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REVIEW 4 major objections 6 minor 46 references

A novel number-theoretic sampling method for neural network solutions of partial differential equations

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that good lattice point sets shrink the expected L2 error bound of physics-informed neural network solutions.

desk verdict A useful empirical study of GLP sampling for PINNs, but the claimed theoretical advantage does not hold at the tested N and d values, and the proofs have gaps that need fixing. read the letter →

arxiv 2411.17039 v7 pith:CQYXIJRP submitted 2024-11-26 math.NA cs.NA

Yu Yang , Pingan He , Xiaoling Peng , Qiaolin He This is my paper · ORCID
classification math.NAcs.NA MSC 65C0565D3068T0765N35
keywords physics-informedneuralnetworksgoodlatticepointsetsquasi-MonteCarlodiscrepancyerrorboundshigh-dimensionalPDEslow-regularityPDEsampling
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 sets out to show that the accuracy of physics-informed neural networks (PINNs) can be improved without any adaptive resampling, simply by feeding the loss function a deterministic good lattice point set instead of independent uniform random points. The theoretical engine is a comparison of quadrature errors: for N interior points, uniform random Monte Carlo estimates the residual integral with error O($N^{{-1/2}}$), while a good lattice point set achieves O((log N)^d / N), which is smaller once N is large. Under standard Lipschitz, gradient-bounded, PL-star and stability assumptions, the paper derives an upper bound on the expected squared L2 error of the trained approximate solution, and shows the lattice-sampling bound is below the uniform-random bound (Theorem 3 and Corollary 1). Numerically, GLP sampling matches or beats uniform random sampling across peaked two-dimensional Poisson problems, a two-dimensional inverse Helmholtz problem, and five- and eight-dimensional linear and nonlinear equations, sometimes using about five times fewer residual points. The practical payoff is a drop-in sampling change that reduces estimation error in PINNs with no extra training cost or hyperparameter tuning.

What carries the argument

The load-bearing object is the good lattice point set (GLP set): deterministic points x_{ij} = (2 q_{ij} - 1)/(2N), where q_{ij} ≡ i h_j (mod N) and (h_1, ..., h_d) is a generating vector chosen so the set has minimal star discrepancy. These points replace i.i.d. uniform samples in the residual-loss quadrature, so the Koksma-Hlawka-type bound of Lemma 1 applies and the discrete loss converges to the integral loss at rate O((log N)^d/N). The proof then chains three standard mechanisms: the PL* condition for over-parameterized networks (Lemma 2) gives convergence of the empirical loss along gradient descent; the variance-growth assumption links the sampling error s(N_r,N_b) to the optimization limit; and the stability Assumption 4 converts residual and boundary losses into the L2 error of the solution.

What would settle it

Train the peaked two-dimensional Poisson example while recording the Hardy-Krause variation of the residual at regular epochs. If the variation diverges during training yet GLP still outperforms uniform random, Lemmas 1(ii) and 3 are not the reason; if a non-smooth activation such as ReLU removes the GLP advantage at the same sample counts, that would indicate the smoothness assumption, not the discrepancy, carries the result.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the estimation error of a PINN is controlled by the discrepancy of the residual training points, and that low-discrepancy lattice points dominate random points in the bound. For a linear PDE satisfying the stability estimate C1||u||2 <= ||Lu||2 + ||Bu||2 <= C2||u||2, Theorem 3 gives ||u* - u($\theta$)||2 <= $\sqrt$(2/C1)( empirical residual loss + empirical boundary loss + O((log N_r)^d/N_r) + O($N_b^{{-1/2}}$) )^{1/2} for GLP interior points, versus a bound with O($N_r^{{-1/2}}$) for uniform random interior points. Combining this with Theorem 1, which says the expected empirical loss converges to (eta L C_V/(2 c mu)) s(N_r,N_b), Corollary 1 states that the limit of the expected squared L2 error is smaller for GLP sampling because (log N_r)^d/N_r is eventually smaller than $N_r^{{-1/2}}$. The paper presents numerical gains as confirmation that the error-bound gap is visible in practice.

Load-bearing premise

The load-bearing premise is that the PDE residual r(x;theta) has bounded Hardy-Krause variation with a constant that remains under control throughout the whole training trajectory; the paper cites the smoothness of tanh and sigmoid activations as justification, but never proves the variation constant stays bounded as network weights grow.

