Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method

Adrian Celaya; Beatrice Riviere; David Fuentes

REVIEW 2 major objections 5 minor 39 references

Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that adaptive collocation point selection driven by QR-DEIM on residual snapshots consistently lowers PINN errors on wave, convection, Allen-Cahn, and Burgers' equations compared with fixed and existing adaptive sampling…

desk verdict A well-run empirical study of a genuinely new QR-DEIM sampling scheme, but the headline accuracy advantage is not clean because baseline point budgets differ. read the letter →

arxiv 2501.07700 v4 pith:OOC3P7I5 submitted 2025-01-13 cs.LG cs.CEcs.NAmath.NA

Adrian Celaya , David Fuentes , Beatrice Riviere This is my paper · ORCID

classification cs.LGcs.CEcs.NAmath.NA

keywords physics-informedneuralnetworkspartialdifferentialequationsQR-DEIMadaptivecollocationpointselectionresidual-basedsamplingreduced-ordermodelingrandomizedSVD

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

Physics-informed neural networks solve PDEs by minimizing a residual loss on a set of collocation points, and the choice of those points strongly affects accuracy. This paper argues that fixed sampling, and even existing adaptive schemes that react to the current residual, miss useful information about how the residual evolves during training. It proposes two adaptive strategies, QR-DEIM and its randomized variant QR-DEIM-R, which build a matrix of residual snapshots over a training window, extract its dominant left singular vectors, and use QR column pivoting to select new collocation points that best represent the residual dynamics. On the wave, convection, Allen-Cahn, and Burgers' equations, both strategies consistently achieve lower relative $\ell^2$ errors than fixed, resampling, and residual-based adaptive baselines, and the randomized variant does so at a lower per-update cost.

What carries the argument

The central object is the residual snapshot matrix $R \in \mathbb{R}^{N_{\text{snapshot}} \times P}$, whose columns are pointwise PDE residuals evaluated over a fixed snapshot set during $P$ training iterations. The method computes its SVD $R = V \Sigma W^{\mathsf T}$, keeps the leading $k$ left singular vectors where $k$ satisfies an energy threshold, and applies QR decomposition with column pivoting to $V_k^{\mathsf T}$. The pivot indices select the snapshot points that form the new training points, mirroring how QR-DEIM selects interpolation indices for nonlinear model reduction. The accompanying convergence-degree vector $d = \log_2(\hat{r}_{\text{old}} / \hat{r}_{\text{new}})$ prunes training points whose residuals have decreased most, and QR-DEIM-R replaces the SVD with a randomized SVD and pivoting with a column-norm-weighted random sample, reducing the update complexity.

What would settle it

On a benchmark with a known solution, after training for a fixed number of iterations, compare the QR-DEIM-selected snapshot points against points with the largest pointwise residual magnitudes and against random points; if the QR-DEIM points do not yield a lower validation error when added as collocation points, or if they systematically miss regions where the error is largest, the central claim fails.

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Extended reading notes

Core claim

The paper's central claim is that the leading singular vectors of a residual snapshot matrix, combined with QR column pivoting, identify the space-time locations at which a PINN most needs new collocation points, and that acting on this information at regular intervals yields more accurate solutions than either fixed sampling or residual-based adaptive methods that only look at the current residual field. Concretely, after every P training iterations the method evaluates the PDE residual at a separate snapshot set, stacks these evaluations into a matrix, truncates its SVD to the modes capturing a specified energy fraction, and applies pivoted QR to the transposed mode matrix to select the snapshot points whose indices become new training points. It simultaneously removes training points whose residuals have decayed most, keeping the set size fixed. A randomized variant replaces the SVD and pivot selection with randomized linear algebra to cut update costs. Across four benchmark PDEs the method is reported to consistently yield lower mean relative $\ell^2$ errors than uniform, Hammersley, resampling, RAR-G, RAR-D, RAD, R3, and PINNACLE baselines.

Load-bearing premise

The load-bearing premise is that the snapshot points singled out by QR-DEIM's pivoted QR on the residual's dominant modes are the locations where adding collocation points will actually reduce the network's error, a connection the paper verifies only empirically on four one-dimensional benchmarks.

Editorial extensions

If this is right

PINN users can adopt QR-DEIM or QR-DEIM-R without changing architecture, loss, or optimizer; the sampling strategy is the only modification.
The methods automatically concentrate points in early-time regions for wave and convection problems and near sharp fronts for Allen-Cahn and Burgers, suggesting they can replace manual time-marching or domain decomposition.
The randomized variant's lower update cost makes it the scalable choice for larger snapshot sets or higher-dimensional PDEs, while matching the accuracy of the full QR-DEIM on the tested benchmarks.
Both methods reach lower validation loss in fewer iterations than fixed sampling, so the modest per-iteration overhead can be offset by faster convergence.
The energy threshold $\varepsilon$ and target rank $k$ have weak influence on accuracy over tested ranges, so default hyperparameters transfer across the four benchmarks.

