Pith. sign in

REVIEW 4 major objections 5 minor 88 references

Backpropagating through a PDE solver during PGD attacks yields stronger, steadier attack losses for neural operators, and training on the resulting hardest samples improves out-of-distribution generalization.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 08:44 UTC pith:EJJZISTC

load-bearing objection The with-solver attack comparison is a real step beyond the dictionary-based baseline for neural operators, but the abstract promises more than the experiments deliver and the solver-gradient path itself is never validated. the 4 major comments →

arxiv 2510.18989 v3 pith:EJJZISTC submitted 2025-10-21 cs.LG

Solver-Integrated Adversarial Attacking and Training of Neural Operators

Yifei Sun This is my paper
classification cs.LG
keywords neural operatorsadversarial trainingPGD attackPDE solversout-of-distribution generalizationdifferentiable solversFourier Neural Operatoractive learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that the right way to attack and harden a neural operator—a network mapping an input function such as an initial condition to a solution function—is to differentiate through both the operator and the numerical PDE solver that provides ground truth. The author formalizes generalization and robustness through a model-solver error operator and shows that when the solver's output also changes with the input, attacks that freeze or approximate the solver mischaracterize worst-case behavior. The proposed 'with-solver' PGD attack, which backpropagates through all solver time steps, produces larger and more steadily increasing losses than detached or dictionary-approximated variants on 1D Burgers and 2D Navier–Stokes; the same attacked samples can then be used in adversarial training, with a round-by-round active-learning scheme lowering out-of-distribution loss for most tested input distributions. The paper also reports that batch-by-batch and random-constant training can degrade original train/test loss, and that solver backpropagation carries large time and memory costs.

Core claim

The central claim is that adversarial robustness and generalization of neural operators should be evaluated against the true solver, not a fixed or surrogate ground truth. Defining the loss as the squared model-solver error, ℓ(a)=‖G(a)−g(a)‖², the gradient involves the difference of Jacobians, 2(J_G(a)−J_g(a))ᵀ(G(a)−g(a)). Omitting the solver Jacobian (detached) or replacing the solver with a nearest-neighbor dictionary proxy can underestimate or mislead attacks, especially in 2D Navier–Stokes, where later solver frames carry most of the information. The paper claims that the solver-integrated attack finds larger final losses and more informative perturbed inputs, and that adversarial traini

What carries the argument

The carrying object is the model-solver error operator with its Fréchet derivative: ℓ(a)=‖G(a)−g(a)‖², whose gradient is 2(J_G−J_g)ᵀ(G−g). Three gradient modes are compared: full backpropagation through both the operator and the solver ('with solver'), detachment of the solver, and a dictionary proxy for the solver. The attack is projected gradient descent in an L2 ball with Adam-style updates, a 10-step cap, and adaptive time-stepping to handle solver blow-up. For the 2D periodic Navier–Stokes case, a soft-DTW loss—a differentiable alignment loss—is used so that pure spatial translations of the solution are not counted as large errors. The neural operators studied are Fourier Neural Operato

Load-bearing premise

Backpropagating through thousands of solver time steps yields gradients that reliably locate worst-case inputs, and the solver's output for those perturbed inputs is accurate enough to serve as ground truth; if long-horizon solver gradients are unstable or uninformative, the claimed advantage of the with-solver attack and the value of its selected samples collapse.

What would settle it

Compute finite-difference directional derivatives of the solver loss at several PGD iterates; if the backpropagated gradient's sign or magnitude disagrees with the finite-difference estimate on a nontrivial fraction of inputs, the with-solver attack is not actually finding worst-case inputs. Alternatively, on a PDE with a computable or enumerable worst-case perturbation, check whether the with-solver attack attains higher true loss than detached or random perturbations of the same norm.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Robustness evaluations that freeze the solver's ground truth can overstate a neural operator's robustness, because the true solver output shifts with the input.
  • Solver-selected adversarial samples can act as automatic active-learning data: retraining on them, round by round, reduces out-of-distribution errors for most tested ranges and kernels relative to vanilla training.
  • The formulation extends to any regression setting with a differentiable ground-truth oracle, so the method is not tied to PDEs.
  • Averaged adversarial perturbations in the Navier–Stokes experiments reflect the external forcing of the system, and become random fields without forcing, suggesting the attack tracks physically meaningful sensitivity.
  • Batch-by-batch adversarial training and random-constant augmentation can worsen in-distribution train/test loss, so the training strategy—not just the attack—determines whether the method helps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The with-solver attack could serve as a probe for which input-space directions a surrogate model gets wrong; the forcing-alignment observation suggests these directions correspond to physically meaningful sensitivities rather than arbitrary noise.
  • The large cost of backpropagating through all solver time steps motivates checkpointed, implicit, or adjoint differentiation of the solver; if long-horizon gradients are unstable, hybrid attacks combining finite-difference and backpropagated information may be needed.
  • In near-linear regimes, detached and with-solver gradients may nearly coincide, so the practical advantage of full solver backpropagation is most likely to appear in strongly nonlinear, sensitive dynamics—this could be tested as a cheaper diagnostic.
  • Random-constant augmentation helped range-shifted inputs but not kernel-shifted ones; combining random augmentation with solver-selected adversarial samples might yield a training distribution with both broad support and hard-case focus.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript (full-text title "Towards Universal Solvers: Using PGD Attack in Active Learning to Increase Generalizability of Neural Operators as Knowledge Distillation from Numerical PDE Solvers") proposes a PGD-style attack in which the attack loss is computed against a differentiable PDE solver (Exponax) and gradients are backpropagated through both the neural operator and all solver time steps. It compares this "with-solver" attack with a detached-solver variant and a dictionary-approximated variant on 1D Burgers and 2D Navier-Stokes, reporting that the with-solver attack gives faster and larger loss increases. It then presents two retraining schemes (round-by-round active learning and batch-by-batch adversarial training) plus a random-constant augmentation baseline, and reports that round-by-round retraining, especially at round 6, reduces OOD test error on many but not all shifted datasets, while the other two schemes hurt in-distribution performance. The paper also includes extensive appendices on spectral GRF generation and pseudo-spectral solvers.