Editorial extensions

If this is right

  • At a fixed sample budget N_r, the gap between the GLP and uniform-random error bounds grows as N_r increases, so lattice sampling pays off most in low-regularity and high-dimensional problems where many residual points are needed.
  • Because the boundary loss still uses uniform random points, the total error rate is capped by O(N_b^{-1/2}); swapping the interior points alone gives the full benefit only when boundary sampling is also improved or boundary conditions are enforced exactly.
  • GLP sampling needs no initial solution, no residual estimator and no adaptive hyperparameters, so it can be inserted into an existing PINN code by replacing only the interior point generator.
  • In high dimension d the factor (log N_r)^d can weaken the advantage unless N_r is large enough; the paper's 5- and 8-dimensional experiments show a clear advantage at the tested budgets, and Remark 3 concedes the limit of small N_r.

Reading between the lines

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

  • The low-discrepancy substitution is generic: any deep PDE solver whose loss is an integral over the domain (Deep Ritz, weak-form, variational, or adversarial) should inherit a similar estimation-error reduction, since the mechanism is simply quasi-Monte Carlo quadrature.
  • A natural hybrid for sharp-peaked problems is to use a GLP base grid for the bulk and add a small number of residual-indicated or failure-region points; the deterministic grid keeps variance low while the extra points address the low-regularity regions the base grid misses.
  • For inverse problems, the observed noise robustness suggests the discrepancy advantage survives noisy observations; a testable extension would be an error bound for parameter identification that includes the data-fitting regression error explicitly.
  • In very high dimensions, the primitive-root construction used here may be replaced by component-by-component generated lattice rules to control (log N)^d/N, which is a straightforward next step the paper does not pursue.
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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 / 6 minor

Summary. The paper proposes to replace uniform random sampling in the PINN empirical loss by good lattice point (GLP) sets generated by number-theoretic methods. It gives an algorithm for constructing the GLP residual points, formulates a PINN loss with GLP residual sampling and uniform random boundary sampling, and proves Theorems 1-3 plus Corollary 1 which are intended to show that the expected squared L2 error of the PINN solution has a smaller upper bound under GLP sampling than under uniform random sampling. Numerical experiments cover a two-dimensional Poisson equation with sharp peaks, a two-dimensional inverse Helmholtz problem with noisy data, and five- and eight-dimensional linear and nonlinear PDEs, comparing GLP sampling with uniform random, LHS, Halton, Hammersley, and Sobol sampling.

Significance. If the theoretical comparison were rigorous, the contribution would be useful: low-discrepancy deterministic sampling is a natural way to reduce the estimation error in PINNs, and the paper includes a concrete algorithm, comparisons against several standard sampling methods, and robustness studies under different sampling budgets and noise levels. The deterministic nature of GLP sets and the explicit construction in Definition 2 and Algorithm 1 are strengths. However, the central theoretical claim is not yet established as stated: the probabilistic-to-deterministic use of Lemma 1(i), the unquantified Hardy-Krause variation constant in the QMC rate, and the mismatch between the asymptotic comparison and the point counts used in the high-dimensional experiments are load-bearing gaps. The paper also ships no code, and one numerical table contains an internally inconsistent entry, so the experimental support needs verification.

