{"id":"dc457e52-bc6d-438b-ba85-816fcf56f18d","arxiv_id":"2506.20441","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Hessian-guided adaptive collocation sampling for PINNs, backed by a 1D quadrature error bound, achieves lower L2 errors than uniform, residual-based, and gradient-based sampling on two test PDEs.","lead":"This paper proposes a quadrature rule that spaces integration points according to the square root of a function's second derivative, then applies the same idea to choose collocation points when training physics-informed neural networks. The authors prove the error bound is no worse than uniform trapezoids and report faster convergence and lower errors on two 2D PDE problems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 bounds a deterministic, interval-maximum-based trapezoid rule, while Algorithm 1 samples i.i.d. from a pointwise Hessian density and trains with equal-weight Monte Carlo sums; the theory therefore does not cover the proposed PINN method.","rationale":"The reader identifies the main gap as the transfer from 1D quadrature theory to the 2D stochastic PINN setting. I agree that the transfer is unproved and load-bearing, and I sharpen it: the mismatch already occurs in 1D, because the theorem's deterministic interval-maximum allocation differs from the algorithm's i.i.d. pointwise-density sampling, and because the PINN loss is an equal-weight Monte Carlo sum rather than a trapezoid rule. I do not find an error in Theorem 3.1 itself; the algebra of the bound is valid, modulo the small-N ceiling gap that is negligible asymptotically. The paper's 1D quadrature contribution can stand as a standalone result, and the experimental section is suggestive but not decisive, especially without error bars. Since the reader already issued a CONDITIONAL verdict for essentially the same reason, my stress-test does not move the verdict; it strengthens the rationale. The concrete test would settle whether the implemented hessian-RAD sampling behaves like the analyzed quadrature rule on a sharp-transition integrand.","tokens_in":7556,"tokens_out":13749,"duration_ms":140031,"concrete_test":"Run a 1D ablation with f(x) = sin(1/(x-0.5+0.01)) on [0,1], using the same N and k as in Figure 1. Compute three quadrature errors: (a) deterministic Theorem 3.1 allocation from Eq. (3), integrated with the trapezoid rule; (b) Algorithm 1-style i.i.d. samples from d(x) proportional to |f''(x)|^{1/2}, sorted and integrated with the trapezoid rule; (c) uniform trapezoids with N intervals. Repeat (b) over many seeds and record median error and per-interval counts. If the median error of (b) exceeds (c), or if its per-interval counts deviate substantially from the n_j of Eq. (3), then Eq. (4) does not implement the method bounded by Theorem 3.1, and the title's claim of a theoretical foundation for the PINN sampler is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is internally sound, but it is not the method used in Section 4. Theorem 3.1 fixes k equal intervals I_j, computes M_j = max_{I_j}|f''|, allocates n_j = ceil(N sqrt(M_j) / sum sqrt(M_p)) points per interval, and bounds the error of the composite trapezoid rule on those deterministic nodes. Algorithm 1 instead draws N points i.i.d. from d(x) proportional to gamma(x)^tau with gamma = Hessian norm and tau = 1/2, and the PINN loss is minimized as an equal-weight Monte Carlo sum over those points, not as a trapezoid quadrature. These differ in three load-bearing ways: (i) the loss is not the trapezoid rule analyzed in the theorem; (ii) the node counts are random, not the deterministic n_j of Eq. (3); and (iii) even the expected counts match only when |f''| is nearly constant on each interval: E[#I_j] = N * int_{I_j}|f''|^{1/2} / int |f''|^{1/2}, which is not N sqrt(M_j)/sum sqrt(M_p) for functions with sharp internal variations, the exact regime of the paper's examples. Additionally, Theorem 3.1 is one-dimensional and scalar, while the PDE loss is two-dimensional and the sampling criterion uses the norm of a Hessian matrix. Thus the theorem does not establish a theoretical foundation for hessian-RAD, even if the 1D bound itself is correct. A secondary finite-N issue: the ceiling in Eq. (3) makes the realized trapezoid count N' = sum n_j >= N, so the comparison is not literally 'same number of trapezoids'; for small N this can make the refined bound larger than a uniform bound at N' points, although the asymptotic comparison survives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes hessian-RAD, an adaptive collocation-point sampling method for physics-informed neural networks (PINNs). The theoretical part (Section 3) analyzes a one-dimensional composite trapezoidal rule in which [a,b] is split into k intervals and the number of trapezoids per interval is chosen proportionally to sqrt(max |f''|); Theorem 