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REVIEW 3 major objections 6 minor 64 references

RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read By treating collocation points and function samples as trainable parameters and moving them up the PDE-residual gradient, RAMS concentrates sampling where models err, improving PINN and operator-learning accuracy without adding data.

desk verdict A genuinely new sample-movement idea for PINNs and operators, broadly benchmarked, but the data-driven operator-learning claim is confounded by a missing RAR-G-only control and the 'first' framing oversells. read the letter →

arxiv 2509.01234 v1 pith:PBAOWENI submitted 2025-09-01 cs.CE cs.LGphysics.comp-ph

Weihang Ouyang , Min Zhu , Wei Xiong , Si-Wei Liu , Lu Lu This is my paper
classification cs.CEcs.LGphysics.comp-ph
keywords adaptivesamplingphysics-informedneuralnetworksoperatorsDeepONetresidual-basedhigh-dimensionalPDEsadversarialgradientscientificmachinelearning
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 tackles the sample-efficiency bottleneck in physics-informed and data-driven learning of PDEs: more collocation points help accuracy but raise cost, and existing adaptive samplers that target high-residual regions become expensive or infeasible in high dimensions and in operator learning. It proposes RAMS, which treats the samples themselves—collocation coordinates in PINNs, input function parameter vectors in DeepONets—as trainable parameters and moves them by gradient ascent on the squared PDE residual. This concentrates samples where the current network most violates the PDE, and a projection step keeps them inside the valid domain or inside the intended random-field class. Integrated with six existing samplers and tested on ten problems, RAMS is claimed to reduce error substantially without increasing sample count, to make random sampling viable up to d=10, and to be the first efficient adaptive sampling method for operator learning. A reader should care because the same cheap mechanism appears to work across PINN, physics-informed operator learning, and data-driven operator learning, turning sample selection into a small optimization problem.

What carries the argument

The central mechanism is the adversarial-gradient sample move: for each sample ξ (a collocation point in PINN, or a function parameter vector in DeepONet), compute the gradient ∇_ξ(F[N_θ](ξ))² by automatic differentiation, update ξ by gradient ascent for n_RAMS steps to maximize the physics residual, then project back with P—a nearest-point or clamping projection for spatial coordinates, and a kernel-smoothing operation P(f)=Kf⊘k′ for Gaussian-random-field-parameterized functions. This converts sample selection from a global search over a high-dimensional domain into a local optimization against the current network, which is cheap because the number of sample parameters is small relative to

What would settle it

Run RAMS on a d=10 PDE whose exact solution has a narrow isolated spike (or several isolated spikes) and compare final test error and achieved residual against uniform random sampling at equal wall-clock time; if the gradient ascent pulls samples into a few sharp maxima while average test error does not improve, or if random sampling matches RAMS once compute is equalized, the load-bearing assumption fails. A second check: in operator learning, compare RAMS with and without the kernel-smoothing projector; if unsmoothed moved functions train just as well, then the projection is not the reason t

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Extended reading notes

Core claim

The paper claims that the costly search for high-residual regions—the bottleneck of adaptive sampling for PINNs and the reason no adaptive sampler existed for operator learning—can be replaced by a cheap local move: treat the samples as trainable parameters and run a short gradient ascent on the squared PDE residual with respect to the sample coordinates. Moving samples along this adversarial gradient concentrates them where the current network most violates the PDE, and a projection step keeps them in the valid domain; for operator inputs, kernel smoothing keeps the moved functions within the Gaussian-random-field family. The paper reports that this mechanism, bolted onto random, LHS, Halto

Load-bearing premise

A short run of gradient ascent on the physics residual with respect to the current sample coordinates finds regions whose denser sampling improves the network's true test error, and for operator learning the kernel-smoothed moved functions remain legitimate samples from the intended random-field distribution; the paper checks this empirically on its test problems but gives no distributional or generalization argument.