major comments (4)
  1. [Section 3.1, Lemma 1(i); Section 3.2, Theorem 3(ii), Eq. (25)] Lemma 1(i) is explicitly identified in Remark 1(1) as a probabilistic error bound, but the proof of Theorem 3 uses it as a deterministic inequality ∫|r|² ≤ (1/N)Σ|r|² + O(N^{-1/2}) for the particular trained network. This is not valid pathwise: for a given random sample the empirical average can lie above or below the integral by more than the stated O term, and the constant depends on the sample through variance-type quantities. Consequently the upper bound in Eq. (25) and the corresponding part of Corollary 1 are not established as stated; the argument needs to be recast in expectation/probability with the random point set made explicit.
  2. [Section 3.1, Lemma 1(ii) and Remark 1; Section 3.2, Theorem 3] The QMC error bound in Lemma 1(ii) requires bounded variation in the sense of Hardy and Krause for the integrand r²(x;θ), and the implicit constant in O((log N)^d/N) is the Hardy-Krause variation of r². Remark 1 argues only that tanh/sigmoid activations are C^∞, which gives smoothness, not a controlled variation constant along the optimization path. Assumption 2 bounds the parameter iterates in a ball B(θ0,R) but does not quantify the variation of r² as a function of R; since the weights can grow during training, the variation constant may blow up. Thus the rate in Eq. (24) and the comparison in Eq. (26) are conditional on an unquantified constant. Please either prove a bound on V_HK(r²(·;θ)) for θ∈B(θ0,R), or state the result with the variation constant made explicit.
  3. [Corollary 1 and Section 4.4/4.5, Tables 5-8] The claimed advantage O((log N)^d/N) < O(N^{-1/2}) in Corollary 1 and Remark 3 is an asymptotic statement, and it fails at the point counts used in the high-dimensional experiments. For d=8 and N_r=11215, (log N)^8/N is on the order of 10^3 while N^{-1/2}≈10^{-2}; for d=5 and N_r=10007, (log N)^5/N is still larger than N^{-1/2}. Since Tables 5-8 are exactly the regime where the theorem's comparison is not active, those experiments do not confirm the theoretical prediction of Corollary 1; the observed GLP advantage must be attributed to suppressed constants or to behavior outside the bound. The paper should either report the actual discrepancy values and variation constants for the trained networks, or explicitly restrict the theoretical claim to the asymptotic regime and remove the unconditional wording in the abstract and the contribution list.
  4. [Appendix A, proof of Theorem 1] The chain of inequalities after Eq. (47), '0<cμη≤ cμ²/[L(M_V+...)] < cμ²/(L μ_G²) ≤1', does not follow from Assumption 3(i). The assumption μ ≤ √L/(c μ_G) implies cμ²/(L μ_G²) ≤ 1/(c μ_G^4) only, which is not ≤1 unless c μ_G^4 ≥1. Since the contraction factor 1-cμη must lie in (0,1) for the recursion to converge to the stated limit, an additional step-size condition (e.g., η ≤ 1/(cμ)) or a corrected inequality is needed.
minor comments (6)
  1. [Throughout] There are several typos: 'postive' in Assumption 1, 'mehtod' and 'Sobel sequence' in Section 4.2.1, and 'It is a well known' in Remark 3.
  2. [Assumption 3] The text uses 'N_d^{-1/2}' where the boundary sample count is N_b; make the notation consistent throughout.
  3. [Figures 25 and 26] The caption of Figure 25 reads 'The performance of errors...' but the figure shows sampling points; Figure 26 appears to contain the error plots. The captions should be corrected.
  4. [Table 6] In Table 6, strategy 3 reports e2(𝒖)=1.172×10^-3, which is identical to the e2 value in Table 5 for the five-dimensional problem, while e∞(𝒖)=5.272×10^-3; this looks like a copy/paste error and should be verified.
  5. [References] Reference [24] is missing publication details; the reference list should be completed.
  6. [Section 3.1] The notation distinguishes Ẽ^{GLP}_{r,N_r} from Ẽ^{UR}_{r,N_r} for the residual loss, but the boundary loss is always denoted Ẽ^{UR}_{b,N_b}; state explicitly that boundary points are always uniform random.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GLP-vs-MC error comparison is imported from standard QMC bounds and explicit assumptions, not from fitted parameters or self-citations.

full rationale

The central theoretical chain (Theorem 1, Theorem 3, Corollary 1) uses Lemma 1, which is a standard Koksma-Hlawka/CLT bound attributed to Niederreiter, and Lemma 2 from Liu-Zhu-Belkin. No parameter is fitted to the numerical results; constants such as C1, C_V, L, mu, and c enter as symbolic assumptions (Assumptions 1-4) and are not tuned against Tables 2-8. The bound s(N_r,N_b) = O((log N_r)^d/N_r) + O(N_b^{-1/2}) for GLP is taken from Lemma 1(ii), so the claimed advantage over O(N_r^{-1/2}) is a direct consequence of a cited external QMC result, not of a quantity defined in terms of the paper's own outputs. The only self-citation is [45] in support of Assumption 4, but Assumption 4 is a stability assumption stated by the authors, not a theorem whose content is imported; the proof would stand unchanged if the citation were removed. Remark 3's concession that (log N_r)^d can weaken the advantage for very large d affects the strength of the claim, not its circularity. The numerical experiments are external validation and are not used to construct the proofs. Hence no circular step can be exhibited.

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

No parameters are fitted to data; the theorem constants (C1, C2, L, c, mu, C_V, M_V) are existence assumptions. The central derivation rests on classical QMC error bounds, standard SGDD convergence analysis, and a coercivity assumption on the PDE. No new entities are introduced.

assumptions (4)
  • standard math Koksma-Hlawka inequality and discrepancy bounds for good lattice point sets (Lemma 1)
    Used to bound the quadrature error of the residual loss in Theorem 3 and Corollary 1.
  • standard math Local PL* condition for over-parameterized feedforward networks (Lemma 2, from Liu et al. [25])
    Used to establish the 2c-PL* condition in Assumption 2, which underpins the SGD convergence rates in Theorems 1 and 2.
  • domain assumption Assumption 4 stability bound C1||u|| <= ||L[u]|| + ||B[u]|| <= C2||u|| for the linear PDE operators
    Needed to convert residual and boundary loss into a bound on the solution error in Theorem 3; this coercivity can fail for operators with resonances (e.g., Helmholtz at eigenvalues).
  • domain assumption Hardy-Krause bounded variation of the residual r(x;theta) with a constant that stays controlled during optimization (Remark 1)
    Required to apply Lemma 1(ii); the paper asserts smooth activations suffice but does not bound the variation constant in terms of network weights.

