{"id":"d53106b7-d752-4825-866e-0352a02f5dcb","arxiv_id":"2505.08487","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An adaptive sampling algorithm that triangulates the response manifold and adds barycenter points guided by a residual threshold produces surrogate-training data that lowers neural network prediction error versus one-shot samplers in a 1D harmonic transport test case.","lead":"This paper presents an adaptive sampling method that adds new simulation points at the centers of triangles on a map of simulation outputs, so a surrogate model can be trained with fewer, better-placed samples. It shows on a wave-propagation equation that this approach reduces prediction error compared with standard Latin Hypercube sampling.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Single-run, no-error-bar MNRE comparison leaves the claimed 26%/15% ASADG improvement statistically unestablished.","rationale":"The reader identified the residual-metric reliability in Eq. (4)-(5) as the weakest assumption. I agree that is a real theoretical gap: no analysis links rho_G to downstream surrogate error, and the threshold is hand-tuned. However, the single most load-bearing concern about the central empirical claim is the absence of repeated trials, error bars, and reproducibility artifacts. Even if the metric were a perfect oracle, the reported 26%/15% MNRE improvements could be within run-to-run noise for this neural-network setup, especially given the huge reported standard deviations and the paper's explicit admission of a single experiment per case. The proof of Lemma 2.1 is also internally flawed: the constructed nu^{-1} does not invert the norm projection and continuity of F does not guarantee the claimed L2 closeness. That invalidates the reduced-manifold theorem, but it is not the primary blocker for the headline empirical claim. The proposed concrete test would settle whether the MNRE gap is statistically real. Since the paper is plausible but the evidence is thin, the verdict should remain CONDITIONAL, matching the reader's stance, pending the requested validation and corrected theory.","tokens_in":10016,"tokens_out":6333,"duration_ms":63064,"concrete_test":"Run K=20 independent repetitions of the low-dimensional experiment: for each k, draw the initial LHS set (which also seeds ASADG's corners), run ASADG to 500 accepted points, train an FCNN with 5 random initializations per sampler, and evaluate MNRE on a fixed held-out uniform test set. Report the mean and 95% bootstrap confidence interval for the paired difference MNRE_LHS - MNRE_ASADG. If the interval contains zero, the claimed improvement is not statistically significant. Also run K=10 repetitions for the high-dimensional case. Provide code, seeds, and raw errors to allow independent reproduction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that ASADG yields better response-manifold representation with the same number of samples rests entirely on Table 3: one FCNN run per sampler per case. The low-dimensional MNRE improves from 8.098 to 5.970 (26%) and the high-dimensional from 1.819 to 1.552 (15%), but there are no repeated trials, no random seeds, and no confidence intervals. The reported standard deviations (409 and 228 for low-dim; 60 and 34 for high-dim) are across test points, not across training runs, and are tens of times larger than the mean MNRE, indicating heavy-tailed errors where a single outlier can dominate the mean. Section 4 explicitly concedes the comparison is 'a single experiment on both cases.' No code or data is provided, so the numbers cannot be independently reproduced. Moreover, the 'same number of input data' comparison ignores ASADG's extra residual solves (563 and 611 residual evaluations) and the cost of triangulation/UMAP; if those overheads were charged to the budget, LHS/uniform could generate more samples. A 15-26% difference in MNRE could easily be within run-to-run variability for a 2-layer FCNN with 500 training examples and a 2048-dimensional output. Thus the central empirical claim is not yet established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces ASADG, an adaptive sampling algorithm for generating training data for surrogate models of expensive physical simulations. Starting from corner points of the input parameter box, the method builds a Delaunay triangulation of the response manifold, computes barycenters of the simplices, and decides whether to add each barycenter's solver-evaluated point based on a residual metric rho_G = F(P_G) and a decaying threshold rho_{epsilon,k}. For high-dimensional inputs, UMAP is used to construct a reduced two-dimensional surface manifold. The method is tested on a one-dimensional harmonic transport problem in two settings: a low-dimensional case (two varying parameters) and a high-dimensional case (3072-dimensional input) with a Chebyshev-parameterized source. The authors compare data distributions and train a two-layer fully connected neural network, reporting MNRE values of 8.098 for LHS versus 5.970 