Reading between the lines

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

If the dominant left singular vectors of residual snapshots are indeed a good proxy for where error concentrates, the same snapshot-subspace strategy could be ported to other neural PDE solvers, such as those using different loss weightings or operator learning, with minimal changes.
The observed early-time concentration for the convection equation resembles the effect of causality-weighted training, but here it arises purely from the sampling rule; an explicit comparison against causal training would reveal whether the two mechanisms are complementary or redundant.
The method's robustness to $\varepsilon$ and $k$ across 1D benchmarks suggests it may generalize to 2D or 3D problems, but the paper only tests 1D; testing on a 2D problem with a known solution would be the natural next step.
Since the pruning rule only removes points whose residual has decreased, it cannot detect regions where the residual is stagnating; a stagnation-aware criterion might further improve the update policy.

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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

First, the novel bit: this is the first use of QR-DEIM for forward PINN collocation sampling. The authors take residual snapshots over training, do an SVD, and use QR column pivoting to pick new points, plus a pruning rule that removes converged points. There is also a randomized variant that cuts update cost. That is a reasonable, well-motivated idea, and the writing is clear.

What the paper does well: the experimental protocol is careful in several ways - 10 runs per method, same test grid, validation-based model selection, and ablations on epsilon, rank, snapshot count, and period. The adaptive behavior is visually plausible: the method concentrates points where the shock or front is. The randomized variant tracks the full QR-DEIM results closely, which is good evidence the randomization is not hurting.

The soft spots are real, and one is load-bearing. Section 2.3 says "We use 2,000 collocation points for all of our benchmark PDEs," but Table 1 shows RAR-G and RAR-D start with 1,000 and add 10 per update every 1,000 iterations. Over a 100,000-iteration run they only reach 2,000 at the final update, so for most of training they have half as many collocation points as QR-DEIM. PINNACLE starts at 1,000 and adds 100 per update, growing to 11,000 points. That means the comparisons do not isolate the sampling strategy, despite the stated intention. The accuracy gap between QR-DEIM and RAR-G/D could partly be the point count. This needs to be fixed by recomputing with equal point budgets, for example by giving RAR-G/D 2,000 initial points or matching total compute.

The other issues are minor by comparison. No code is released, which matters because PINN sampling papers live or die on reproducibility of the exact schedule. The theoretical grounding is thin - there is no error bound connecting residual subspace selection to solution error - but the paper does not claim one, and the ablations show robustness to hyperparameters. The benchmarks are all 1D, so calling the method "generalizable" is overreach, but that is a scope limit, not a flaw.

Who is this for? PINN practitioners who want an adaptive sampling method that tracks temporal dynamics. It deserves a serious referee, but the manuscript needs a revision that fixes the point budget confound and releases code.

My recommendation: send to peer review, with the explicit expectation that the comparison be redone under equal point budgets. If the advantage survives that, it is a useful contribution.

Referee Report

2 major / 5 minor

Summary. The paper proposes two adaptive collocation point selection strategies for physics-informed neural networks, based on the QR-DEIM algorithm and a randomized variant. The methods build a residual snapshot matrix over a separate snapshot set during training, use an SVD to identify dominant residual modes, and select new collocation points via QR column pivoting, while pruning converged training points. The authors evaluate the methods on the wave, convection, Allen-Cahn, and Burgers' equations, compare against fixed and adaptive baselines, and report mean relative L2 errors over ten runs, together with ablations on the energy threshold, rank, snapshot count, and snapshot period. The central claim is that the proposed strategies consistently achieve lower errors than existing techniques.

Significance. If the empirical claim holds, the paper makes a useful contribution by importing reduced-order modeling tools into PINN collocation sampling, with a concrete algorithm and extensive ablations. The experimental protocol has strengths: ten runs per configuration, a common test grid, validation-based model selection, and ablations over several hyperparameters. The main limitations are that the comparison does not hold the collocation point budget constant across baselines, that the margins on two of the four benchmarks are small relative to the reported standard deviations, and that the results are restricted to one-dimensional problems with a single optimizer and network architecture. The proposed heuristic is plausible but is not supported by an error bound or convergence analysis; its value rests on the empirical evidence.

major comments (2)