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Cite this review

Pith. "Pith review of A novel number-theoretic sampling method for neural network solutions of partial differential equations." pith.science (2026). https://pith.science/paper/CQYXIJRP

@misc{pith2026241117039,
  author       = {Pith},
  title        = {Pith review of: A novel number-theoretic sampling method for neural network solutions of partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQYXIJRP}},
  note         = {Machine review of arXiv:2411.17039}
}
read the original abstract

Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics-Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low-regularity Poisson equations, two-dimensional inverse Helmholtz problems, and high-dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities.

Figures

Figures reproduced from arXiv: 2411.17039 by the authors.

Figure 1
Figure 1. Flow chart of our method. Corollary 1. Suppose that Theorem 1 and Theorem 3 hold, then we have (i) If { 𝒙𝑘 }𝑁𝑟 𝑘=1 is a good lattice point set, lim 𝑖→∞ 𝔼 ( ‖𝒖 ∗ (𝒙) − 𝒖(𝒙; 𝜃𝑖 )‖ 2 2 ) ≤ 2 𝐶1 2 ( 𝜂𝐋𝐶𝑉 2𝑐𝜇 ( 𝑶 ( (log 𝑁𝑟 ) 𝑑 𝑁𝑟 ) + 𝑶 ( 𝑁 − 1 2 𝑏 ))2 + 𝑶 ( (log 𝑁𝑟 ) 𝑑 𝑁𝑟 ) + 𝑶 ( 𝑁 − 1 2 𝑏 )) . (26) (ii) If { 𝒙𝑘 }𝑁𝑟 𝑘=1 is a uniform random point set, lim 𝑖→∞ 𝔼 ( ‖𝒖 ∗ (𝒙) − 𝒖(𝒙; 𝜃𝑖 )‖ 2 2 ) ≤ 2 𝐶1 2 ( 𝜂𝐋𝐶𝑉 2𝑐𝜇 ( 𝑶 ( 𝑁 − 1… view at source ↗
Figure 2
Figure 2. The result of uniform random sampling for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by uniform random sampling method; (b) the approximate solution 𝑢 UR; (c) the absolute error |𝑢 ∗ − 𝑢 UR|. (a) sampling points (b) approximate solution (c) absolute error [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The result of LHS method for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by LHS method; (b) the approximate solution 𝑢 LHS; (c) the absolute error |𝑢 ∗ − 𝑢 LHS|. (a) sampling points (b) approximate solution (c) absolute error [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: The result of Halton sequence for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by Halton sequence; (b) the approximate solution 𝑢 Halton; (c) the absolute error |𝑢 ∗ − 𝑢 Halton|. is poor. However, in a similar comparative …
Figure 5
Figure 5. Figure 5: The result of Hammersley sequence for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by Hammersley sequence; (b) the approximate solution 𝑢 Hammersley; (c) the absolute error |𝑢 ∗ − 𝑢 Hammersley|. (a) Sampling points (b) app…
Figure 6
Figure 6. Figure 6: The result of Sobol sequence for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by Sobol sequence; (b) the approximate solution 𝑢 Sobol; (c) the absolute error |𝑢 ∗ − 𝑢 Sobol|. (a) Sampling points (b) approximate solution (c…
Figure 7
Figure 7. Figure 7: The result of GLP sampling for the two-dimensional Poisson equation with the solution Eq (30). (a) training points sampled by GLP sampling; (b) the approximate solution 𝑢 GLP; (c) the absolute error |𝑢 ∗ − 𝑢 GLP|. To verify the robustness of GLP sampling under varying …
Figure 8
Figure 8. Figure 8: The performance of errors for the two-dimensional Poisson equation with the solution Eq (30). (a) the relative error 𝑒∞(𝑢) with different training epochs; (b) the relative error 𝑒2 (𝑢) with different training epochs [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: The performance of errors for the two-dimensional Poisson equation with the solution Eq (30). (a) the relative error 𝑒∞(𝑢) with different sampling budgets; (b) the relative error 𝑒2 (𝑢) with different sampling budgets. 4.2.2. Two-dimensional Poisson equation with two p…
Figure 10
Figure 10. Figure 10: The sampling points of six different sampling methods for the two-dimensional Poisson equation with the exact solution Eq (31). Similar to the previous subsection, we also design comparative experiments to observe the performance of GLP sampling and uniform random sam…
Figure 11
Figure 11. Figure 11: The absolute errors of six different sampling methods for the two-dimensional Poisson equation with the exact solution Eq (31). (a) performance of 𝑒∞(𝑢) (b) performance of 𝑒2 (𝑢) [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The performance of errors for the two-dimensional Poisson equation with the solution Eq (31). (a) the relative error 𝑒∞(𝑢) with different training epochs; (b) the relative error 𝑒2 (𝑢) with different training epochs. where Ω = (0, 1)2 and 𝑘 2 is the unknown parameter,…