Significance. If the central empirical claim is correct, the differentiable-solver PGD attack offers a concrete way to mine worst-case samples for neural operators and to use them in active learning, potentially improving OOD robustness. The paper should be credited for: (i) clearly distinguishing with-solver, detached, and dictionary-approximated attack gradients; (ii) reporting timing/memory breakdowns (Tables 1-3); (iii) documenting a falsifiable observation that adversarial perturbations reflect the Navier-Stokes external forcing; and (iv) making explicit that relative FNO loss can mask larger absolute errors. These are useful contributions. However, the manuscript currently resembles an extended technical report: the formalization advertised in the abstract is absent, the training evaluation is statistically thin, and the key gradient-fidelity assumption is unchecked. The contribution is promising but not yet established.

major comments (4)
  1. [Title/Abstract vs. Methods] The arXiv abstract claims the paper "formalizes the definition of generalization and robustness through a model-solver error operator" and identifies norm-bounded attack-loss increase and Jacobian-error function norm as robustness metrics. The full text contains no such formalization: no model-solver error operator is defined, and no theorem or definition links the proposed MSE loss to the claimed generalization/robustness metrics. The Methods section defines Fréchet derivatives and batch derivatives (the equations for ℓ(a)) but never uses them to state the advertised formal definitions. Since the abstract's formalization is a central advertised contribution, this unbacked claim must either be removed or actually provided in a revision.
  2. [Adversarial Training Using Above PGD Attack] The central training claim is not supported by the reported numbers. The text says batch-by-batch and random-constant training "cause greater loss in the original train and test datasets," and that round-by-round achieves lower loss for "nearly all datasets except range (-1.5,-1)" only at the 6th round; it also says results "increase the loss of many datasets while decrease the loss of others." Figure 5 is a single loss-change plot with no numerical values, no error bars, no multiple seeds, and no statistical tests. Moreover, batch-by-batch training is forced to batch size 2 while the original FNO training uses batch size 20, confounding the comparison. The conclusion that solver-integrated adversarial training "clearly improves both generalizability and robustness" is therefore overstated and requires quantitative tables and repeated-run statistics.
  3. [Challenges in the Above PGD Attack] The with-solver attack's advantage rests on gradients backpropagated through all ~4000 solver time steps (Tables 1-3). The paper does not validate the fidelity of these gradients. It concedes the need for adaptive time-step shrinking, a 10-step limit, and fallback to the last non-blow-up input. Without a check against a non-gradient search (e.g., random or coordinate ascent within the same ε-ball) or a finite-difference verification on a reduced problem, the larger attack losses in Figures 17-36 and the samples selected for round-by-round training may reflect numerical artifacts of the unrolled solver Jacobian rather than true worst-case model-solver discrepancy. This is load-bearing for the with-solver advantage and for active-sample selection.
  4. [Experiments and Evaluation] The empirical evaluation is too thin to support the strong claims. OOD results are reported as plots (Figures 1-2) without dataset sizes, confidence intervals, or repeated training; the qualitative claim that value range dominates is not quantified or compared against kernel-type effects. No code or data release is mentioned, and the model/training hyperparameters for FNO, DeepONet, and Exponax are not specified. The paper's own text notes that attacks on NS require 30-50 GB memory and only batch size 7 fits, yet the effect of these constraints on results is not discussed. In addition, the OOD evaluation uses the same model-solver RMSE/MSE that the training objective optimizes; alternative metrics (e.g., spectral errors or physics-based invariants) are not reported, so it is unclear how much of the improvement is specific to the chosen metric.
minor comments (5)
  1. [Title/Abstract] The arXiv abstract (with the "Solver-Integrated..." title) and the full-text title/abstract differ substantially. Please reconcile the two and ensure the abstract claims match the content.
  2. [References] The reference list contains duplicates and incomplete entries: [39] and [40] are the same paper; [9]/[10] are both Soft-DTW; [47]/[48] are both Madry et al.; several entries are incomplete (e.g., "Kong" et al., "et al." placeholders).
  3. [Algorithm 1] The note "Normalizing d(step 4)" in Algorithm 1 is incorrect; normalization occurs in step 14. The L2 PGD pseudocode and Algorithm 1 also duplicate but differ in Adam details; unify them.
  4. [Notation and typos] There are numerous typos and inconsistent notations, e.g., "Orzsag" (should be Orszag), "V orticity-Streamfucntion F ormulation", "negand" / "All pos", and inconsistent naming of norms ("norm=2" vs. L2). These should be cleaned up.
  5. [Figures] Many figure captions omit units or full parameter settings (e.g., Figures 17-36), and several figures lack error bars or repeated-run information. The appendix is also very long and contains tutorial material that could be condensed or moved to supplementary material.