Editorial extensions

If this is right

  • With the same number of collocation points, RAMS reduces PINN relative L2 error by about an order of magnitude for non-adaptive samplers and by roughly 40–95% for adaptive samplers on the Burgers, wave, and Poisson tests.
  • On high-dimensional PDEs, random sampling with RAMS stays below 10^-2 relative error at d=10 where plain random sampling saturates near error 1, and its measured training cost grows approximately linearly in dimension rather than exponentially.
  • For operator learning, RAMS provides the first efficient adaptive sampling strategy, with physics-informed DeepONets matching the accuracy of a flow-based sampler using roughly 3% of its training samples on the dynamic-system benchmark.
  • In data-driven operator learning, RAMS acts as an active-learning rule, reducing error by up to 63% on the wave-equation problem and about 40% on both 2D Burgers cases with the same total dataset size.
  • The extra cost of RAMS is small relative to network training: at n_RAMS=400 the reported overhead is about 2.1% of training time.

Reading between the lines

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

  • An implication left implicit in the paper is that the same residual-gradient moving-sample loop should transfer to other operator architectures and to boundary or initial-condition residual losses, since it only requires differentiability with respect to sample inputs; a cheap test would be running RAMS on a Fourier-neural-operator training loop with the same physics loss.
  • The gain presumably depends on the residual landscape being smooth enough for local gradient ascent to locate informative regions; in problems with oscillatory or measure-zero residual maxima, a wall-clock-matched comparison against uniform sampling would show whether the move pays for itself.
  • For operator inputs, RAMS optimizes only the sensor values and then projects by kernel smoothing, which biases moved functions toward the covariance kernel's smoothness; a testable extension is to move samples in a whitened or latent function representation that respects the Gaussian-random-field prior exactly.
  • If the heuristic is right, it suggests a broader principle: sample adaptation can be framed as an inner maximization against the current model, so the same loop could drive active data collection in other scientific machine learning settings, not just collocation-point selection.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces RAMS (residual-based adversarial-gradient moving sample), a sampling strategy for physics-informed and data-driven neural PDE solvers. Samples—collocation points in PINN or input functions in operator learning—are treated as trainable parameters and moved by gradient ascent on the PDE residual, optionally projected back to the admissible domain (e.g., by kernel smoothing for GRF-based operator inputs). The method is integrated with random, LHS, Halton, RAR-G, RAR-D, and R3 sampling, and is evaluated on a broad set of problems: 1D Burgers, 1D wave, 2D Poisson, a high-dimensional (up to d=10) Poisson-type PDE, physics-informed operator learning for four PDEs, and data-driven operator learning for two PDEs. The central claim is that RAMS improves accuracy without increasing the number of training samples, and that it is the first efficient adaptive sampling approach for operator learning. The paper reports large error reductions in many settings, e.g., random PINN Burgers error dropping from 0.181 to 0.010, and high-dimensional random sampling remaining below 10^-2 at d=10 with 20,000 points.

Significance. If the claims hold, RAMS would be a valuable, easily integrable plug-in for several sampling schemes, with demonstrated benefits across PINN, physics-informed operator learning, and data-driven operator learning. The manuscript's strengths are its broad benchmark suite, repeated independent runs, explicit hyperparameter tables, the projector-ablation study in Appendix D, and the direct with/without-RAMS comparisons in the PINN and physics-informed operator-learning sections. These elements support the accuracy-improvement claim more strongly than is typical for a new sampling heuristic. However, the data-driven operator-learning experiments contain a missing control that prevents attribution of the reported gains to the RAMS movement step specifically. The high-dimensional cost comparison is also weakened by per-method normalization and early termination of the baseline. No code or data are provided, which is a reproducibility concern given the number of tunable hyperparameters. Overall, the central idea is plausible and the empirical evidence is substantial for two of the three claimed application areas, but the third pillar and the cost-scaling claim need additional support befor