Figure 13
Figure 13. Figure 13: The performance of errors for the two-dimensional Poisson equation with the solution Eq (31). (a) the relative error 𝑒∞(𝑢) with different sampling budgets; (b) the relative error 𝑒2 (𝑢) with different sampling budgets. We sample 6765 points within Ω as the training se…
Figure 14
Figure 14. Figure 14: The training points for the inverse problem of two-dimensional Helmholtz equation. (a) the residual training points by uniform random sampling; (b) the residual training points of GLP sampling. After completing 50,000 epochs of Adam training and an additional 50,000 e…
Figure 15
Figure 15. Figure 15: The results of uniform random sampling for the inverse problem of two-dimensional Helmholtz equation. (a) the approximate solution 𝑢 UR; (b) the absolute error |𝑢 ∗ − 𝑢 UR|. (a) approximate solution (b) absolute error [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: The results of GLP set for the inverse problem of two-dimensional Helmholtz equation. (a) the approximate solution 𝑢 GLP; (b) the absolute error |𝑢 ∗ − 𝑢 GLP|. (a) performance of 𝑒∞(𝑢) (b) performance of 𝑒2 (𝑢) (c) performance of |𝑘 − 𝑘 ∗ | [PITH_FULL_IMAGE:figures/f…
Figure 17
Figure 17. Figure 17: The performance of errors for the inverse problem of two-dimensional Helmholtz equation. (a) the relative error 𝑒∞(𝑢) with different training epochs; (b) the relative error 𝑒2 (𝑢) with different training epochs; (c) the absolute error |𝑘−𝑘 ∗ | with different training …
Figure 18
Figure 18. Figure 18: The performance of errors under different noise scales for the inverse problem of two-dimensional Helmholtz equation. (a) the relative error 𝑒∞(𝑢) under different noise scales; (b) the relative error 𝑒2 (𝑢) under different noise scales [PITH_FULL_IMAGE:figures/full_f…
Figure 19
Figure 19. Figure 19: The training points for the five-dimensional linear problem. (a) the residual training points by uniform random sampling; (b) the residual training points of GLP sampling. (a) performance of 𝑒∞(𝑢) (b) performance of 𝑒2 (𝑢) [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: The performance of errors for the five-dimensional linear problem. (a) the relative error 𝑒∞(𝑢) during training process; (b) the relative error 𝑒2 (𝑢) during training process [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: The training points for the eight-dimensional linear problem. (a) the residual training points by uniform random sampling; (b) the residual training points of GLP sampling. (a) performance of 𝑒∞(𝑢) (b) performance of 𝑒2 (𝑢) [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]
Figure 22
Figure 22. Figure 22: The performance of errors for the eight-dimensional linear problem. (a) the relative error 𝑒∞(𝑢) during training process; (b) the relative error 𝑒2 (𝑢) during training process [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]
Figure 23
Figure 23. Figure 23: The training points for the five-dimensional nonlinear problem. (a) the residual training points by uniform random sampling; (b) the residual training points of GLP sampling. In [PITH_FULL_IMAGE:figures/full_fig_p021_23.png]
Figure 24
Figure 24. Figure 24: The performance of errors for the five-dimensional nonlinear problem. (a) the relative error 𝑒∞(𝑢) during training process; (b) the relative error 𝑒2 (𝑢) during training process. We also developed different training strategies by the number of training points and cond…
Figure 25
Figure 25. Figure 25: The performance of errors for the eight-dimensional nonlinear problem. (a) the residual training points by uniform random sampling; (b) the residual training points of GLP sampling. In [PITH_FULL_IMAGE:figures/full_fig_p022_25.png]
Figure 26
Figure 26. Figure 26: The performance of errors for the eight-dimensional nonlinear problem. (a) the relative error 𝑒∞(𝑢) during training process; (b) the relative error 𝑒2 (𝑢) during training process. 5. Conclusions In this work, we propose a number-theoretic method sampling neural networ…

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