Circularity Check

0 steps flagged

No significant circularity: the solver is an external oracle and the OOD generalization benchmarks are held out from the training objective.

full rationale

The derivation chain is not circular in the sense that matters here. The adversarial objective and all reported losses are computed against solver outputs g(a) produced by Exponax, an external numerical solver; the learned neural operator is trained to match that solver, and no fitted parameter is later renamed as a prediction. The paper defines generalization as fixed-input model-solver loss, but the OOD test sets use different initial-condition distributions (different GRF kernels, correlation lengths, value ranges) that are not inputs to the minimax objective. Thus the round-by-round improvements on those held-out distributions are not forced by construction. The with-solver attack naturally maximizes the exact loss used to evaluate it, so its larger final attack loss is partly an expected property of first-order optimization rather than an independent discovery; however, this is not a fitted-input-called-prediction circularity, and the central OOD generalization claim does not reduce to this. The acknowledged numerical fragility of backpropagating through roughly 4000 solver timesteps ('The gradient has to be back propagated through all 4000 layers', Challenges section) is a real validation limitation for the method's effectiveness, but it is a correctness/robustness risk, not a circular step. There are no load-bearing self-citations and no imported uniqueness theorems from the authors' own prior work.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The method does not introduce new physical entities. The free parameters are experimental hyperparameters rather than fitted physical constants. The two load-bearing domain assumptions—solver reliability and gradient meaningfulness through long solver trajectories—are not independently validated. The paper's empirical claims rest on these assumptions.

free parameters (5)
  • Attack norm-ball radius epsilon (1D L2=10, L∞=0.5; 2D L2=13.1–78.6, L∞=0.5–1) = varies per experiment
    Chosen by hand; controls perturbation size and directly affects whether attacks and training improve OOD results. No sensitivity analysis is reported for final training outcomes.
  • PGD step size alpha (0.0025–50) and number of PGD steps T (10–200) = varies per experiment
    Manual choices; ablations only show loss-progression effects, not the effect on final generalization or training stability.
  • Active-learning round count (best at 6th round) = 6
    Selected as the best round among several; no stopping rule or multiple-comparison correction is reported.
  • Soft-DTW smoothing parameter gamma = 0.01
    Chosen for the 1D soft-DTW experiment; not swept. It changes the attack loss landscape and can affect the reported behavior.
  • Dictionary size for approximated baseline (20 to 20000) = varied
    Used to show surrogate loss decreases with dictionary size; not a fitted parameter of the central with-solver method.
axioms (4)
  • domain assumption Exponax pseudo-spectral solver output g(a) is an accurate ground-truth oracle and is differentiable with respect to input a.
    Used throughout Methods and Experiments; if the solver is inaccurate or its differentiability is compromised by blow-up, the attack and training labels/gradients are unreliable. The text acknowledges blow-up handling but does not validate solver accuracy against an independent reference.
  • domain assumption Backpropagating through all 4000 solver timesteps yields gradients that reliably identify worst-case perturbations.
    Central to Algorithm 1 and Tables 1–3. For chaotic or long-horizon PDEs, such gradients may be unstable or uninformative; the paper does not test gradient reliability.
  • domain assumption Small L2/L∞ perturbations of initial conditions are physically meaningful and sufficient to expose OOD failures.
    Used to construct PGD attacks and training data. The only physical plausibility check is smoothness and periodic boundary behavior, not a broader validity criterion.
  • ad hoc to paper Out-of-distribution failure of neural operators is primarily controlled by input value range rather than kernel/spectral shape.
    Motivates the OOD test design and the training augmentation strategy. The paper observes empirically that range matters more than kernel type, but this observation is not a theorem and may not generalize.

pith-pipeline@v1.3.0-alltime-deepseek · 29609 in / 14405 out tokens · 130473 ms · 2026-08-04T08:44:41.138142+00:00 · methodology