major comments (3)
  1. [§3.3, Algorithm 6, Figs. 8–9] The data-driven operator-learning experiments compare 'RAR-G with RAMS' against vanilla random sampling only. Algorithm 6 bundles two mechanisms: RAR-G's residual-based selection of m candidates from a pool of M, and RAMS's gradient-ascent movement of those selected candidates. With no arm that runs RAR-G without RAMS (n_RAMS=0) under the same nini, M, m, and tr, the reported reductions—up to 63% in Fig. 8C and about 40% in Fig. 9—cannot be attributed specifically to the RAMS movement step. The residual-selection step alone may account for a large share of the gain. Since the abstract and title credit RAMS, not the combined RAR-G+RAMS pipeline, a proper ablation (RAR-G without RAMS, with n_RAMS=0) is load-bearing and should be added.
  2. [§3.2.3, §3.3.1, Table 3–8] The method's performance depends on several hyperparameters whose optimal values vary by problem: n_RAMS gives best results at 300 in §3.2.3 (Fig. 6A) but at 200 in §3.3.1 (Fig. 8B), while tr, p, and the trainable fraction differ across all experiments. The paper notes that tuning is important but gives no selection criterion or default heuristic. This makes the 'consistent improvement' claim conditional on problem-specific tuning and complicates fair comparison, especially because the 'same sample count' comparisons do not count the extra computational cost of the RAMS gradient-ascent iterations. The paper should provide a sensitivity analysis or a principled way to set these parameters, and report the relative wall-clock overhead of RAMS for each experiment (not only Fig. 5D).
  3. [§3.1.4, Fig. 3B] The claim that RAMS reduces computational cost and scales 'approximately linearly' in dimension is based on a comparison where each method's cost is normalized by its own cost at d=2, the stopping criteria differ (doubling |T| versus tuning tr and n_RAMS), and the random-sampling baseline was not run for d≥7 due to prohibitive runtime. These choices make the normalized growth curves not directly comparable and leave the linear-scaling claim without a direct wall-clock comparison. Reporting absolute runtimes or total numbers of network forward/backward passes for both methods across d would provide the needed evidence.
minor comments (6)
  1. [§2.1] Typo: 'spatial-temperate coordinate' should be 'spatial-temporal coordinate'.
  2. [§3.2.1, equation] The diffusion-reaction equation is written as ∂u/∂t = D ∂²u/∂t² + ku² + v(x); the second derivative should be with respect to x, not t.
  3. [Appendix D, Fig. 14] The text in Appendix D says the ablation is for the diffusion-reaction equation, but the caption of Fig. 14 says 'advection equation'. Please correct the inconsistency.
  4. [§3.3.1, Fig. 8] The claim of 'up to 63%' error reduction is based on three independent runs with error bars only for RAR-G with RAMS. Given the small number of runs, adding the mean and spread for the random baseline (or reporting the underlying values) would improve the presentation.
  5. [Abstract] The statement that RAMS is 'the first efficient adaptive sampling approach for operator learning' is strong. The paper compares with DAS2 only in one setting and does not survey all prior adaptive operator-learning samplers; consider softening to 'the first efficient' or adding a more systematic comparison.
  6. [Reproducibility] No code or data are provided. Given the many hyperparameter choices and the custom algorithms, releasing code and the generated datasets would substantially strengthen reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: RAMS's residual-maximization objective is the algorithm itself and all reported claims are tested on independent test sets; the §3.3 missing-control issue is an attribution concern, not circularity.