[Section 2.3, Table 1] The stated goal 'To isolate the effects of the sampling strategies' is not met because the baseline methods are not given the same collocation point budget as the proposed methods. RAR-G and RAR-D begin with 1,000 points and add 10 every 1,000 iterations, so over a 100,000-iteration run they reach 2,000 points only at the final update; PINNACLE starts at 1,000 and adds 100 per update, ending at 11,000 points. The proposed QR-DEIM methods maintain a fixed budget of 2,000 points throughout. The sentence 'We use 2,000 collocation points for all of our benchmark PDEs' is therefore contradicted by Table 1. As a result, the accuracy differences in Table 2 may reflect the number of collocation points rather than the sampling strategy. Please rerun the comparison with matched point budgets, or provide an explicit control showing that the budget difference does not drive the reported improvements.
[Table 2, Section 4] The central claim that QR-DEIM and QR-DEIM-R 'consistently yield lower errors than existing techniques' is not supported by appropriate statistical evidence. On the Allen-Cahn benchmark, QR-DEIM-R gives 3.85e-03 (1.87e-03) versus RAR-G at 5.14e-03 (9.35e-04); on Burgers' equation, QR-DEIM gives 5.96e-04 (7.07e-05) versus RAR-D at 7.31e-04 (1.78e-04). In both cases the difference is smaller than the combined standard deviation with n=10, so the advantage may not be statistically significant. The paper reports only means and standard deviations and does not perform significance tests or effect-size calculations. Please add such analyses or temper the 'consistently' claim for benchmarks where the improvement is marginal.

minor comments (5)

[Section 2.2, Eq. (15)] The notation 'min {i = 1,...,P | ...}' should be phrased as 'the smallest i in {1,...,P} such that ...' to avoid ambiguity about whether the set is over i or over a condition.
[Section 3.5] The statement 'We observe similar patterns of robustness across the other benchmark problems' is not accompanied by tables or figures for the other problems; please provide the data or explicitly label this as a qualitative observation.
[Section 3.6] The runtime comparison with baseline adaptive methods is informal; reporting measured wall-clock times for each method would make the claim 'roughly in the middle of the spectrum' concrete.
[Figures 3 and 5] Adding error bars or shaded regions across the ten runs would help the reader assess whether the temporal concentration behavior is consistent.
[General] The manuscript does not include a reproducibility statement or code release; providing code and random seeds would strengthen the empirical claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: QR-DEIM sampling is a fixed residual-driven heuristic, and the reported accuracy gains are empirical rather than derived from a fitted input.

full rationale

Walking the derivation chain, QR-DEIM and QR-DEIM-R select collocation points from residual snapshot matrices via SVD and pivoted QR, with pruning based on pointwise residual decrease. None of these quantities is constructed from the test errors or target solutions used in Table 2, and no parameter is fitted to the reported errors; the hyperparameters epsilon, k, and z are fixed defaults with ablation robustness checks. The method borrows QR-DEIM from external reduced-order modeling literature, and the comparison baselines are external published methods, so there is no load-bearing self-citation chain. The central claim is therefore empirical, not circular. One experimental-design concern should be stated separately: Section 2.3 says 'We use 2,000 collocation points for all of our benchmark PDEs,' but Table 1 shows RAR-G and RAR-D start at 1,000 points and add only 10 per update, while PINNACLE grows to 11,000 points, so the baseline point budgets are not controlled throughout training. This undermines the strength of the 'consistently yield lower errors' comparison, but it is a confounding-variable issue, not a reduction of a prediction to its inputs by construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The method's central mechanism rests on the heuristic that residual dynamics, summarized by a truncated SVD and QR pivoting, reveal where new collocation points are most needed. No new physical or mathematical entities are introduced. The listed hyperparameters are manually chosen defaults, not fitted to the reported test errors.

free parameters (5)

energy threshold epsilon = 0.005
Controls how many singular vectors of the residual snapshot matrix are retained, hence how many new points are added per update. Chosen as default; ablation in Table 3 shows modest sensitivity.
target rank k (QR-DEIM-R) = 100
Rank of the randomized SVD approximation. Chosen as default; ablation in Table 4 shows moderate variation in error.
oversampling parameter z = 50
Extra columns sampled in the randomized pivot selection. Authors note minimal effect on error.
number of snapshot points N_snapshot = 1000
Size of the candidate pool for new collocation points. Default; ablation in Tables 5-6 shows robustness.
snapshot period P = 1000
Number of iterations between updates of the training set. Default; ablation shows modest effect.

assumptions (4)

domain assumption The dominant left singular vectors of the residual snapshot matrix correspond to regions where the PINN residual is persistently large or dynamically active.
This heuristic motivates point selection in Section 2.2; it has no theoretical error bound.
domain assumption Removing points with high convergence degree (large residual decrease) does not harm final accuracy.
Pruning rule in Algorithm 1, Steps 8-13 assumes that converged points can be safely replaced by new under-resolved points.
domain assumption Strong enforcement of initial and boundary conditions via an output transform (Equation 7) is valid for the benchmarks considered.
Section 2.1.1 explicitly notes this simplifying assumption, used for all methods in the comparison.
standard math SVD, QR with column pivoting, and randomized SVD provide numerically reliable decompositions of the residual matrices.
Standard numerical linear algebra; referenced to [22,33].

