{"id":"d37bb731-92d2-4cff-9d95-77a887480e3f","arxiv_id":"2411.17039","paper_version":7,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using good lattice points as training points for physics-informed neural networks gives lower errors on low-regularity and high-dimensional PDEs, with a quadrature error of O((log N)^d/N) versus O(N^{-1/2}) for random sampling.","lead":"Swap uniform random sampling for deterministic 'good lattice point' sets when training physics-informed neural networks. The change yields smaller solution errors on rough and high-dimensional partial differential equations, backed by error bounds that improve on Monte Carlo rates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The asymptotic comparison behind Corollary 1 does not hold at the paper's own high-dimensional settings: at d=8, N≈10^4, (log N)^d/N≈5×10^3, while N^{-1/2}≈10^{-2}, so the claimed smaller error bound is not supported in the tested regime.","rationale":"The reader's weakest_assumption correctly identified the uncontrolled Hardy-Krause variation constant as a soft spot in Lemma 1(ii). My stress-test adds a more direct and, in my view, more load-bearing problem: even if the variation constant were benign, the asymptotic rate comparison in Corollary 1 is not realized at the N_r and d values used in the paper's high-dimensional experiments. At d=5 and d=8 with N_r≈10^4, the quantity (log N_r)^d/N_r is orders of magnitude larger than N_r^{-1/2}, so the claimed 'smaller upper bound' is not established in the regime where the paper claims high-dimensional success. This does not mean the numerical method is wrong; the empirical gains in Tables 5–8 could be real. It means the theoretical guarantee, as stated, does not cover those experiments, and the paper should either supply the missing constants and generating vectors or temper the theoretical claim. The reader's verdict of conditional acceptance remains appropriate: the work is plausible and the experiments are suggestive, but the central theoretical claim needs revision or re-scoping before acceptance.","tokens_in":22581,"tokens_out":10874,"duration_ms":107474,"concrete_test":"Reconstruct the GLP sets using the paper's primitive-root construction for the exact values used in Tables 5–8 (d=5,8; N_r=10007,11215,3001), compute their star discrepancy D*, and compare D* with N_r^{-1/2}. Also evaluate the Koksma-Hlawka bound V(r^2)·D* for the trained residuals from §4.4–4.5. If D*≤N_r^{-1/2} for these sets, the concern is resolved; if D*≫N_r^{-1/2}, Corollary 1 does not support the high-dimensional claims and the paper should report these constants or limit its theoretical conclusion to low-dimensional or asymptotic cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 1's central comparison is O((log N_r)^d/N_r) < O(N_r^{-1/2}) for large N_r. This is only an asymptotic statement with all constants suppressed, and it fails at the point counts used in the high-dimensional experiments. In §4.4 and §4.5, the paper uses N_r=10007 and N_r=11215 for d=5 and d=8 (and N_r=3001 for d=5). For d=8, N=11215, (log N)^8/N ≈ 5.1×10^3, whereas N^{-1/2} ≈ 9.4×10^{-3}; for d=5, N=10007, (log N)^5/N ≈ 6.6, still larger than N^{-1/2}. Thus Corollary 1 cannot explain the observed GLP advantage in Tables 5–8; that advantage must come from uncontrolled constants or from behavior outside the stated error bound. Lemma 1(ii) adds a second uncontrolled constant: the Hardy-Krause variation of r^2. Even under Assumption 2, this variation depends on the radius R, the PDE operator, and the loss weights, and no quantitative bound is given, so the asymptotic advantage is conditional on an unknown constant. Remark 3 concedes the log-term issue only for 'very large' d, while the abstract and contribution list assert rigorous lower error bounds without this caveat. The paper should either report the actual discrepancies and variation constants for the GLP sets used, or restrict the theoretical claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22922,"tokens_out":6787,"duration_ms":64264,"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":[{"comment":"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.","section":"Section 3.1, Lemma 1(i); Section 3.2, Theorem 3(ii), Eq. (25)"},{"comment":"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.","section":"Section 3.1, Lemma 1(ii) and Remark 1; Section 3.2, Theorem 3"},{"comment":"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.","section":"Corollary 1 and Section 4.4/4.5, Tables 5-8"},{"comment":"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.","section":"Appendix A, proof of Theorem 1"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"The text uses 'N_d^{-1/2}' where the boundary sample count is N_b; make the notation consistent throughout.","section":"Assumption 3"},{"comment":"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.","section":"Figures 25 and 26"},{"comment":"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.","section":"Table 6"},{"comment":"Reference [24] is missing publication details; the reference list should be completed.","section":"References"},{"comment":"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.","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant topic and the numerical comparisons are extensive, but the theoretical core needs substantial repair before the claims as stated can be accepted. The probabilistic/deterministic gap and the uncontrolled Hardy-Krause variation constant affect the main theorem, and the high-dimensional experiments are in a regime where the asymptotic comparison does not apply. The authors should be encouraged to revise, but the revised manuscript must either fix these issues or carefully restrict the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's genuine contribution is the systematic numerical comparison: six sampling methods, low-regularity and high-dimensional PDEs, an inverse Helmholtz problem with noise, and multiple sampling budgets. GLP consistently wins, and the training curves look honest. That is a useful data point for the PINN sampling literature.\n\nThe theory, however, is significantly overstated. The headline result, Corollary 1, rests on the asymptotic comparison O((log N_r)^d/N_r) < O(N_r^{-1/2}). At the paper's own high-dimensional settings this inequality is false: for d=8, N_r=11215, (log N)^8/N is on the order of 5e3, while N^{-1/2} is about 1e-2. For d=5, N=10007, (log N)^5/N is still around 6.6. So the observed GLP advantage in Tables 5–8 cannot be explained by the stated bound; it must come from uncontrolled constants or from behavior outside the theorem. Remark 3 concedes the log-term issue only for \"very large\" d, but the abstract and contribution list assert rigorous lower error bounds without that caveat.