full rationale

No load-bearing circular step is present. RAMS is defined as gradient ascent on the physics residual with respect to sample coordinates (Algorithm 2 and Section 2.4.1: "we treat the samples ... as trainable parameters. These trainable samples are optimized to maximize L_phy via gradient ascent based on the gradient ∇ξ(F[Nθ](ξ))²"). This is the method's definition, not a fitted target relabeled as a prediction. The central accuracy claims are measured on independent test data, e.g., Section 3.1.4 evaluates on two disjoint test sets (“E computed over two disjoint test sets [51]”), and the operator-learning examples test on held-out correlation lengths not used in training. No equation reduces by construction to a quantity fitted from the data, and no pre-fit parameter is renamed as a prediction. The many self-citations (DeepXDE [16], DeepONet [28], the sampling study [47], and related operator-learning papers) supply infrastructure, baselines, and background rather than a uniqueness theorem or a forced choice; none is invoked to rule out alternative explanations for RAMS's performance. The one legitimate methodological concern is in Section 3.3: Algorithm 6 bundles RAR-G's residual-based selection with RAMS's gradient movement, and the baselines in Figures 8-9 are vanilla random sampling only, so the marginal contribution of the movement step is not isolated. That is an experimental attribution weakness, not circularity, because no step in the derivation is equivalent to its input by construction. A score of 1 reflects the presence of ordinary self-citation in related work, not load-bearing circularity.

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

The method introduces no new physical entities and no theoretical free constants. The free parameters are algorithmic hyperparameters that must be tuned per problem, and the central claim depends on the residual-as-informativeness assumption plus the projection assumption for operator inputs.

free parameters (4)
  • n_RAMS (sample-movement iterations) = 5 to 400, tuned per problem
    Controls how far each sample is moved; the paper shows the best value varies by problem (300 for Poisson, 200 for the wave example, higher is better for advection).
  • tr (number of resampling stages) = 20 to 160; tuned from 10 upward in the high-dimensional cost test
    Controls how often sample movement is applied; tuned per benchmark and used as a stopping knob in the high-dimensional cost experiment.
  • trainable sample fraction (|T2| or p) = e.g., 10-50% of samples; p=0.1, 0.2, 0.4 for 2D Burgers
    Determines how many samples RAMS moves; the authors report p=0.1 is best for Burgers, and performance varies with the split.
  • RAMS and network Adam learning rates = 1e-2 for sample movement, 1e-3 for network training
    Fixed by hand across experiments; no sensitivity study is reported for the sample-movement learning rate.
assumptions (4)
  • domain assumption PDE residual magnitude is a reliable proxy for where additional training samples reduce generalization error.
    The motivating assumption of all residual-based adaptive sampling (RAR, RAR-D, R3) and of RAMS; stated in Section 2.4.1 and used in Algorithms 4-6.
  • ad hoc to paper Gradient ascent on the residual with respect to sample coordinates finds globally informative high-residual regions.
    RAMS's core update rule; no convergence or coverage guarantee is provided for high-dimensional residual landscapes.
  • domain assumption Kernel smoothing keeps optimized input functions in the Gaussian random field prior class.
    Used as projection P in Section 2.4.1 for operator learning; supported only by the Appendix D ablation and not by a distributional argument.
  • domain assumption Numerical solvers that generate labels for data-driven operator learning are exact enough.
    Data-driven RAMS uses solver outputs as ground truth in Section 3.3; any solver error transfers to training and evaluation.

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Cite this review

Pith. "Pith review of RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations." pith.science (2026). https://pith.science/paper/PBAOWENI

@misc{pith2026250901234,
  author       = {Pith},
  title        = {Pith review of: RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBAOWENI}},
  note         = {Machine review of arXiv:2509.01234}
}
read the original abstract

Physics-informed neural networks (PINNs) and neural operators, two leading scientific machine learning (SciML) paradigms, have emerged as powerful tools for solving partial differential equations (PDEs). Although increasing the training sample size generally enhances network performance, it also increases computational costs for physics-informed or data-driven training. To address this trade-off, different sampling strategies have been developed to sample more points in regions with high PDE residuals. However, existing sampling methods are computationally demanding for high-dimensional problems, such as high-dimensional PDEs or operator learning tasks. Here, we propose a residual-based adversarial-gradient moving sample (RAMS) method, which moves samples according to the adversarial gradient direction to maximize the PDE residual via gradient-based optimization. RAMS can be easily integrated into existing sampling methods. Extensive experiments, ranging from PINN applied to high-dimensional PDEs to physics-informed and data-driven operator learning problems, have been conducted to demonstrate the effectiveness of RAMS. Notably, RAMS represents the first efficient adaptive sampling approach for operator learning, marking a significant advancement in the SciML field.