3.1 claims that the resulting error bound is no larger than the uniform-trapezoid bound for the same nominal number of trapezoids, and is strictly smaller when |f''| varies. Section 4 embeds this idea in the RAD framework by sampling N collocation points i.i.d. from a density proportional to a power of the Hessian norm of the residual, and periodically resampling during training. Experiments on a 2D Poisson equation and a 2D diffusion-reaction equation compare unif-RAD, res-RAD, grad-RAD, and hessian-RAD in terms of final L2 error and convergence speed.","tokens_in":7908,"tokens_out":7572,"duration_ms":78612,"significance":"If stated and proved correctly, the one-dimensional quadrature result is a clean and useful elementary observation: for a fixed C^2 integrand, allocating trapezoids according to sqrt(|f''|) gives a worst-case error bound at least as good as uniform allocation. The paper is also commendable for making the RAD family explicit through Eq. (4) and for releasing code. However, the claimed theoretical foundation for the PINN sampling method is not currently established: the theorem concerns deterministic interval-max allocation for a fixed one-dimensional integrand, whereas Algorithm 1 uses stochastic i.i.d. sampling from a pointwise Hessian density and trains a time-varying network. The experimental evidence is also based on single runs without statistical repetitions. The core quadrature theorem is potentially publishable as a short theoretical note, but the PINN contribution needs reframing or additional analysis.","major_comments":[{"comment":"Theorem 3.1 analyzes a deterministic composite trapezoidal rule on an interval [a,b] for a fixed C^2 function, with interval lengths l=(b-a)/k and node counts proportional to sqrt(max_{I_j}|f''|). Algorithm 1, by contrast, draws N points i.i.d. from d(x) proportional to gamma(x)^tau, uses equal-weight Monte Carlo sums to estimate the PINN loss, and periodically resamples during training as the residual changes. None of the three components of the theorem -- fixed integrand, interval-max node allocation, and trapezoid weights -- is present in the PINN procedure. In particular, the expected number of Monte Carlo samples in interval I_j is proportional to int_{I_j}|f''|^{1/2}, which equals N sqrt(M_j) / sum_p sqrt(M_p) only when |f''| is nearly constant on I_j; the paper's own examples (Fig. 1 and Section 4) are precisely in the sharp-variation regime where this equality fails. Moreover, Theorem 3.1 is scalar and one-dimensional, while the PINN loss is a two-dimensional integral and the sampling criterion uses the norm of a Hessian matrix. Consequently, the abstract's claim that the PINN sampling is 'fondée théoriquement' on the theorem is not established. The authors should either prove a stochastic/discrete analogue or explicitly reframe Theorem 3.1 as motivation rather than as a guarantee for the PINN algorithm.","section":"§3.2, §4, Algorithm 1"},{"comment":"The ceiling in Eq. (3) implies that the realized total number of trapezoids N' = sum_j n_j is at least N, so the comparison in Theorem 3.1 is not literally between rules with the same number of trapezoids. The text acknowledges an excess of at most k, but the theorem statement says 'pour un même nombre de trapèzes.' For small N or large k this overhead can be material. Please either state and prove the comparison with N' = sum_j n_j on both sides, or use a rounding scheme (for example, largest remainder) that fixes the total number of trapezoids to exactly N.","section":"§3.2, Eq. (3)"},{"comment":"The displayed algebra in the proof sketch appears to contain an error. Substituting n_j from Eq. (3) into the bound gives a factor (sum_p sqrt(M_p))^2 / M_j, whereas the manuscript displays (sum_p sqrt(M_p) sqrt(M_j))^2. As printed, the chain of inequalities does not follow from the preceding line. Please correct the displayed formula and verify whether the proof uses the rounded or unrounded value of n_j; the comparison in the theorem statement depends on this distinction.","section":"§3.2, proof of Theorem 3.1"},{"comment":"The experimental claims that hessian-RAD converges much faster and reaches a lower final error are based on what appears to be a single run per method: no error bars, no repeated seeds, and no statistical test are reported. Given that the PINN advantage is the main application claim of the paper, this is insufficient support. Please provide multiple independent runs with reported means and variances, or clearly label the figures as illustrative single runs.","section":"§4.1, §4.2"}],"minor_comments":[{"comment":"The formula for d(x) is ambiguous as printed: the proportional symbol and the division by E[gamma(x)^tau] are mixed with the additive constant c. Please write the normalized density explicitly, e.g., d(x) = (gamma(x)^tau / E[gamma(X)^tau] + c) / Z, and clarify whether c is a smoothing constant added before or after normalization.","section":"§4, Eq. (4)"},{"comment":"The statement that the cost of computing M_j 'does not matter' is only valid for the one-dimensional illustration. In the PINN context, computing pointwise Hessian norms over tens of thousands of candidates at each resampling step is a real computational overhead, and the paper should report or at least discuss this cost in the complexity comparison of Figure 2.","section":"§3.3"},{"comment":"The caption of Figure 3 says 'des méthodes basées sur les résiduels' but the figure includes unif-RAD, grad-RAD, and hessian-RAD; please correct the caption to reflect the methods actually shown.","section":"§4, Figure 3"},{"comment":"The experimental comparison is limited to instantiations of the proposed generic Eq. (4) family. Since R3 sampling and gPINN are cited in the introduction and conclusion, a comparison with at least one of these methods, or an explicit statement that they are outside the scope, would strengthen the empirical claims.","section":"§4, Related work"},{"comment":"In Proposition 3.1 and Theorem 3.1, the notation uses f^2 and f'' interchangeably for the second derivative; please standardize the notation to avoid confusion with the square of f.","section":"§3.1, Eq. (2)"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress-test concern is valid and matches my reading: Theorem 3.1 is a correct-looking one-dimensional quadrature result, but it does not cover the stochastic, training-dependent, two-dimensional sampling scheme used in Section 4. The manuscript should either add a genuine analysis of the PINN sampling procedure or substantially soften the theoretical claims. A major revision focused on reframing, fixing the proof details, and adding experimental repetitions would be appropriate. Editors should also verify whether a French-language submission fits the journal's usual language policy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper's actual contribution is a practical adaptive sampling scheme for PINNs based on the Hessian of the residual, plus a clean 1D quadrature bound showing that allocating trapezoid nodes proportionally to sqrt(|f''|) beats uniform spacing in worst-case error. The theorem is textbook but correct; the proof sketch is coherent. The empirical results on two 2D PDEs show real improvements, especially on the Poisson problem where hessian-RAD converges much faster and reaches L2 error 7e-7 vs 6e-5 for uniform. That's worth something.\n\nThe soft spot is exactly what the stress-test note says. Theorem 3.1 analyzes a deterministic composite trapezoid rule with interval-wise node counts n_j = ceil(N sqrt(M_j)/sum sqrt(M_p)), where M_j is the max of |f''| on each interval. The PINN method uses i.i.d. sampling from a continuous density proportional to |∇²f|^τ, equal-weight Monte Carlo loss, and the node count per interval is a random variable whose expectation is proportional to ∫_{I_j}|f''|^{1/2}, not to sqrt(max|f''|). These differ when |f''| varies sharply inside intervals—which is exactly the regime in the examples. Also, the theorem is 1D scalar, while the PINN loss is 2D and uses the Hessian norm. So the paper's central claim of a 'theoretical foundation' for hessian-RAD is not actually established by Theorem 3.1. The quadrature bound itself is fine, but it doesn't transfer.\n\nOther soft spots: only two PDEs, no error bars or multiple seeds, and the hyperparameters τ and c are chosen without much analysis. The related work on adaptive sampling is cited properly.\n\nOverall: it's a decent empirical paper with a correct but mismatched theory. The reviewer should not reject on the theorem being wrong—it isn't—but should ask for the gap to be addressed or the claims scaled back. I'd send it for peer review; a good referee can push the authors to either prove a stochastic version or reposition the theory as motivation rather than justification. I'd cite the empirical method if I worked on PINN sampling.","headline":"A correct but mismatched theory: the 1D quadrature bound is solid, yet it does not cover the 2D stochastic PINN sampling scheme it is claimed to justify.","tokens_in":8490,"tokens_out":1558,"would_cite":true,"duration_ms":15520,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","41A55","62D05","35A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"A quadrature rule that spaces nodes in proportion to the square root of the local second-derivative magnitude has a provably no-worse error bound than uniform spacing, and it improves PINN collocation sampling.","keywords":["PINN","collocation points","adaptive sampling","quadrature rule","Hessian","trapezoidal rule","residual-based sampling","PDE surrogate"],"falsifier":"Compute $B_{\\mathrm{tot,refined}}$ and $B_{\\mathrm{tot,unif}}$ from the theorem's own formulas on a $C^2$ function with varying second derivative, say $f(x)=\\sin(1/\\sqrt{x})$ on $[0.1,1]$ with $N=25$, $k=10$: any valid choice with $B_{\\mathrm{tot,refined}}>B_{\\mathrm{tot,unif}}$ would refute Theorem 3.1. For the PINN claim, train Hessian-guided and uniform sampling on a PDE whose residual has spatially constant $|f''|$: the claimed strict advantage should disappear, and if it persists that would indicate the benefit comes from something other than the proven bound.","tokens_in":7354,"feed_emoji":"📐","tokens_out":10215,"duration_ms":93810,"temperature":0.7,"pith_summary":"Physics-informed neural networks replace a PDE's residual integral by a sum over collocation points, and how those points are chosen largely determines training quality. This paper proposes a quadrature rule that spaces the points according to the local magnitude of the integrand's second derivative, and proves a theorem: for the same total number of trapezoids, the error bound of this refined rule is never worse than the uniform rule's, and is strictly better as soon as the second derivative varies. The paper then instantiates the rule inside an adaptive collocation scheme where the sampling density is proportional to the Hessian of the residual, and reports faster convergence and lower L2 errors than uniform, residual-based, and gradient-based sampling on two 2D PDEs. The reason this matters is that adaptive collocation methods have so far come without a theoretical reason to prefer them; this gives a first such guarantee, albeit for a one-dimensional idealization of the loss integral.","feed_headline":"Curvature-aware sampling beats uniform quadrature — and guides PINNs","feed_subtitle":"A proven 1D quadrature bound gives adaptive PINN collocation a first theoretical edge over uniform sampling.","key_machinery":"The engine of the argument is the composite trapezoidal rule with subinterval counts $n_j=\\left\\lceil N\\sqrt{M_j}/\\sum_{p=1}^k\\sqrt{M_p}\\right\\rceil$, where $M_j=\\max_{x\\in I_j}|f''(x)|$. The local trapezoid error on a subinterval of length $l$ with $n_j$ trapezoids is $l^3|f''(\\xi_j)|/(12n_j^2)$; summing these local errors and applying the allocation formula plus $|f''(\\xi_j)|\\le M_j$ converts the sum into the global uniform bound, so the inequality becomes strict exactly when the second derivative is not constant. In the PINN application, the same object appears as $\\gamma(x)=|f''(x)|$ (or the Hessian norm) in the sampling density of Eq. (4), so the quadrature insight directly dictates where collocation points are redrawn during training.","core_discovery":"The central claim, Theorem 3.1, is that for any $C^2$ function $f:[a,b]\\to\\mathbb{R}$, splitting the interval into $k$ pieces and placing $n_j$ trapezoids in piece $j$ in proportion to $\\sqrt{M_j}$, where $M_j=\\max_{x\\in I_j}|f''(x)|$, yields a total quadrature error bound $B_{\\mathrm{tot,refined}}$ that is at most the uniform-trapezoid bound $B_{\\mathrm{tot,unif}}=(b-a)^3/(12N^2)\\max_x|f''(x)|$ for the same total number $N$ of trapezoids, with the inequality strict when $|f''|$ varies. The proof bounds each subinterval's error by $l^3/(12n_j^2)|f''(\\xi_j)|$ and then uses the allocation formula and $|f''(\\xi_j)|\\le M_j$ to telescope the sum into the uniform bound. On the PINN side, the paper specializes the generic sampling distribution $d(x)\\propto \\gamma(x)^\\tau/\\mathbb{E}[\\gamma(x)^\\tau]+c$ to $\\gamma(x)=|f''(x)|$ (the Hessian norm in 2D), giving a Hessian-guided collocation strategy, and its experiments show this strategy matching or beating uniform, residual-based, and gradient-based sampling in test error on 2D Poisson and diffusion-reaction equations.","pith_inferences":["Editorial inference: the theorem's strict-improvement condition (non-constant $|f''|$) suggests the practical gain of Hessian-guided sampling grows with how 'bumpy' the residual landscape is; on globally smooth residual problems the method should degrade gracefully toward uniform sampling rather than hurt.","Editorial inference: the same curvature-proportional allocation is a variance-reduction recipe for any Monte Carlo loss whose integrand has concentrated curvature; one testable extension is whether replacing uniform collocation with the density $d(x)\\propto |f''(x)|^\\tau+ c$ reduces stochastic gradient variance during early PINN training.","Editorial inference: because the transfer from Theorem 3.1 to the continuous 2D sampling scheme is not proven, a natural next step is to check numerically whether the bound's inequality still holds when the residual is evaluated at i.i.d. samples from this density rather than on a fixed refined grid."],"forward_implications":["With a fixed budget of $N$ collocation points, a 1D residual integral is approximated at least as well by