for ASADG (low-dimensional) and 1.819 for uniform sampling versus 1.552 for ASADG (high-dimensional). They conclude that ASADG produces a more representative response manifold with the same number of samples than one-shot samplers.","tokens_in":10249,"tokens_out":5310,"duration_ms":53353,"significance":"If the central empirical claim were established, ASADG would be a useful contribution to adaptive sampling for surrogate modeling of PDE-based systems: it is metamodel-agnostic, uses a physical residual rather than a purely geometric criterion, and addresses the expensive-solver regime where sample count must be kept low. The paper offers qualitative evidence that ASADG concentrates samples in high-curvature regions of the response manifold, and it honestly acknowledges in Section 4 that the benchmarks are single experiments. The method's core idea is clearly articulated and appears implementable. However, the current evidence is preliminary rather than conclusive: the empirical comparison rests on one training run per sampler without seeds or confidence intervals, the computational-cost comparison ignores the extra residual solves, and the theoretical appendix contains a proof gap that underlies the high-dimensional reduction. The magnitude of the claimed improvement, when corrected against the paper's own numbers, is about 26% and 15% rather than the stated 33% and 16%.","major_comments":[{"comment":"The central claim that ASADG improves MNRE over LHS and uniform sampling rests on a single FCNN training run per sampler, with no random seeds and no confidence intervals. The reported standard deviations (409.007 and 228.511 for the low-dimensional case; 60.367 and 33.571 for the high-dimensional case) are across test points, not across repeated training runs, and are tens of times larger than the mean MNRE, indicating heavy-tailed error distributions in which one outlier can dominate the mean. The paper's own Section 4 concedes that the comparison is 'a single experiment on both cases.' Consequently, the claimed 26% and 15% improvements are not statistically established; please provide multiple independent runs with different seeds and initializations, report means with error bars or confidence intervals, and perform a paired comparison.","section":"Section 3.4, Table 3"},{"comment":"The 'same number of input data' comparison does not charge ASADG's additional residual evaluations to the sampling budget. Table 1 reports 563 residual computations for the low-dimensional case and 611 for the high-dimensional case in addition to the 500 solver solves. If residual evaluations are counted at their reported cost, the same total time would allow one-shot samplers to generate roughly 700 solver-evaluated samples, a 40% larger training set, which could plausibly offset the MNRE gap. The fairness of the comparison depends on whether the target is a fixed sample count or a fixed computational budget; please discuss this explicitly or re-run the benchmark under an equal-cost comparison.","section":"Section 3.3.3, Table 1"},{"comment":"The selection metric rho_G = F(P_G) is evaluated at the simplex barycenter, where the output is the linear interpolation of the vertex outputs rather than a solver solution. The paper provides no analysis or experiment showing that this interpolated residual predicts where adding a solver-evaluated point will most reduce the surrogate error. Because this criterion drives all point additions, the method's efficacy hinges on this assumption. Please add a correlation or ablation study that relates rho_G to actual error reduction after adding the barycenter's solver-evaluated point, or provide a theoretical justification.","section":"Section 2.1, Eq. (4)-(5)"},{"comment":"The proof of Lemma 2.1 is incomplete. The mapping nu(y) = ||y||_{L2} from R^M to R is not injective, so the pseudo-inverse nu^{-1} is not well-defined without an additional selection rule. The proof then asserts that 'by continuity of the manifold M or implicit function F' there exists tilde y with F(tilde p, tilde y) <= epsilon, but this is not an immediate consequence of continuity; one needs an implicit-function or projection argument that is not supplied. Since Theorem 2.1 and the high-dimensional reduced-manifold construction rely on this lemma, the theoretical support for the high-dimensional case is currently incomplete. Please repair the proof or state the additional assumptions (e.g., regularity of F and a well-defined projection onto M) under which the lemma holds.","section":"Appendix A, proof of Lemma 2.1"},{"comment":"The reported improvement percentages in the text do not match the numbers in Table 3. The low-dimensional MNRE improves from 8.098 to 5.970, which is a relative reduction of approximately 26%, not the stated '33%'; the high-dimensional MNRE improves from 1.819 to 1.552, approximately 15%, not 'nearly 16%.' Please correct these values in the text and re-check whether any other reported percentages are affected.","section":"Section 3.4"}],"minor_comments":[{"comment":"The