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Cite this review

Pith. "Pith review of Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method." pith.science (2026). https://pith.science/paper/OOC3P7I5

@misc{pith2026250107700,
  author       = {Pith},
  title        = {Pith review of: Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOC3P7I5}},
  note         = {Machine review of arXiv:2501.07700}
}

read the original abstract

Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs demonstrate that our QR-DEIM-based approaches improve PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.

Figures

Figures reproduced from arXiv: 2501.07700 by the authors.

**Figure 1.** Visualization of u(x, y) = 0.0005x 2 (x − 1)2y 2 (y − 1)2 exp(10x 2 + 10y) with randomly sampled collocation points shown as red dots. While fixed sampling methods work well for simple PDEs, they may not be sufficient for more complex problems. Inspired by adaptive mesh refinement in finite element methods, [2] introduced the residual-based adaptive refinement method, which adds new residual points in locations wit… view at source ↗

**Figure 2.** Visualization of a typical PINN training loop. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Fraction of collocation points in the first half of the time domain over the course [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Evolution of points for the QR-DEIM (left) and QR-DEIM-R (right) methods [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Fraction of collocation points in the first half of the time domain over the course [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Training collocation points (black dots) overlaid on top of the true solution for [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: Training collocation points (black dots) overlaid on top of the true solution for [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

**Figure 8.** Figure 8: Average validation loss curves for the wave equation for QR-DEIM, QR-DEIM [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]

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M. Penwarden, A. D. Jagtap, S. Zhe, G. E. Karniadakis, and R. M. Kirby, “A unified scalable framework for causal sweeping strategies for physics-informed neural networks (pinns) and their temporal decompo- sitions,” Journal of Computational Physics, vol. 493, p. 112464, 2023

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Extended physics-informed neu- ral networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equa- tions,

A. D. Jagtap and G. E. Karniadakis, “Extended physics-informed neu- ral networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equa- tions,” Communications in Computational Physics, vol. 28, no. 5, 2020

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Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks,

C. L. W. Zhao and Jia, “Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks,” Communications in Computational Physics, vol. 29, pp. 930–954, June 2021. 34

2021

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Physics-informed machine learning,

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,”Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

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DeepXDE: A deep learning library for solving differential equations,

L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, “DeepXDE: A deep learning library for solving differential equations,”SIAM review, vol. 63, no. 1, pp. 208–228, 2021

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Physics-informed neu- ral networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neu- ral networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,”Journal of Computational physics, vol. 378, pp. 686–707, 2019

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Physics- informed neural networks (PINNs) for fluid mechanics: A review,

S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, “Physics- informed neural networks (PINNs) for fluid mechanics: A review,”Acta Mechanica Sinica, vol. 37, no. 12, pp. 1727–1738, 2021

2021

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Multiphysics- informed neural networks for coupled soil hydrothermal modeling,

Y. Wang, L. Shi, X. Hu, W. Song, and L. Wang, “Multiphysics- informed neural networks for coupled soil hydrothermal modeling,” Water Resources Research, vol. 59, no. 1, p. e2022WR031960, 2023

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3D multi-physics uncertainty quantification using physics-based machine learning,

D. Degen, M. Cacace, and F. Wellmann, “3D multi-physics uncertainty quantification using physics-based machine learning,”ScientificReports, vol. 12, no. 1, p. 17491, 2022

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Systems biology informed deep learning for inferring parameters and hidden dynamics,

A. Yazdani, L. Lu, M. Raissi, and G. E. Karniadakis, “Systems biology informed deep learning for inferring parameters and hidden dynamics,” PLoS computational biology, vol. 16, no. 11, p. e1007575, 2020. 30

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A comprehensive study of non-adaptive and residual-based adaptive sampling for physics- informed neural networks,

C. Wu, M. Zhu, Q. Tan, Y. Kartha, and L. Lu, “A comprehensive study of non-adaptive and residual-based adaptive sampling for physics- informed neural networks,” Computer Methods in Applied Mechanics and Engineering, vol. 403, p. 115671, 2023

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