0 comments
read the original abstract

Neural operators are widely used as fast surrogates for numerical PDE solvers, mapping input functions to solution functions. However, their generalizability and robustness are not yet clearly defined in the operator-learning setting, which differs from traditional adversarial robustness definitions. This paper studies the generalizability and robustness of a learned neural operator from a solver-integrated perspective, addressing the challenge that the output of a learned operator and a numerical solver tends to change in tandem under input perturbation. First, we formalize the definition of generalization and robustness through a model-solver error operator, identifying fixed-input model-solver loss as generalization metric, and norm-bounded adversarial attack loss increase and Jacobian-error function norm as robustness metric. Second, we identify the solver-integrated adversarial attack as appropriate for PDE operator learning and show why model-only or fixed-ground-truth attacks can be insufficient when the solver output also changes with the input. Third, we develop solver-integrated adversarial training methods for neural operators. Experiments on representative PDE benchmarks show that this solver-integrated adversarial training clearly improves both generalizability and robustness. Deeper solver integration yields more effective attacks, more informative samples, and more efficient training than less integrated alternatives. These results provide a general framework for robust operator training and automatic sample selection without heavy manual intervention. More broadly, the formulation applies to adversarial regression whenever a ground-truth oracle can evaluate, and ideally differentiate, the true input-output map; PDE operator learning is one such case.

Figures

Figures reproduced from arXiv: 2510.18989 by Yifei Sun.