first sentence is ungrammatical: 'Physical models classically involved PDE and depending of their underlying complexity and the level of accuracy required, and known to be computationally expensive...' should be rewritten for clarity and correctness.","section":"Abstract"},{"comment":"There is a duplicated word in 'The learning capability of surrogate models models depend on different factors'; please remove the duplicate 'models'.","section":"Section 1"},{"comment":"'This construction allows to proof the lemma 2.1' should read 'allows us to prove Lemma 2.1.'","section":"Section 2.2"},{"comment":"The word 'proove' should be 'prove' in the proof of Lemma 2.1.","section":"Appendix A"},{"comment":"The captions of Figures 4 and 5 are identical and both mention 'LHS', but Figure 5 compares ASADG with the uniform sampler in the high-dimensional case; please correct the captions to reflect the actual samplers shown.","section":"Figures 4 and 5 captions"},{"comment":"The column header 'Number of computation' is ambiguous; please clarify that the rows correspond to the number of solver solves and the number of residual evaluations, respectively.","section":"Table 1"},{"comment":"The term 'Chebeshev polynomial' should be 'Chebyshev polynomial', and 'mach number' should be 'Mach number' in the description of the harmonic transport equation.","section":"Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the core idea is potentially useful, but the empirical foundation is currently too weak for publication: the headline comparison rests on a single run per sampler with no seeds or error bars, the equal-sample comparison ignores the extra residual cost, and the theoretical appendix has a proof gap that affects the high-dimensional case. The authors should also be asked to provide code and data for reproducibility, since none are currently available. Additionally, the incorrect percentage claims (33% versus the actual 26%) in Section 3.4 suggest that the quantitative reporting needs careful checking throughout. I recommend a major revision requiring reruns with uncertainty quantification and a repaired or qualified Lemma 2.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proposes ASADG, which triangulates the response manifold (input-output pairs), computes barycenters, and adds a new solver point when the linear-interpolation residual at the barycenter exceeds a decaying threshold. For high-dimensional inputs it uses UMAP with a grid-based inverse to triangulate in a 2D projection. That combination is genuinely not in the cited prior art: refs [7,8] triangulate the input space, not the response manifold. The exposition is clear, and the authors explicitly state in Section 4 that the comparisons are a single experiment, which is honest.\n\nThe soft spots are exactly where the evidence is thin. The central comparison rests on Table 3: one FCNN run per sampler per case. There are no repeated trials, no random seeds, no confidence intervals. The reported standard deviations are over test points, not training runs, and are tens of times larger than the MNRE mean, so the 26% and 15% improvements could easily be within run-to-run variability. The residual metric rho_G is never analyzed; the paper doesn't show it tracks surrogate error, and the threshold rho_eps,k is hand-tuned. The 500-point ASADG configuration is selected post-hoc from the parameter study. The comparison also ignores the algorithm's own overhead: the residual solves and triangulation cost are listed in Table 1, and in the same wall-clock budget LHS/uniform could generate around 700 points, not 500. So 'same number of input data' is not the same cost budget. The theoretical Lemma 2.1 proof is a sketch—the inverse construction for the output map is assumed, not built—but the algorithm doesn't depend on its rigor.\n\nThat said, the idea is worth pursuing. The paper is written for researchers in sequential experimental design for PDE surrogate modeling, and it gives a concrete, readable baseline variant that improves on one-shot samplers in a toy but non-trivial problem. It deserves a serious referee, but only if the authors add repeated trials, compare against the sequential baselines they cite, and analyze the residual metric. Without those, the empirical claim is not supported.