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REVIEW 4 major objections 5 minor 37 references

A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2502.01938 v1 pith:6BSIV4WD submitted 2025-02-04 math.NA cs.NA

Yaqin Zhang , Ke Li , Zhipeng Chang , Xuejiao Liu , Yunqing Huang , Xueshuang Xiang This is my paper · ORCID
classification math.NAcs.NA MSC 68T9935Q6865N99
keywords Kolmogorovsuperpositiontheoremhigh-orderneuralnetworkscurseofdimensionalityhigh-frequencyPDEsphysics-informedHelmholtzequationhigh-dimensionalapproximation
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. 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.

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 (4)
  1. [Section 3 (Theorem 2) and Section 4.3-4.4] 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.
  2. [Appendix E, Theorem 4] 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+.
  3. [Section 3 and Section 4.2.5] 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.
  4. [Section 4.2.5, Tables 8 and 10] 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.
minor comments (5)
  1. [Abstract and Section 2.3] 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.
  2. [Section 3, definition of K^{sigma1}_{p,n}] 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}.
  3. [Section 4.1] 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.
  4. [Table 3] 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.
  5. [Section 2.3, Eq. (2.9)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CoD bound is derived from external KST, Jackson, and spline approximation lemmas; self-citations are architectural precedents, not load-bearing.

full rationale

The central derivation chain is self-contained. Theorem 2 combines the external Kolmogorov Superposition Theorem (Theorem 1), Jackson's polynomial approximation bound (Lemma 1), and a standard linear-spline estimate (Lemma 2, from Lai and Shen [26]) to bound the error of the special network class Kσ_{p,n}, whose definition (Eqs. (3.1)-(3.3)) is exactly the KST formula with polynomial-clipped inner approximants and a ReLU-spline outer approximant. The hypothesis class K_L is a regularity condition on the KST outer function, not on the network, and the stated error O(d^2(n^{-1}+p^{-α})) with parameter count 2n+d+(p+1)(2d+1) is proved rather than assumed. The citations to the authors' earlier HOrderDNN work ([15], [16]) supply the high-order transformation layer as an architectural ingredient, but the CoD argument does not reduce to those papers. The numerical results in Section 4 are benchmark comparisons against PINN and HOrderDNN, not fitted parameters relabeled as predictions. Section 4.2.5 explicitly admits that the observed p-convergence is faster than the theoretical O(p^{-α}) and attributes this to possibly higher smoothness of the K-inner functions; that is an acknowledged scope limitation, as is the unverified membership of the Section 4 PDE solutions in K_L. These are correctness or applicability concerns, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central theoretical result is a KST-based approximation bound; its main assumptions are the KST representation and membership in K_L. The implemented network adds trainable weights far beyond the idealized parameter count, and no evidence links PDE solutions to K_L.

free parameters (4)
  • p (polynomial degree/order) = 1,3,5,7,9,40 in experiments
    Controls the number of univariate high-order basis functions; selected by hand and swept in experiments rather than derived.
  • n (spline knots, approximated by gw) = gw = 45 to 315 in experiments
    Controls outer-function approximation; the idealized class K_sigma_p,n uses n knots, but the implemented GNN width is set independently.
  • lambda_i (KST arithmetic coefficients) = learned during training
    KST guarantees existence of irrational lambda_i; the network optimizes them along with weights, so the implemented approximation depends on these fitted values.
  • All trainable DNN weights in hp and GNN = not reported (Xavier init, Adam)
    The actual network has far more parameters than the idealized count in Theorem 2; Table 3 shows about 9.4E+04 parameters for d=10.
assumptions (5)
  • standard math Kolmogorov Superposition Theorem (Theorem 1) holds for every continuous f on [0,1]^d with inner functions in Lip(log_10 2).
    Assumed as the foundation; used in Section 2.2 and all approximation theorems.
  • standard math The class K_L of functions with Lipschitz KST outer function is dense in C([0,1]^d).
    Cited to Lai-Shen [26]; used to claim a dense subset is tractable.
  • domain assumption PDE solutions in the experiments belong to K_L or have controlled outer-function modulus of continuity.
    Never verified; without it, Theorem 2 and Theorem 3 error bounds do not attach to the tested problems.
  • domain assumption Training by Adam finds a network close to the best approximation in K_sigma_p,n.
    The theory is an existence result in function space; nonconvex optimization is not analyzed.
  • ad hoc to paper The unstated bound |phi_q - bar_L_q| <= 3|phi_q - L_q| (missing Lemma E3) is true.
    Appendix E Step 2 relies on this to prove Theorem 4; the bound is plausible but not proved in the text.