Figure 1
Figure 1. Figure 1: Unnormalized and Normalized FNO 1D Burgers OOD testing RMSE and MAE [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Unnormalized FNO 2D NS OOD testing RMSE PGD attack FNO1d on 1D Burgers FNO1d on 1D Burgers is a relatively simple case, with one intial condition and a final condition, all of them being 1D tensors [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 1D Burgers PGD attack illustration (with solver) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 2D NS PGD attack illustration (with solver) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Loss change of round-by-round method Generally I feel this is a difficult task due to the fact that we are trying to sample inputs in the very high dimensional function space. Because of the data-driven nature of the deep-learning model, without ingenious mathematical construction, it is hard for deep learning method to generalize well on OOD and unseen datasets. This is aggravated by the fact that using s… view at source ↗
Figure 6
Figure 6. Figure 6: Loss change of batch-by-batch method [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Loss change of random constant method 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normalized and Unnormalized FNO 1D Burgers Testing Sample 1 (in-distribution/train and test [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Normalized and Unnormalized FNO 1D Burgers OOD Testing Sample 10 [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized and Unnormalized FNO 1D Burgers OOD Testing Sample 11 [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalized and Unnormalized FNO 1D Burgers OOD Testing Sample 13 [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Unnormalized FNO 2D NS Testing Sample (in-distribution/train) [PITH_FULL_IMAGE:figures/full_fig_p039_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Unnormalized FNO 2D NS Testing Sample (in-distribution/test) [PITH_FULL_IMAGE:figures/full_fig_p039_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Unnormalized FNO 2D NS OOD Testing Sample 1 [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Unnormalized FNO 2D NS OOD Testing Sample 7 [PITH_FULL_IMAGE:figures/full_fig_p040_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Unnormalized FNO 2D NS OOD Testing Sample 9 [PITH_FULL_IMAGE:figures/full_fig_p041_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: 1D Burgers nu=0.0005 norm=2 index=0 steps=100 epsilon=10.0 alpha=0.05 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p041_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: 1D Burgers nu=0.0005 norm=inf index=0 steps=100 epsilon=0.5 alpha=0.025 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p042_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: 1D Burgers nu=0.01 norm=2 index=0 steps=100 epsilon=10.0 alpha=0.1 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p042_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: 1D Burgers nu=0.01 norm=2 index=0 steps=100 epsilon=10.0 alpha=0.5 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p042_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: 1D Burgers nu=0.01 norm=inf index=0 steps=100 epsilon=0.5 alpha=0.005 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p043_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: 1D Burgers nu=0.01 norm=inf index=0 steps=100 epsilon=0.5 alpha=0.05 approxNs(dict [PITH_FULL_IMAGE:figures/full_fig_p043_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: 1D Burgers nu=0.01 norm=inf index=0 steps=100 epsilon=0.5 alpha=0.0025 [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: 1D Burgers nu=0.01 norm=2 index=0 steps=100 epsilon=10 alpha=0.01 [PITH_FULL_IMAGE:figures/full_fig_p045_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: 1D Burgers nu=0.01 norm=inf index=0 steps=100 epsilon=0.5 alpha=0.025 [PITH_FULL_IMAGE:figures/full_fig_p046_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: 1D Burgers nu=0.01 norm=2 index=0 steps=200 epsilon=20 alpha=0.6 Soft-DTW as loss [PITH_FULL_IMAGE:figures/full_fig_p047_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Test RMSE and MAE for FNO of different variants and parameters [PITH_FULL_IMAGE:figures/full_fig_p048_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Test RMSE T test for FNO of different variants and parameters [PITH_FULL_IMAGE:figures/full_fig_p049_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: 2D NS PGD attack illustration (detached) [PITH_FULL_IMAGE:figures/full_fig_p050_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: 2D NS PGD attack illustration (approximated) [PITH_FULL_IMAGE:figures/full_fig_p051_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: 2D NS PGD attack illustration (constant) [PITH_FULL_IMAGE:figures/full_fig_p051_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: 2D NS PGD attack loss progression alpha=5.0 epsilon=13.1072 steps=100 [PITH_FULL_IMAGE:figures/full_fig_p052_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: 2D NS PGD attack FNO and PDE solver output final difference alpha=5.0 epsilon=13.1072 [PITH_FULL_IMAGE:figures/full_fig_p052_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: 2D NS PGD attack final perturbed initial condition alpha=5.0 epsilon=13.1072 steps=100 [PITH_FULL_IMAGE:figures/full_fig_p053_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: 2D NS PGD attack loss progression alpha=5.0 epsilon=39.3216 steps=100 [PITH_FULL_IMAGE:figures/full_fig_p053_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: 2D NS PGD attack FNO and PDE solver output final difference alpha=5.0 epsilon=39.3216 [PITH_FULL_IMAGE:figures/full_fig_p053_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: 2D NS PGD attack results norm=inf alpha=0.05 epsilon=0.5 steps=100 index=2 [PITH_FULL_IMAGE:figures/full_fig_p054_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: 2D NS PGD attack results norm=inf alpha=0.05 epsilon=0.5 steps=100 index=2 final step [PITH_FULL_IMAGE:figures/full_fig_p055_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: 2D NS PGD attack results norm=inf alpha=0.01 epsilon=1 steps=100 index=3 [PITH_FULL_IMAGE:figures/full_fig_p056_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: 2D NS PGD attack results norm=inf alpha=0.01 epsilon=1 steps=100 index=3 final step gradient’s [PITH_FULL_IMAGE:figures/full_fig_p057_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: 2D NS PGD attack (detach5to9) alpha ablation study results norm=2 epsilon=13.1072 steps=100 [PITH_FULL_IMAGE:figures/full_fig_p058_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: 2D NS PGD attack (detach1to9) alpha ablation study results norm=2 epsilon=13.1072 steps=100 [PITH_FULL_IMAGE:figures/full_fig_p058_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: 2D NS PGD attack alpha=5 epsilon=13.1072 steps=100 loss progression [PITH_FULL_IMAGE:figures/full_fig_p059_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: 2D NS PGD attack alpha=5 epsilon=39.3216 steps=100 loss progression [PITH_FULL_IMAGE:figures/full_fig_p059_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: 2D NS PGD attack examples alpha=10 epsilon=78.6432 steps=10 loss progression [PITH_FULL_IMAGE:figures/full_fig_p060_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: 2D NS PGD attack examples alpha=50 epsilon=78.6432 steps=10 loss progression [PITH_FULL_IMAGE:figures/full_fig_p061_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: 2D NS dataset external forcing pattern (diagonal) [PITH_FULL_IMAGE:figures/full_fig_p061_47.png] view at source ↗
Figure 48
Figure 48. Figure 48: 2D NS dataset and perturbed result evolution difference [PITH_FULL_IMAGE:figures/full_fig_p062_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: 2D NS dataset and perturbed result evolution difference (upsampled 256 [PITH_FULL_IMAGE:figures/full_fig_p062_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: 2D NS dataset initial condition spectrum and temporal evolution of vorticity spectrum [PITH_FULL_IMAGE:figures/full_fig_p063_50.png] view at source ↗
Figure 51
Figure 51. Figure 51: 2D NS dataset temporal evolution of enstrophy [PITH_FULL_IMAGE:figures/full_fig_p063_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: 2D NS dataset temporal evolution of average vorticity (orginal 64 [PITH_FULL_IMAGE:figures/full_fig_p064_52.png] view at source ↗
Figure 53
Figure 53. Figure 53: 2D NS dataset temporal evolution of average log absolute Fourier coefficients of vorticity (orginal [PITH_FULL_IMAGE:figures/full_fig_p064_53.png] view at source ↗
Figure 54
Figure 54. Figure 54: All forcing patterns tested in the 2D NS datasets [PITH_FULL_IMAGE:figures/full_fig_p065_54.png] view at source ↗
Figure 55
Figure 55. Figure 55: Evolution of original and perturbed datasets and the difference of perturbed and original datasets [PITH_FULL_IMAGE:figures/full_fig_p066_55.png] view at source ↗
Figure 56
Figure 56. Figure 56: Evolution of original and perturbed datasets and the difference of perturbed and original datasets [PITH_FULL_IMAGE:figures/full_fig_p066_56.png] view at source ↗
Figure 57
Figure 57. Figure 57: Evolution of original and perturbed datasets and the difference of perturbed and original datasets [PITH_FULL_IMAGE:figures/full_fig_p066_57.png] view at source ↗
Figure 58
Figure 58. Figure 58: Entropy of singular value of the complex matrices on every frequency point [PITH_FULL_IMAGE:figures/full_fig_p067_58.png] view at source ↗
Figure 59
Figure 59. Figure 59: Outputs of trained FNO when truncated in terms of number of Fourier layers [PITH_FULL_IMAGE:figures/full_fig_p068_59.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

