An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling

Axel TahmasebiMoradi; Chetra Mang; David Danan; Mouadh Yagoubi

REVIEW 5 major objections 7 minor 19 references

An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling

T0 review · 5 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.08487 v1 pith:AGXOKUZJ submitted 2025-05-13 cs.LG cs.AIstat.ML

Chetra Mang , Axel TahmasebiMoradi , David Danan , Mouadh Yagoubi This is my paper · ORCID

classification cs.LGcs.AIstat.ML

keywords adaptivesamplingsurrogatemodelingresponsemanifoldtriangulationbarycenterLatinhypercubeneuralnetworkharmonictransport

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

That 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.

I 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.

Referee Report

5 major / 7 minor

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.

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 (5)

[Section 3.4, Table 3] 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 3.3.3, Table 1] 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 2.1, Eq. (4)-(5)] 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.
[Appendix A, proof of Lemma 2.1] 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 3.4] 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.

minor comments (7)

[Abstract] 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 1] There is a duplicated word in 'The learning capability of surrogate models models depend on different factors'; please remove the duplicate 'models'.
[Section 2.2] 'This construction allows to proof the lemma 2.1' should read 'allows us to prove Lemma 2.1.'
[Appendix A] The word 'proove' should be 'prove' in the proof of Lemma 2.1.
[Figures 4 and 5 captions] 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.
[Table 1] 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 3.1] The term 'Chebeshev polynomial' should be 'Chebyshev polynomial', and 'mach number' should be 'Mach number' in the description of the harmonic transport equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ASADG advantage is established by an external MNRE benchmark, not by its own sampling criterion.

full rationale

The paper's central claim—that ASADG yields a better response-manifold representation than LHS/uniform for equal sample counts—is tested by training a fully connected neural network on the respective 500-point training sets and comparing MNRE on a separately generated test set (Section 3.4, Table 3). The MNRE numbers are measurements, not algebraic consequences of the ASADG criterion. The residual metric rho_G = F(P_G) is used only to decide where to add solver points and to stop the iteration; it is not the reported accuracy metric, and no parameter in the MNRE comparison is fitted to that metric. The Appendix A proof of Lemma 2.1 does contain a non-rigorous step: nu^{-1} is defined by selecting a manifold point with a prescribed norm, and the closeness bound is then asserted by continuity. That is a proof gap or correctness concern, not a circular reduction of the empirical claim, and it does not enter the MNRE comparison. Section 4's admission that the comparisons are 'a single experiment on both cases' is a statistical robustness limitation, not circularity. There are no load-bearing self-citations, and no fitted input is renamed as a prediction. The ASADG derivation is therefore self-contained against an external benchmark rather than equivalent to its own inputs.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim depends on several hand-chosen hyperparameters (initial threshold, decay factor, UMAP parameters, grid size) and on geometric assumptions (smoothness and triangulability of the response manifold, local fidelity of the UMAP embedding) that are asserted rather than verified. The paper provides no formal or empirical justification that the residual at a simplex barycenter is a reliable proxy for surrogate error reduction.

free parameters (5)

initial residual threshold rho_eps,0 = not stated, selected from Fig. 3b
Controls how many barycenters are added; the paper shows a strong effect on sample count but does not give the value used in the final FCNN comparison.
decay factor lambda = 1.5
Set by hand; no sensitivity analysis is reported for the neural network results.
UMAP hyperparameters = not stated
No n_neighbors, min_dist, or metric is given, so the high-dimensional experiments cannot be reproduced exactly.
grid size h for UMAP inverse projection = not stated
The grid-based inverse has an error bounded by h, but h is never reported.
number of generation iterations = implicitly chosen so the mean metric converges; yields 500 samples
The stopping rule is convergence of the same metric the algorithm minimizes, so the final sample count is not chosen by an independent criterion.

assumptions (3)

domain assumption The response manifold M defined by F is smooth and triangulable in the considered parameter range.
The algorithm triangulates M; for the high-dimensional case it reduces to a 2D surface via UMAP. Smoothness is asserted citing Whitehead and Moise, but not verified for the harmonic transport data manifold.
standard math F is continuous, so small input perturbations produce small output perturbations.
Used in the proof of Lemma 2.1 (Appendix A) to claim that ||tilde(y) - y|| < epsilon follows from ||tilde(p) - p|| < epsilon.
domain assumption UMAP 2D embedding preserves local manifold structure sufficiently for barycenter-based sampling to transfer to the original space.
The grid-inverse maps a reduced barycenter to the nearest grid point in original space; no bound on the resulting bias is provided beyond a heuristic dependence on grid size h.

