function. Additionally, p^{-0.301} is so slow that it cannot explain Table 10's 4.4e-3 error at p=7. So the abstract's \"CoD is mitigated\" should really read \"CoD is mitigated for K_L, a dense but non-constructive class, and we have empirical evidence the method works well on a range of high-frequency PDEs.\" That is an honest conditional result, but the unconditional phrasing oversells it.\n\nSmaller issues: Theorem 4's proof invokes an unstated Lemma E3, the parameter count in the theorem's idealized class does not match the implemented network architecture, and there is no code or seed information, so the strong numbers are single-run observations. None of these are fatal, but they need fixing before this is archival.\n\nBottom line: the paper is worth refereeing. The architecture is useful, the experiments are convincing that K-HOrderDNN is a practical improvement over HOrderDNN in high dimensions, and the theoretical claim is sound conditional on K_L membership. A serious referee should push for a softened CoD claim, a check or discussion of K_L membership for the test problems, a statement of the missing lemma, and code/data release. I would bring this up in reading group and would likely cite it in work on neural PDE solvers.","headline":"A real architectural fix for HOrderDNN's exponential basis count, backed by broad experiments, but the headline CoD claim is conditional on a class K_L that the paper never checks for its own test problems.","tokens_in":29611,"tokens_out":1550,"would_cite":true,"duration_ms":18072,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T99","35Q68","65N99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a neural network built from the Kolmogorov superposition theorem approximates high-frequency, high-dimensional PDE solutions with parameter count polynomial in the dimension, breaking the curse of dimensionality for…","keywords":["Kolmogorov superposition theorem","high-order neural networks","curse of dimensionality","high-frequency PDEs","physics-informed neural networks","Helmholtz equation","high-dimensional approximation"],"falsifier":"Take a ten-dimensional target function whose Kolmogorov outer function is explicitly non-Lipschitz, for example by constructing the KST representation with $g(z)=\\sqrt{z}$, and train K-HOrderDNN with increasing $p$; if the error continues to decay at roughly $p^{-0.30}$ or faster, the Lipschitz assumption is not the operative constraint, while if the error stalls or decays much slower, the result is confined to the dense subclass $\\mathcal{K}_L$ and does not extend to arbitrary continuous targets.","tokens_in":28495,"feed_emoji":"🧮","tokens_out":7235,"duration_ms":58301,"temperature":0.7,"pith_summary":"This paper introduces K-HOrderDNN, a neural network architecture that combines Kolmogorov's Superposition Theorem with high-order polynomial basis layers to solve high-frequency PDEs in high dimensions without the exponential parameter growth that limits earlier HOrderDNNs. The central claim is that for a dense subclass of continuous functions, the approximation error decays as $O(d^2(n^{-1}+p^{-\\log_{10}2}))$ using only $2n+d+(p+1)(2d+1)$ parameters, so the curse of dimensionality is broken. If correct, this means high-frequency problems like the ten-dimensional Helmholtz equation become tractable for physics-informed training; the paper reports a relative L2 error of $4.40\\times10^{-3}$ for K-HOrderDNN($p=7$) where the previous method only reaches $9.93\\times10^{-1}$. The practical payoff would be a neural PDE solver whose parameter count grows polynomially while retaining the high-order basis advantage for oscillatory solutions.","feed_headline":"Kolmogorov trick tames high-frequency PDEs in high dimensions","feed_subtitle":"A 10D Helmholtz error drops from 0.99 to 0.0044 by replacing exponential basis growth with univariate nets.","key_machinery":"The load-bearing mechanism is the Kolmogorov Superposition Theorem in Lorentz's form, which represents any continuous $d$-variate function as a sum over $2d+1$ univariate inner functions $\\varphi_q$ composed with a single outer function $g$: $f(x_1,\\ldots,x_d)=\\sum_{q=0}^{2d}g(\\sum_{i=1}^d\\lambda_i\\varphi_q(x_i))$. The paper's architecture makes this concrete: the inner functions are learned by a high-order network $h_p$ (polynomial basis of degree $p$, applied to each coordinate with shared weights), and the outer function by a fully connected network $G_{\\mathrm{NN}}$, giving the whole ansatz $k_p(x)=G_2\\circ G_{\\mathrm{NN}}\\circ\\sigma\\circ G_1([h_p(x_1),\\ldots,h_p(x_d)]^T)$. The approximation argument then splits the error into two parts: Jackson's theorem bounds the polynomial approximation of the H\\\"older-$\\alpha$ inner functions by $p^{-\\alpha}$, and a linear-spline (ReLU) or tanh network bound of the Lipschitz outer function by $n^{-1}$; the product of the two error terms with the $2d+1$ sums produces the $O(d^2(\\cdot))$ rate.","core_discovery":"K-HOrderDNN replaces the $(p+1)^d$ tensor-product basis functions of HOrderDNN with $d(p+1)$ univariate basis functions arranged through the Kolmogorov–Lorentz representation $f(x)=\\sum_{q=0}^{2d}g(\\sum_{i=1}^d\\lambda_i\\varphi_q(x_i))$. The inner functions $\\varphi_q$ are approximated by a shared high-order subnetwork $h_p$ applied coordinate-wise, and the outer function $g$ by a fully connected subnetwork; the construction mirrors the proof of Theorem 2, which shows that when $g$ is Lipschitz the error is bounded by a constant times $d^2(1/n+1/p^\\alpha)$ with $\\alpha=\\log_{10}2$. The paper claims this breaks the curse of dimensionality for the dense subset $\\mathcal{K}_L$ of continuous functions, and demonstrates numerically that in dimensions 10, 20, and 50 K-HOrderDNN($p>1$) remains trainable and accurate while HOrderDNN($p>1$) becomes intractable.","pith_inferences":["The reported experiments show faster-than-$p^{-0.30}$ decay, which the paper attributes to smoothness; measuring the empirical $p$-exponent across problems with controlled smoothness would separate the worst-case guarantee from the smooth case, and is a natural test the paper does not perform.","Because the KST inner functions are universal (independent of $f$), the shared subnetwork $h_p$ could potentially be pre-trained on one target and fine-tuned for another high-dimensional PDE at nearly constant cost, an extension the paper leaves implicit.","The paper's comparison with KAN shows comparable accuracy at roughly twelve times faster training, but it does not test highly oscillatory outer functions; a natural extension would be to check whether the architecture retains its advantage when $g$ itself carries the high-frequency content.","The method inherits the Monte Carlo sampling error of the PINN loss, and the paper suggests insufficient sample size at $d=50$; a hybrid with quadrature-based losses, as used in tensor neural networks, is an untested direction that could tighten the gap between theory and practice at larger dimensions."],"forward_implications":["For target functions whose KST outer function is Lipschitz, the parameter count $2n+d+(p+1)(2d+1)$ is polynomial in the dimension $d$, so the method stays trainable as $d$ grows.","K-HOrderDNN($p>1$) inherits the high-order basis advantage of HOrderDNN for oscillatory problems but removes the exponential $(p+1)^d$ blow-up, so the order $p$ can be increased in high dimensions without an explosion in parameters.","In the ten-dimensional Helmholtz benchmark the reported relative L2 error drops from $9.93\\times10^{-1}$ (HOrderDNN $p=1$ and PINN) to $4.40\\times10^{-3}$ (K-HOrderDNN $p=7$), suggesting orders-of-magnitude gains for high-frequency high-dimensional wave problems.","With tanh activation the same CoD-avoidance result holds (Theorem 4), making the construction compatible with the second-derivative losses used in PINN training."],"supporting_citations":[{"why":"Kolmogorov's original superposition theorem, the source of the decomposition that the architecture is built on.","marker":"[17]"},{"why":"Lorentz's sharpened KST with a single outer function and $2d+1$ inner functions, which is the representation Theorem 1 uses.","marker":"[18]"},{"why":"The HOrderDNN method being improved, supplying the high-order basis layer