\n\nI would not cite it yet in my own work, but I'd bring it to a reading group to discuss the method and the reproducibility problems.","headline":"A plausible incremental adaptive-sampling idea, but the headline improvement numbers come from a single run with no error bars, so the central claim is not yet established.","tokens_in":10804,"tokens_out":2905,"would_cite":false,"duration_ms":29214,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that an adaptive barycenter-sampling algorithm, ASADG, produces a more faithful response manifold from the same number of physical solver calls as one-shot sampling, and that surrogates trained on its data have lower…","keywords":["adaptive sampling","surrogate modeling","response manifold","triangulation","barycenter","Latin hypercube sampling","neural network surrogate","harmonic transport"],"falsifier":"Take a fixed PDE problem and a fixed initial corner-point set, run ASADG, but at each step also solve the PDE at several barycenters rejected by the threshold. If adding those rejected points to the training set improves the surrogate's test error as much as or more than the accepted points, the residual criterion is not selecting the right locations. A quantitative version would compute the correlation between $\\rho_G$ at candidate barycenters and the reduction in test error after solving and adding each one; a weak or negative correlation would refute the central claim.","tokens_in":9800,"feed_emoji":"🎯","tokens_out":8664,"duration_ms":75921,"temperature":0.7,"pith_summary":"This paper proposes an adaptive sampling algorithm, ASADG, for generating training data for surrogate models of expensive physical simulations. The claim is that by iteratively triangulating the response manifold and adding solved points at simplex barycenters where an unevaluated residual is large, the same number of input samples as a one-shot Latin Hypercube or uniform sampler can represent the response surface more faithfully. In the paper's harmonic-transport test problem, a fully connected neural network trained on 500 ASADG points reaches a mean normalized relative error of 5.970 versus 8.098 for LHS in the low-dimensional case, and 1.552 versus 1.819 for uniform sampling in the high-dimensional case, with roughly half the error spread. If correct, the practical payoff is that expensive PDE solver calls can be concentrated where they most improve surrogate accuracy.","feed_headline":"Adaptive barycenter sampling cuts surrogate error by 26%","feed_subtitle":"Triangulating the response manifold and adding points at high-residual barycenters beats one-shot sampling.","key_machinery":"The load-bearing mechanism is the barycentric residual test: for each simplex in a triangulation of the response manifold, compute the barycenter $P_G$, approximate the output there as the linear interpolation of the vertex outputs, and evaluate $\\rho_G = F(P_G)$, the residual of the physical equation at that interpolated point. A new solved point is added only when $\\rho_G$ exceeds the decaying threshold $\\rho_{\\epsilon,k} = \\rho_{\\epsilon,0}/\\lambda^k$, which keeps the sample count from exploding while directing new samples toward regions where the interpolated manifold is far from a true solution. For high-dimensional inputs, the method projects input and output coordinates onto a reduced surface manifold with a learned projector and a grid-based inverse, so every suggested barycenter maps back to a valid point in the original parameter domain. The triangulation turns the continuous manifold into a finite list of candidate locations, and the thresholded residual selects among them.","core_discovery":"The central discovery is that a response manifold can be grown adaptively from a minimal corner-point initialization without knowing the manifold in advance. At each step the current solved points are triangulated into simplices; for each simplex, the barycenter is formed and the residual of the physical equation at that barycenter is estimated using linearly interpolated vertex outputs rather than a new solver run. If that residual exceeds a relaxation threshold that decays with iteration number, the point is solved and added. Iterating concentrates samples in high-curvature regions of the response manifold, which is where a surrogate most needs them. The paper demonstrates that with equal sample counts this geometry-driven distribution lowers the error and error variance of a trained fully connected network compared with one-shot samplers in both low- and high-dimensional settings.","pith_inferences":["A natural extension the paper does not develop is to use the plateau in the mean residual as an automatic stopping rule: the paper notes the mean metric stabilizes around iteration 20 while the point count keeps growing, so a fixed solver budget could be saved by stopping earlier.","The barycentric residual score could serve as an acquisition function for other surrogate families, such as Gaussian-process or polynomial-chaos models, because it selects locations by geometric under-representation rather than by the surrogate's own uncertainty.","Benchmarking against adaptive sequential samplers such as Sobol sequences or adaptive kriging would be the sharper test; the paper compares only with one-shot samplers and itself flags this as open work, so the size of the advantage over other adaptive methods is not yet established.","The threshold parameters $\\rho_{\\epsilon,0}$ and $\\lambda$ are hand-tuned in the experiments; a testable variant would set them from a hold-out residual target, making the algorithm parameter-free."],"forward_implications":["At equal sample counts, ASADG-trained surrogates have lower mean normalized relative error than surrogates trained on