curvature-proportional spacing as by uniform spacing, and strictly better when the residual's second derivative varies.","Adaptive collocation methods for PINNs gain a theoretical justification: this is the first among them to prove a bound relative to uniform sampling, in the 1D setting underlying the loss.","In the paper's 1D illustration on $f(x)=\\sin(1/\\sqrt{x})$, the refined rule cuts quadrature error from 16.4% to 1.89% with the same 25 trapezoids.","On the 2D Poisson benchmark, Hessian-guided sampling reaches a stabilized L2 error of about $7\\times10^{-7}$ after 20,000 iterations, versus $6\\times10^{-5}$ for uniform sampling; on diffusion-reaction it matches gradient-based sampling at $3\\times10^{-3}$ while residual-based sampling is 5 to 10 times worse.","Because Hessian computation is the expensive part, the authors suggest stochastic estimation of Hessian entries and combination with gradient-enhanced PINNs as low-cost ways to scale the approach."],"supporting_citations":[{"why":"Supplies Hamming's Theorem 20.5.1, the local trapezoid error bound used in Proposition 3.1 and Theorem 3.1.","marker":"[5]"},{"why":"Defines the PINN residual-loss formulation whose collocation integrals the paper samples.","marker":"[6]"},{"why":"Provides the residual-based adaptive distribution (RAD) framework and the uniform/residual comparison baselines.","marker":"[9]"},{"why":"Gives the gradient-based adaptive sampling baseline (grad-RAD) compared in the experiments.","marker":"[8]"},{"why":"Introduces gradient-enhanced PINNs, cited as a route to reuse Hessian computations cheaply.","marker":"[10]"},{"why":"Supplies convergence and error analysis of PINNs under uniform sampling, the theoretical baseline the paper positions itself against.","marker":"[3]"}],"fun_headline_variants":["Hessian quadrature proven to beat uniform for PINNs","Curvature-aware quadrature gives PINNs a proven edge","Quadrature theorem sharpens PINN collocation strategy","Curvature-guided collocation: first proof for PINNs","Adaptive PINN sampling gets a tight Hessian-based bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a worst-case bound proven for a fixed twice-differentiable function on a one-dimensional interval continues to describe what happens when the same curvature-proportional heuristic is used to draw random collocation points in a two-dimensional, time-dependent, and parameter-dependent residual loss during network training; the paper does not prove this transfer.","fun_headline_variants_meta":{"raw":{"variants":["Hessian quadrature proven to beat uniform for PINNs","Curvature-aware quadrature gives PINNs a proven edge","Quadrature theorem sharpens PINN collocation strategy","Curvature-guided collocation: first proof for PINNs","Adaptive PINN sampling gets a tight Hessian-based bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001352,"raw_usage":{"total_tokens":5498,"prompt_tokens":963,"completion_tokens":4535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":4450}},"tokens_in":579,"tokens_out":4535,"duration_ms":34580,"temperature":1.0,"reasoning_tokens":4450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:48:29.510044+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $B_{\\mathrm{tot,refined}}$ and $B_{\\mathrm{tot,unif}}$ from the theorem's own formulas on a $C^2$ function with varying second derivative, say $f(x)=\\sin(1/\\sqrt{x})$ on $[0.1,1]$ with $N=25$, $k=10$: any valid choice with $B_{\\mathrm{tot,refined}}>B_{\\mathrm{tot,unif}}$ would refute Theorem 3.1. For the PINN claim, train Hessian-guided and uniform sampling on a PDE whose residual has spatially constant $|f''|$: the claimed strict advantage should disappear, and if it persists that would indicate the benefit comes from something other than the proven bound.","supporting_citations":[{"cited_title":"France 2030","cited_arxiv_id":null,"evidence_quote":"Supplies Hamming's Theorem 20.5.1, the local trapezoid error bound used in Proposition 3.1 and Theorem 3.1."},{"cited_title":"Bajaj, L","cited_arxiv_id":null,"evidence_quote":"Defines the PINN residual-loss formulation whose collocation integrals the paper samples."},{"cited_title":"Girault, R","cited_arxiv_id":null,"evidence_quote":"Provides the residual-based adaptive distribution (RAD) framework and the uniform/residual comparison baselines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces gradient-enhanced PINNs, cited as a route to reuse Hessian computations cheaply."},{"cited_title":"1 a Mp V (3) avec Mj “ max xPIj |f2pxq|. En raison de la fonction r.s, řk j“1 nj « řk j“1 N ? Mj řk p“1 ? Mp “ N avec un écart au plusk, celui-ci devenant négligeable quandN","cited_arxiv_id":null,"evidence_quote":"Supplies convergence and error analysis of PINNs under uniform sampling, the theoretical baseline the paper positions itself against."}],"review_version":1}