88 extracted references · 2 canonical work pages

  1. [1]

    Wikipedia, 2025

    Dirac comb. Wikipedia, 2025. URLhttps://en.wikipedia.org/wiki/Dirac_comb

    2025

  2. [2]

    Wikipedia, 2025

    Mat´ ern covariance function. Wikipedia, 2025. URLhttps://en.wikipedia.org/wiki/Mat%C3%A9rn_ covariance_function

    2025

  3. [3]

    Evaluating the adversarial robustness for fourier neural operator

    Adedotun Adesoji et al. Evaluating the adversarial robustness for fourier neural operator. SRML Workshop at ICLR 2022, 2022. URLhttps://arxiv.org/abs/2204.04259

    Pith/arXiv arXiv 2022

  4. [4]

    Implicit neural differential model for spatiotemporal dynamics (im-pindiff).arXiv preprint arXiv:2504.02260, 2025

    Deepak Akhare, Pan Du, Tengfei Luo, and Jian-Xun Wang. Implicit neural differential model for spatiotemporal dynamics (im-pindiff).arXiv preprint arXiv:2504.02260, 2025

    Pith/arXiv arXiv 2025

  5. [5]

    Ham, and Ado Farsi

    Nacime Bouziani, David A. Ham, and Ado Farsi. Differentiable programming across the pde and machine learning barrier.arXiv preprint arXiv:2409.06085, 2024

    Pith/arXiv arXiv 2024

  6. [6]

    Burns, Geoffrey M

    Keaton J. Burns, Geoffrey M. Vasil, Jeffrey S. Oishi, Daniel Lecoanet, and Benjamin P. Brown. Dedalus: A flexible framework for numerical simulations with spectral methods.Physical Review Research, 2020. doi: 10.1103/PhysRevResearch.2.023068. URLhttps://arxiv.org/abs/1905.10388

    Pith/arXiv arXiv 2020

  7. [7]

    Canuto, M.Y

    C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang.Spectral Methods in Fluid Dynamics. Springer, 1988

    1988

  8. [8]

    Clercx et al

    Herman J.H. Clercx et al. A spectral solver for the navier–stokes equations in cylindrical geometries. Journal of Computational Physics, 1997. AB2 for advection, Crank–Nicolson for diffusion

    1997

  9. [9]

    Soft-DTW: a differentiable loss function for time-series

    Marco Cuturi and Mathieu Blondel. Soft-DTW: a differentiable loss function for time-series. InPro- ceedings of the 34th International Conference on Machine Learning (ICML), pages 894–903, Sydney, Australia, 2017. PMLR

    2017

  10. [10]

    Soft-dtw: a differentiable loss function for time-series

    Marco Cuturi and Mathieu Blondel. Soft-dtw: a differentiable loss function for time-series. InProceedings of the 34th International Conference on Machine Learning (ICML), 2017

    2017

  11. [11]

    Physics-informed re- duced order modeling of time-dependent pdes via differentiable solvers.arXiv preprint arXiv:2505.14595, 2025

    Nima Hosseini Dashtbayaz, Hesam Salehipour, Adrian Butscher, and Nigel Morris. Physics-informed re- duced order modeling of time-dependent pdes via differentiable solvers.arXiv preprint arXiv:2505.14595, 2025

    arXiv 2025

  12. [12]

    C. R. Dietrich and G. N. Newsam. Fast and exact simulation of stationary gaussian processes through circulant embedding.SIAM Journal on Scientific Computing, 1997

    1997

  13. [13]

    The kernel cookbook: Advice on covariance functions, 2014

    David Duvenaud. The kernel cookbook: Advice on covariance functions, 2014. URLhttps://www.cs. toronto.edu/~duvenaud/cookbook/

    2014

  14. [14]

    Fourier series representation of periodic signals

    Michael Dzhibladze. Fourier series representation of periodic signals. Purdue University lecture notes,

  15. [15]

    GitHub / open source

    Tum-PBS / Philipp Holl et al.ϕflow: A differentiable pde solving framework. GitHub / open source. Integration with PyTorch, TensorFlow, JAX

  16. [16]

    Advanced gaussian processes (scribed notes, lecture 21)

    Konstantin Genin and Yutong Zheng. Advanced gaussian processes (scribed notes, lecture 21). CMU 10-708 Spring 2015, 2015. URLhttps://www.cs.cmu.edu/ ~epxing/Class/10708-15/notes/10708_ scribe_lecture21.pdf

    2015

  17. [17]

    Goodfellow, Jonathon Shlens, and Christian Szegedy

    Ian J. Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples.arXiv preprint arXiv:1412.6572, 2015. URLhttps://arxiv.org/abs/1412.6572

    Pith/arXiv arXiv 2015

  18. [18]

    Discrete fourier transform and wavelet transforms (math 357 notes)