LHS or uniform one-shot samples: 5.970 versus 8.098 in the low-dimensional test and 1.552 versus 1.819 in the high-dimensional test.","Error standard deviation is roughly halved, so surrogate predictions are not only better on average but more consistent across test points.","Because rejected barycenters use only interpolated residuals, most candidate evaluations cost a small fraction of a solver run, keeping total data-generation time comparable to one-shot sampling at 500 points.","The method is metamodel-agnostic and applies to any physical model whose solver can be queried pointwise, since it only needs solved input-output pairs, a triangulation, and a residual check.","For high-dimensional parameters, the reduced surface manifold construction lets ASADG work in a two-dimensional embedded space while still returning physically valid input points through the grid-based inverse."],"supporting_citations":[{"why":"Survey of adaptive sampling for global metamodeling; supplies the taxonomy of explorative and exploitative search and the motivation for sequential design.","marker":"[6]"},{"why":"Adaptive exploration-exploitation algorithm using Delaunay triangulation; the closest prior construction that ASADG extends with barycentric residual selection.","marker":"[7]"},{"why":"Adaptive kriging strategy based on Delaunay triangulation and TOPSIS; an example of scoring triangles to choose new points.","marker":"[8]"},{"why":"Delaunay triangulation of manifolds, including a perturbed-point algorithm for higher dimensions; underpins the triangulation step used by ASADG.","marker":"[15]"},{"why":"UMAP dimension-reduction method used as the projector onto the reduced surface manifold for high-dimensional inputs.","marker":"[18]"},{"why":"Goldstein's model for time-harmonic acoustics; supplies the harmonic transport equation used in the numerical experiments.","marker":"[19]"}],"fun_headline_variants":["Adaptive barycenter sampling beats LHS for surrogates","Simplex barycenters guide iterative sampling for surrogates","Geometry-driven sampling trims surrogate error vs one-shot","Barycenter-wise data generation improves surrogate learning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the residual $\\rho_G = F(P_G)$ computed at a simplex barycenter from linearly interpolated vertex outputs is a reliable signal for where a new solved point will most improve the surrogate's representation of the response manifold; the paper does not measure how strongly this residual predicts surrogate error, and the threshold that decides whether to add a point is hand-tuned.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive barycenter sampling beats LHS for surrogates","Simplex barycenters guide iterative sampling for surrogates","Geometry-driven sampling trims surrogate error vs one-shot","Barycenter-wise data generation improves surrogate learning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000928,"raw_usage":{"total_tokens":3973,"prompt_tokens":943,"completion_tokens":3030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":2964}},"tokens_in":559,"tokens_out":3030,"duration_ms":22652,"temperature":1.0,"reasoning_tokens":2964,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:53:35.376847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed PDE problem and a fixed initial corner-point set, run ASADG, but at each step also solve the PDE at several barycenters rejected by the threshold. If adding those rejected points to the training set improves the surrogate's test error as much as or more than the accepted points, the residual criterion is not selecting the right locations. A quantitative version would compute the correlation between $\\rho_G$ at candidate barycenters and the reduction in test error after solving and adding each one; a weak or negative correlation would refute the central claim.","supporting_citations":[{"cited_title":"A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design","cited_arxiv_id":null,"evidence_quote":"Survey of adaptive sampling for global metamodeling; supplies the taxonomy of explorative and exploitative search and the motivation for sequential design."},{"cited_title":"An adaptive exploration-exploitation algorithm for constructing metamodels in random simulation using a novel sequential experimental design","cited_arxiv_id":null,"evidence_quote":"Adaptive exploration-exploitation algorithm using Delaunay triangulation; the closest prior construction that ASADG extends with barycentric residual selection."},{"cited_title":"Delaunay triangulation of manifolds","cited_arxiv_id":null,"evidence_quote":"Delaunay triangulation of manifolds, including a perturbed-point algorithm for higher dimensions; underpins the triangulation step used by ASADG."},{"cited_title":"Mathematical analysis of goldstein’s model for time-harmonic acoustics in flows","cited_arxiv_id":null,"evidence_quote":"Goldstein's model for time-harmonic acoustics; supplies the harmonic transport equation used in the numerical experiments."}],"review_version":1}