\n\nThere are also two technical gaps. First, Lemma 1(i) is a probabilistic error bound, but Theorem 3 uses it as a deterministic inequality. That is not fatal if the probability statement is tracked, but it should be stated correctly. Second, Lemma 1(ii) requires bounded Hardy–Krause variation of the squared residual, and the variation constant depends on the network weights. The paper never controls this constant along the optimization path, so the O((log N)^d/N) rate is conditional on something not established.\n\nWhat holds up: the SGD convergence analysis in Theorems 1 and 2 is standard and correctly assembled, and the experimental methodology is reasonable, including multiple seeds for random sampling and comparisons across sampling budgets. The lack of released code and precise generating vectors is a practical weakness; so is the absence of a direct comparison with the most relevant prior work, [3] on QMC sampling for PINNs.\n\nThis paper is for researchers working on sampling strategies in PINNs. The empirical evidence is worth having, but the theoretical claims need major revision. A referee should ask for the actual discrepancies/variation constants or a restriction of the theoretical claim to regimes where it genuinely applies.\n\nI would send it to peer review, but with the expectation of heavy revision.","headline":"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.","tokens_in":23446,"tokens_out":2197,"would_cite":true,"duration_ms":21684,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C05","65D30","68T07","65N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that good lattice point sets shrink the expected L2 error bound of physics-informed neural network solutions.","keywords":["physics-informed neural networks","good lattice point sets","quasi-Monte Carlo","discrepancy","error bounds","high-dimensional PDEs","low-regularity PDEs","PDE sampling"],"falsifier":"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.","tokens_in":22391,"feed_emoji":"🧮","tokens_out":8636,"duration_ms":76330,"temperature":0.7,"pith_summary":"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.","feed_headline":"Number-theoretic sampling shrinks neural-network PDE errors","feed_subtitle":"Good lattice point sets replace random loss samples, provably lowering expected error and winning with fewer points.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Lemma 1: the O(N^{-1/2}) Monte Carlo integration error and the O((log N)^d/N) bound for low-discrepancy lattice points that drive the comparison.","marker":"[33]"},{"why":"Defines discrepancy and the good lattice point set construction used to generate residual training points.","marker":"[7]"},{"why":"Provides Lemma 2, the PL* condition for over-parameterized networks that turns loss convergence into parameter convergence.","marker":"[25]"},{"why":"Introduces the PINN loss and training setup that this paper modifies by replacing uniform random interior points with GLP sets.","marker":"[34]"},{"why":"Supplies the stochastic-gradient variance and expectation assumptions used in Theorem 1 for quasi-Monte Carlo based deep PDE solvers.","marker":"[3]"},{"why":"Gives the stability bound assumption (Assumption 4) linking residual and boundary losses to the L2 error of the solution.","marker":"[42]"},{"why":"Also cited for the stability bound assumption used to connect empirical losses to solution error.","marker":"[45]"},{"why":"Provides the Hardy-Krause bounded-variation sufficient condition that makes Lemma 1(ii) applicable to neural-network residuals.","marker":"[15]"}],"fun_headline_variants":["Low-discrepancy sampling beats random for PINN accuracy","Number-theoretic sampling cuts PINN error bounds","Lattice points beat random sampling for PINN accuracy","Deterministic sampling wins for PINNs: lower error bounds","Lattice sampling shrinks PINN error beyond random"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Low-discrepancy sampling beats random for PINN accuracy","Number-theoretic sampling cuts PINN error bounds","Lattice points beat random sampling for PINN accuracy","Deterministic sampling wins for PINNs: lower error bounds","Lattice sampling shrinks PINN error beyond random"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000621,"raw_usage":{"total_tokens":2851,"prompt_tokens":889,"completion_tokens":1962,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":1884}},"tokens_in":505,"tokens_out":1962,"duration_ms":14143,"temperature":1.0,"reasoning_tokens":1884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:35:25.717707+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Random number generation and quasi-Monte Carlo methods","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 1: the O(N^{-1/2}) Monte Carlo integration error and the O((log N)^d/N) bound for low-discrepancy lattice points that drive the comparison."},{"cited_title":"Number-theoretic methods in statistics","cited_arxiv_id":null,"evidence_quote":"Defines discrepancy and the good lattice point set construction used to generate residual training points."},{"cited_title":"Losslandscapesandoptimizationinover-parameterizednon-linearsystemsandneuralnetworks","cited_arxiv_id":null,"evidence_quote":"Provides Lemma 2, the PL* condition for over-parameterized networks that turns loss convergence into parameter convergence."},{"cited_title":"Numerical Mathematics: Theory, Methods & Applications 14","cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic-gradient variance and expectation assumptions used in Theorem 1 for quasi-Monte Carlo based deep PDE solvers."},{"cited_title":"Das-pinns: A deep adaptive sampling method for solving high-dimensional partial differential equations","cited_arxiv_id":null,"evidence_quote":"Gives the stability bound assumption (Assumption 4) linking residual and boundary losses to the L2 error of the solution."},{"cited_title":"Moving sampling physics-informed neural networks induced by moving mesh pde","cited_arxiv_id":null,"evidence_quote":"Also cited for the stability bound assumption used to connect empirical losses to solution error."},{"cited_title":"Funktionen von beschränkter variatiou in der theorie der gleichverteilung","cited_arxiv_id":null,"evidence_quote":"Provides the Hardy-Krause bounded-variation sufficient condition that makes Lemma 1(ii) applicable to neural-network residuals."}],"review_version":1}