    Roe Goodman. Discrete fourier transform and wavelet transforms (math 357 notes). Technical re- port, Rutgers University, 2008. URLhttps://sites.math.rutgers.edu/ ~goodman/pub/math_357_ lectures.pdf. Supplementary class notes, revised May 3, 2008. 20

    2008

  19. [19]

    Finite fourier transform, circulant matrices, and the fast fourier transform

    Roe Goodman. Finite fourier transform, circulant matrices, and the fast fourier transform. Techni- cal report, Rutgers University, 2015. URLhttps://sites.math.rutgers.edu/ ~yzhuang/rci/math/ fftnotes.pdf. Supplementary notes for Math 642/550

    2015

  20. [20]

    Gou et al

    J. Gou et al. Knowledge distillation: A survey.arXiv preprint arXiv:2006.05525, 2020

    Pith/arXiv arXiv 2006

  21. [21]

    Robert M. Gray. Toeplitz and circulant matrices: A review.Foundations and Trends in Communications and Information Theory, 2(3):155–239, 2006

    2006

  22. [22]

    A review: Applications of the spectral finite element method.Archives of Computational Methods in Engineering, 2023

    Muhammad Bilal Hafeez and Marek Krawczuk. A review: Applications of the spectral finite element method.Archives of Computational Methods in Engineering, 2023. doi: 10.1007/s11831-023-09911-2. URLhttps://link.springer.com/article/10.1007/s11831-023-09911-2

    work page doi:10.1007/s11831-023-09911-2 2023

  23. [23]

    Spectral kernels (gpss 2021)

    Markus Heinonen. Spectral kernels (gpss 2021). Gaussian Process Summer School slides, 2021. URL https://gpss.cc/gpss21/slides/Heinonen2021.pdf

    2021

  24. [24]

    Distilling the knowledge in a neural network

    Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015

    Pith/arXiv arXiv 2015

  25. [25]

    Hou and Ruo Li

    Thomas Y. Hou and Ruo Li. Computing nearly singular solutions using pseudo-spectral methods. Communications in Computational Physics, 2007

    2007

  26. [26]

    Show, attend and distill: Knowledge distillation via attention-based feature matching.arXiv preprint arXiv:2102.02973, 2021

    Mingi Ji, Byeongho Heo, and Sungrae Park. Show, attend and distill: Knowledge distillation via attention-based feature matching.arXiv preprint arXiv:2102.02973, 2021

    Pith/arXiv arXiv 2021

  27. [27]

    Unsupervised learning of full-waveform inversion: Connecting cnn and partial differential equation in a loop.Preprint / Conference (ICLR), 2022

    Peng Jin, Xitong Zhang, Yinpeng Chen, Sharon Huang, Zicheng Liu, and Youzuo Lin. Unsupervised learning of full-waveform inversion: Connecting cnn and partial differential equation in a loop.Preprint / Conference (ICLR), 2022

    2022

  28. [28]

    , and Youzuo Lin

    Peng Jin, Yinan Feng, . . . , and Youzuo Lin. An empirical study of large-scale data-driven full waveform inversion.Scientific Reports, 14, 2024

    2024

  29. [29]

    Perceptual losses for real-time style transfer and super-resolution.European Conference on Computer Vision (ECCV) Workshops, 2016

    Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and super-resolution.European Conference on Computer Vision (ECCV) Workshops, 2016

    2016

  30. [30]

    Cambridge University Press, 3rd edition, 2004

    Yitzhak Katznelson.An Introduction to Harmonic Analysis. Cambridge University Press, 3rd edition, 2004

    2004

  31. [31]

    Elastic image matching is np-complete.Pattern Recognition Letters, 24(1-3):445–453, 2003

    Daniel Keysers and Walter Unger. Elastic image matching is np-complete.Pattern Recognition Letters, 24(1-3):445–453, 2003

    2003

  32. [32]

    Exponax: Fourier spectral etdrk time-steppers in jax

    Florian K¨ ohler. Exponax: Fourier spectral etdrk time-steppers in jax. Online, 2024.https://fkoehler. site/exponax/

    2024

  33. [33]

    et al. Kong. Physics informed adversarial training for solving partial differential equations (piat). arXiv:2207.XXXX, 2022. physics-informed adversarial training (representative early work)

    2022

  34. [34]

    A selective survey on versatile knowledge distillation paradigm for neural network models.arXiv preprint arXiv:2011.14554, 2020

    Jeong-Hoe Ku, JiHun Oh, YoungYoon Lee, et al. A selective survey on versatile knowledge distillation paradigm for neural network models.arXiv preprint arXiv:2011.14554, 2020

    Pith/arXiv arXiv 2011

  35. [35]

    Laga and W

    I. Laga and W. Kleiber. The modified mat´ ern process.Applied Mathematics and Computation, 2017. URLhttps://amath.colorado.edu/faculty/kleiberw/papers/Laga2017.pdf

    2017

  36. [36]