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Cite this review

Pith. "Pith review of An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling." pith.science (2026). https://pith.science/paper/AGXOKUZJ

@misc{pith2026250508487,
  author       = {Pith},
  title        = {Pith review of: An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGXOKUZJ}},
  note         = {Machine review of arXiv:2505.08487}
}

read the original abstract

Physical models classically involved Partial Differential equations (PDE) and depending of their underlying complexity and the level of accuracy required, and known to be computationally expensive to numerically solve them. Thus, an idea would be to create a surrogate model relying on data generated by such solver. However, training such a model on an imbalanced data have been shown to be a very difficult task. Indeed, if the distribution of input leads to a poor response manifold representation, the model may not learn well and consequently, it may not predict the outcome with acceptable accuracy. In this work, we present an Adaptive Sampling Algorithm for Data Generation (ASADG) involving a physical model. As the initial input data may not accurately represent the response manifold in higher dimension, this algorithm iteratively adds input data into it. At each step the barycenter of each simplicial complex, that the manifold is discretized into, is added as new input data, if a certain threshold is satisfied. We demonstrate the efficiency of the data sampling algorithm in comparison with LHS method for generating more representative input data. To do so, we focus on the construction of a harmonic transport problem metamodel by generating data through a classical solver. By using such algorithm, it is possible to generate the same number of input data as LHS while providing a better representation of the response manifold.

Figures

Figures reproduced from arXiv: 2505.08487 by the authors.

**Figure 2.** Evolution of surface manifold according iteration number: (a) ASADG with 3000 points and (b) LHS with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Evolution of (a) the sampling number (red curve) and the metric mean value at the barycenters (blue curve) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Distribution of sample points by ASADG (red points) and LHS (blue points) of (a) input data and (b) manifold [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Distribution of sample points by ASADG (red points) and LHS (blue points) of (a) input data and (b) manifold [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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Surrogate models coupled with machine learning to approximate complex physical phenomena involving aerodynamic and aerothermal simulations

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work page 2013

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work page 2012

[4] [4]

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Qi Zhou, Xinyu Shao, Ping Jiang, Hui Zhou, and Leshi Shu. An adaptive global variable fidelity metamodeling strategy using a support vector regression based scaling function. Simulation Modelling Practice and Theory, 59:18–35, 2015

work page 2015

[5] [5]

State-of-the-art and comparative review of adaptive sampling methods for kriging

Jan N Fuhg, Amélie Fau, and Udo Nackenhorst. State-of-the-art and comparative review of adaptive sampling methods for kriging. Archives of Computational Methods in Engineering, 28:2689–2747, 2021

work page 2021

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A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design

Haitao Liu, Yew-Soon Ong, and Jianfei Cai. A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design. Structural and Multidisciplinary Optimization, 57:393–416, 2018. 10 Theoretical study of performances of numerical and automatic differentiation physical informed neural network

work page 2018

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An adaptive exploration-exploitation algorithm for constructing metamodels in random simulation using a novel sequential experimental design

Ali Ajdari and Hashem Mahlooji. An adaptive exploration-exploitation algorithm for constructing metamodels in random simulation using a novel sequential experimental design. Communications in Statistics-Simulation and Computation, 43(5):947–968, 2014

work page 2014

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An adaptive sampling strategy for kriging metamodel based on delaunay triangulation and topsis

Ping Jiang, Yahui Zhang, Qi Zhou, Xinyu Shao, Jiexiang Hu, and Leshi Shu. An adaptive sampling strategy for kriging metamodel based on delaunay triangulation and topsis. Applied Intelligence, 48:1644–1656, 2018

2018