and the high-frequency motivation.","marker":"[15]"},{"why":"Introduces the dense class $\\mathcal{K}_L$, the linear-spline approximation of the outer function, and the claim that KST can break the curse of dimensionality; the paper directly extends this to high-order inner approximations.","marker":"[26]"},{"why":"The Jackson theorem that bounds polynomial approximation of the H\\\"older-$\\alpha$ inner functions by $p^{-\\alpha}$ in the proof of Theorem 2.","marker":"[31]"},{"why":"Supplies the tanh neural network approximation result (Lemma E1) used for the tanh version of the theorem.","marker":"[38]"},{"why":"The Kolmogorov-Arnold network baseline compared in the experiments, also KST-inspired.","marker":"[36]"},{"why":"The tensor neural network comparison showing where the method is complementary, handling non-tensor-product source terms.","marker":"[35]"}],"fun_headline_variants":["Kolmogorov structure cuts basis count: high-dim PDEs solved","From exponential to linear: Kolmogorov net solves high-dim PDEs","d(p+1) basis beats (p+1)^d: high-dim PDEs","K-HOrderDNN: solving high-dim PDEs with fewer basis functions","Smaller basis, bigger accuracy: K-HOrderDNN for high-dim PDEs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theoretical guarantee of breaking the curse of dimensionality holds only for functions whose Kolmogorov outer function is Lipschitz continuous, and the paper does not establish that the high-frequency PDE solutions used in the experiments actually lie in that dense subclass.","fun_headline_variants_meta":{"raw":{"variants":["Kolmogorov structure cuts basis count: high-dim PDEs solved","From exponential to linear: Kolmogorov net solves high-dim PDEs","d(p+1) basis beats (p+1)^d: high-dim PDEs","K-HOrderDNN: solving high-dim PDEs with fewer basis functions","Smaller basis, bigger accuracy: K-HOrderDNN for high-dim PDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001409,"raw_usage":{"total_tokens":5768,"prompt_tokens":1098,"completion_tokens":4670,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":714,"completion_tokens_details":{"reasoning_tokens":4566}},"tokens_in":714,"tokens_out":4670,"duration_ms":31452,"temperature":1.0,"reasoning_tokens":4566,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:56:52.434647+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a ten-dimensional target function whose Kolmogorov outer function is explicitly non-Lipschitz, for example by constructing the KST representation with $g(z)=\\sqrt{z}$, and train K-HOrderDNN with increasing $p$; if the error continues to decay at roughly $p^{-0.30}$ or faster, the Lipschitz assumption is not the operative constraint, while if the error stalls or decays much slower, the result is confined to the dense subclass $\\mathcal{K}_L$ and does not extend to arbitrary continuous targets.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kolmogorov's original superposition theorem, the source of the decomposition that the architecture is built on."},{"cited_title":"Lorentz, Metric entropy, widths, and superpositions of functions, The American Mathe- matical Monthly 69 (6) (1962) 469–485","cited_arxiv_id":null,"evidence_quote":"Lorentz's sharpened KST with a single outer function and $2d+1$ inner functions, which is the representation Theorem 1 uses."},{"cited_title":"Chang, K","cited_arxiv_id":null,"evidence_quote":"The HOrderDNN method being improved, supplying the high-order basis layer and the high-frequency motivation."},{"cited_title":"Wang, Numerical Approximation, 2nd Edition, Beijing Higher Education Press, Beijing, 2012","cited_arxiv_id":null,"evidence_quote":"The Jackson theorem that bounds polynomial approximation of the H\\\"older-$\\alpha$ inner functions by $p^{-\\alpha}$ in the proof of Theorem 2."},{"cited_title":"De Ryck, S","cited_arxiv_id":null,"evidence_quote":"Supplies the tanh neural network approximation result (Lemma E1) used for the tanh version of the theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The tensor neural network comparison showing where the method is complementary, handling non-tensor-product source terms."}],"review_version":1}