Figures

Figures reproduced from arXiv: 2509.01234 by the authors.

Figure 1
Figure 1. Workflow of ML for solving PDEs. The proposed RAMS treats the samples, either the spatial-temporal coordinates or the functions of interest, as trainable parameters and moves them through the gradient-based optimization to maximize the physics loss defined by the underlying PDEs. Both data-driven and PI training paradigms are fundamentally reliant on training samples, for either collecting data samples or computing … view at source ↗
Figure 2
Figure 2. Comparison of different sampling methods with and without RAMS for PINN problems. (A) Burgers’ equation. (B) Wave equation. (C) Poisson equation. Vertical error bars denote one standard deviation. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. RAMS for high-dimensional PDEs. (A) Relative MSE of random sampling with and without RAMS for different dimensions d. (B) Minimal training computational cost Td required to reach RMSE ≤ 10−3 for each method. The cost is normalized by each method’s cost at d = 2 (Td=2). The inset panel reports the minimal number of collocation points needed by random sampling without RAMS, which grows exponentially with d. increase i… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: PI operator learning of the diffusion-reaction equation. The correlation length ltrain of training data is randomly sampled from U(0.1, 0.8). Relative L 2 errors for different test data correlation lengths ltest are computed from eight independent runs. (A) Random samp…
Figure 5
Figure 5. Figure 5: PI operator learning of the advection equation. (A) Relative L 2 error on the test dataset with the correlation length ltest = 0.2. (B) ltest = 0.1. (C) ltest = 0.05. The red lines are the mean values, and vertical bars indicate one standard deviation from eight indepe…
Figure 6
Figure 6. Figure 6: PI operator learning of the Poisson equation. (A) Relative L 2 error for different sampling methods with different sample-training iterations in RAMS. (B) Relative L 2 error for different sampling methods with different training dataset sizes. 3.2.4 One-dimensional dyn…
Figure 7
Figure 7. Figure 7: PI operator learning of a dynamic system. RAR-G with RAMS and random sampling with RAMS achieve comparable accuracy to DAS2 [52] while using significantly fewer samples. 1,000 and 2,500 samples in our methods match the performance of DAS2 trained with 25,000 and 75,000…
Figure 8
Figure 8. Figure 8: Data-driven operator learning of the wave equation with discontinuous ve￾locity. (A) Relative L 2 error for DeepONets trained with 50 samples. (B) Relative L 2 error for DeepONets trained with 100 samples. (C) Relative L 2 error for DeepONets trained with 200 sam￾ples.…
Figure 9
Figure 9. Figure 9: Data-driven operator learning of the two-dimensional Burgers’ equation. (A) Relative L 2 errors of Case 1 for different trainable sample sizes p and numbers of resampling stages tr. (B) Relative L 2 errors of Case 2. For clarity, the one standard deviation from three i…
Figure 10
Figure 10. Figure 10: Visualization of PINN results. (A) Burgers’ equation. (B) Wave equation. (C) Poisson equation. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Representative examples of PI operator learning. (A) Diffusion-reaction equa￾tion. (B) Advection equation. (C) Poisson equation. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Representative examples of the wave equation with discontinuous velocity by data-driven operator learning. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: Representative examples of two-dimensional Burgers’ equation by data￾driven operator learning. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: Function samples optimized by RAMS with and without projector for the advection equation in PI operator learning. For each case, five function samples are visual￾ized. References [1] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ron￾…

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