REVIEW 4 major objections 5 minor 15 references
M\'ethode de quadrature pour les PINNs fond\'ee th\'eoriquement sur la hessienne des r\'esiduels
T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (4)
- [§3.2, §4, Algorithm 1] 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.
- [§3.2, Eq. (3)] 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.
- [§3.2, proof of Theorem 3.1] 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.
- [§4.1, §4.2] 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.
minor comments (5)
- [§4, Eq. (4)] 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.
- [§3.3] 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.
- [§4, Figure 3] 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.
- [§4, Related work] 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.
- [§3.1, Eq. (2)] 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.
Circularity Check
No circularity: Theorem 3.1 is a self-contained comparison against uniform trapezoids, and the PINN sampling heuristic is not a fitted prediction of the theorem.
full rationale
The paper's central theoretical result, Theorem 3.1, derives an upper bound for a deterministic composite trapezoid rule whose subinterval counts are proportional to sqrt(M_j), where M_j is the maximum of |f''| on each interval, and compares it to the uniform trapezoid bound of Proposition 3.1. Both bounds come from the same standard quadrature error formula (Hamming's Theorem 20.5.1), and the comparison is a direct inequality; no parameter is fitted to the target result and no result is assumed that is equivalent to the conclusion. The method's extension to PINNs in Section 4 uses a sampling distribution d(x) proportional to |f''|^tau (Eq. 4) as an algorithmic heuristic derived from the theorem, but the paper does not claim that Algorithm 1 achieves the theorem's bound; it only says the theoretical result 'leads us to conceive' the adaptive sampling method. The experiments use analytical solutions (Poisson and diffusion-reaction) as external ground truth, so the reported errors are not a renaming of the inputs. The only self-citation is [4], cited in the introduction as an example of prior work on uniform-sampling theory; it plays no role in deriving Theorem 3.1 or the hessian-RAD algorithm, so it is not load-bearing. The gap between the 1D deterministic trapezoid analysis and the 2D i.i.d. Monte Carlo sampling used in Algorithm 1 is a limitation in the transfer of theory, but it is a correctness/scope concern, not circularity. No circular step can be exhibited from the paper's equations or citations.
Assumptions & free parameters
free parameters (2)
- tau (concentration exponent) =
1/2
- c (smoothing constant) =
0
assumptions (5)
- standard math Lagrange interpolation error formula for degree-1 polynomials, including the existence of xi in the interval.
- standard math Composite trapezoidal rule error formula from Hamming [5, Th.20.5.1].
- standard math Cauchy-Schwarz inequality for the bound on the sum of sqrt(M_p).
- domain assumption The PDE residual and its hessian are well-defined and differentiable enough on the domain (C2 or smooth).
- domain assumption The neural network has sufficient capacity and the optimizer converges to a low-loss solution.
Cite this review
Pith. "Pith review of M\'ethode de quadrature pour les PINNs fond\'ee th\'eoriquement sur la hessienne des r\'esiduels." pith.science (2026). https://pith.science/paper/H3CIERVM
@misc{pith2026250620441,
author = {Pith},
title = {Pith review of: M\'ethode de quadrature pour les PINNs fond\'ee th\'eoriquement sur la hessienne des r\'esiduels},
year = {2026},
howpublished = {\url{https://pith.science/paper/H3CIERVM}},
note = {Machine review of arXiv:2506.20441}
}
read the original abstract
Physics-informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.
Figures
Reference graph
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Introduction Malgré des avancées scientifiques importantes en simulation numérique, résoudre efficace- ment des EDPs demeure un problème complexe et coûteux. En incorporant l’équation dans la fonction de perte et en minimisant ses résiduels en des points decollocation, les réseaux de neurones informés par la physique (PINNs) [6] sont apparus comme une sol...
work page Pith review arXiv 2025
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Notations On considère les EDPs de la formeBu Bt ` Nrus“ 0, où Nr¨s est un opérateur différentiel en temps et en espace, et oùupt, xq est la solution avectPr 0, Ts et xP Ω. Cette équation est généralement accompagnée de conditions initiales et limites :@x P Ω, Irusp0, xq “0, et@xPB Ω, tPr 0, Ts, Bruspt, xq“ 0, où B est l’opérateur de frontière appliqué àB...
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Bornes d’erreurs de quadrature Soient a, bP R et une fonction f : ra, bs ÝÑR. Pour résoudre la quadrature de f, on choisit x0, . . . , xN Pr a, bs et les poidsw0, . . . , wN peuvent être obtenus en approximantf par des fonctions polynomiales (méthode dite de Newton-Cotes). Afin d’éviter le phénomène de Runge, où l’interpolation polynomiale présente des pi...
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[4]
Échantillonnage adaptatif et PINNs Pour comparer les principales méthodes d’échantillonnage adaptatif pour les PINNs, nous utilisons le cadre de la méthode RAD [9], où lesN points de collocation sont tirés selon une distribution proportionnelle à un critère d’intérêt. Ce dernier peut prendre la forme des 6 A. CARADOT, R. EMONET, A. HABRARD, A.R. MEZIDI, E...
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[5]
Conclusion Nousavonsprésentéuneméthodedequadraturebaséesurlesdérivéessecondes. L’exploitation de la hessienne des résiduels montre également des résultats prometteurs dans une méthode d’échantillonnage adaptatif pour les PINNs. Mais l’utilisation de f2 peut devenir coû- teuse en haute dimension. Une direction porte sur une approche stochastique, où à chaq...
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N. Doumèche, G. Biau et C. Boyer : Convergence and error analysis of PINNs. arXiv:2305.01240 (2023). 10 A. CARADOT, R. EMONET, A. HABRARD, A.R. MEZIDI, ET M. SEBBAN
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Reviewed August 6, 2026 · model on record in the stance chip above.
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