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Cite this review

Pith. "Pith review of A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions." pith.science (2026). https://pith.science/paper/6BSIV4WD

@misc{pith2026250201938,
  author       = {Pith},
  title        = {Pith review of: A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BSIV4WD}},
  note         = {Machine review of arXiv:2502.01938}
}
abstract

This paper proposes a Kolmogorov high order deep neural network (K-HOrderDNN) for solving high-dimensional partial differential equations (PDEs), which improves the high order deep neural networks (HOrderDNNs). HOrderDNNs have been demonstrated to outperform conventional DNNs for high frequency problems by introducing a nonlinear transformation layer consisting of $(p+1)^d$ basis functions. However, the number of basis functions grows exponentially with the dimension $d$, which results in the curse of dimensionality (CoD). Inspired by the Kolmogorov superposition theorem (KST), which expresses a multivariate function as superpositions of univariate functions and addition, K-HOrderDNN utilizes a HOrderDNN to efficiently approximate univariate inner functions instead of directly approximating the multivariate function, reducing the number of introduced basis functions to $d(p+1)$. We theoretically demonstrate that CoD is mitigated when target functions belong to a dense subset of continuous multivariate functions. Extensive numerical experiments show that: for high-dimensional problems ($d$=10, 20, 50) where HOrderDNNs($p>1$) are intractable, K-HOrderDNNs($p>1$) exhibit remarkable performance. Specifically, when $d=10$, K-HOrderDNN($p=7$) achieves an error of 4.40E-03, two orders of magnitude lower than that of HOrderDNN($p=1$) (see Table 10); for high frequency problems, K-HOrderDNNs($p>1$) can achieve higher accuracy with fewer parameters and faster convergence rates compared to HOrderDNNs (see Table 8).

Figures

Figures reproduced from arXiv: 2502.01938 by the authors.

Figure 1
Figure 1. Illustration of the architecture of HOrderDNN( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustration of K-HOrderDNN(p=2) for d=2. Compared to HOrderDNN(p=2) decipted in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The change in relative L2 errors with respect to hw (the first row) and the number of trainable parameters (the second row) for PINN, HOrderDNNs, and K-HOrderDNNs on problem (4.1) when d=2. Each column corresponds to a different setting of hd, specifically hd = 1, 2, and 3. We set hw = 45, hd = 1 to further investigate the effect of increasing gw and gd on error reduction. As we can observe, in almost all cases, whe… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: The change in relative L2 errors with respect to gw (the first row) and the number of trainable parameters (the second row) for K-HOrderDNN(p) on the problem (4.1) when d=2. Each column corresponds to a different setting of gd, specifically gd = 2, 3, and 4. 4.2.2 Comp…
Figure 5
Figure 5. Figure 5: Convergence processes of HOrderDNNs (the first row), K-HOrderDNNs (the second row), and PINN [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Absolute pointwise errors of PINN (the first row), HOrderDNNs (the second row), and K-HOrderDNNs [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Frequency analysis at different epochs for PINN (the first row), HOrderDNNs (the second row), and [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: The relative L2 error Trends for ReLU activation function (first row) and Tanh activation func￾tion(second row): (a) Loss vs n for fixed p=40; (b) Loss vs p for fixed n=200; (c) Loss vs N for fixed p=30 with n=6N; (d) Loss vs p for fixed n=250 [PITH_FULL_IMAGE:figures…
Figure 9
Figure 9. Figure 9: Convergence Processes of PINN, HOrderDNNs and K-HOrderDNNs on problem ( [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Exact (a) and numerical solutions of PINN (b), HOrderDNNs (c−g), and K-HOrderDNNs (h−l) on problem (4.4) when d=2. Here, hd=3, hw=45, gd=2, gw=90 for K-HOrderDNNs, and L=6, W=220 for HOrderDNNs and PINN [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Exact solution and absolute pointwise errors of PINN, HOrderDNN(9), and K-HOrderDNN(9) in [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Comparative analysis of absolute pointwise errors for PINN, HOrderDNN( [PITH_FULL_IMAGE:figures/full_fig_p024_12.png]
Figure 13
Figure 13. Figure 13: The absolute pointwise errors obtained by PINN, HOrderDNN( [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: Convergence processes of PINN, HOrderDNNs and K-HOrderDNNs on problem ( [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
Figure 15
Figure 15. Figure 15: Exact solution (a) and numerical solutions given by PINN (b), HOrderDNNs (the second row), K-HOrderDNNs (the third row) on problem (4.9) when d=2. Here, hd=3, hw=45, gd=2, gw=90 for K￾HOrderDNNs, and L=6, W =220 for HOrderDNNs and PINN. approximate high-frequency solu…
Figure 16
Figure 16. Figure 16: Comparative analysis of absolute pointwise errors for PINN, HOrderDNN( [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]
Figure 17
Figure 17. Figure 17: Comparative analysis of absolute pointwise errors for PINN, HOrderDNN( [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]

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