    Direct image matching by dynamic warping

    Hansheng Lei and Venu Govindaraju. Direct image matching by dynamic warping. InProceedings of the International Conference on Pattern Recognition (ICPR), 2003

    2003

  37. [38]

    Adversarial training for physics-informed neural networks (at-pinns), 2023

    Yao Li, Shengzhu Shi, Zhichang Guo, and Boying Wu. Adversarial training for physics-informed neural networks (at-pinns), 2023. URLhttps://arXiv.org/abs/2310.11789. Preprint arXiv:2310.11789, https://arxiv.org/abs/2310.11789. 21

    arXiv 2023

  38. [40]

    Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Andrew Stuart, Kaushik Bhat- tacharya, and Anima Anandkumar

    Zongyi Li, Nikola B. Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Andrew Stuart, Kaushik Bhat- tacharya, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv:2010.08895, 2020

    Pith/arXiv arXiv 2010

  39. [41]

    Fourier neural operator with learned deformations for pdes on general geometries.Journal of Machine Learning Research, 2023

    Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural operator with learned deformations for pdes on general geometries.Journal of Machine Learning Research, 2023. Preprint / JMLR version

    2023

  40. [42]

    et al. Liu. Adaptive movement sampling physics-informed residual network. arXiv:2504.03244, 2025. movement-based residual sampling and domain-coverage constraints

    Pith/arXiv arXiv 2025

  41. [44]

    Deeponet: Learning nonlinear operators for identi- fying differential equations based on the universal approximation theorem of operators.arXiv preprint arXiv:1910.03193, 2019

    Lu Lu, Pengzhan Jin, and George Em Karniadakis. Deeponet: Learning nonlinear operators for identi- fying differential equations based on the universal approximation theorem of operators.arXiv preprint arXiv:1910.03193, 2019. URLhttps://arxiv.org/abs/1910.03193. Accessed: 2025-09-27

    Pith/arXiv arXiv 1910

  42. [45]

    Karniadakis

    Lu Lu, Pengzhan Jin, and George E. Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, 2021

    2021

  43. [46]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, 2021

    Lu Lu, Pengzhan Jin, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, 2021. Preprint / arXiv:2104.09481

    Pith/arXiv arXiv 2021

  44. [47]

    Towards deep learning models resistant to adversarial attacks.arXiv preprint arXiv:1706.06083, 2017

    Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks.arXiv preprint arXiv:1706.06083, 2017. URL https://arxiv.org/abs/1706.06083. Accessed: 2025-09-27

    Pith/arXiv arXiv 2017

  45. [48]

    To- wards deep learning models resistant to adversarial attacks

    Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. To- wards deep learning models resistant to adversarial attacks. InInternational Conference on Learning Representations (ICLR), 2018. URLhttps://openreview.net/pdf?id=rJzIBfZAb

    2018

  46. [49]

    Majda and Andrea L

    Andrew J. Majda and Andrea L. Bertozzi.Vorticity and Incompressible Flow. Cambridge University Press, 2002

    2002

  47. [50]

    Giometto, Marc B

    Fabien Margairaz, Marco G. Giometto, Marc B. Parlange, and Marc Calaf. Comparison of dealiasing schemes in large-eddy simulation of neutrally stratified atmospheric flows.Geoscientific Model Devel- opment, 11:4069–4084, 2018. doi: 10.5194/gmd-11-4069-2018. URLhttps://gmd.copernicus.org/ articles/11/4069/2018/

    work page doi:10.5194/gmd-11-4069-2018 2018

  48. [51]

    Circulant matrices

    Department of Mathematics / Course 18.06 MIT. Circulant matrices. Technical report, Massachusetts In- stitute of Technology, 2017. URLhttps://web.mit.edu/18.06/www/Spring17/Circulant-Matrices. pdf. Lecture notes, available at MIT OCW (PDF)

    2017

  49. [52]

    Molenaar

    D. Molenaar. Forced navier–stokes flows on a bounded 2d domain: Abcn scheme overview. Technical report, 2004

    2004

  50. [53]

    Gladstone, and Hadi Meidani

    Mohammad Amin Nabian, Ryan J. Gladstone, and Hadi Meidani. Efficient training of physics-informed neural networks via importance sampling.Computer-Aided Civil and Infrastructure Engineering, 36(8): 962–977, 2021. doi: 10.1111/mice.12685

    doi:10.1111/mice.12685 2021

  51. [54]

    Learning dtw global constraint for time series classification

    Vit Niennattrakul and Chotirat Ann Ratanamahatana. Learning dtw global constraint for time series classification. InProceedings of the 2009 International Conference on Data Mining, 2009. 22

    2009

  52. [55]

    Steven A. Orszag. Numerical methods for the simulation of turbulence. InFluid Dynamics, 1972. Introduces the 2/3 dealiasing rule

    1972

  53. [56]

    K. V. Park et al. Al-pinn: Active learning-driven physics-informed neural networks for efficient sample selection. arXiv:2502.03963, 2025

    Pith/arXiv arXiv 2025

  54. [57]

    Karniadakis

    Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations.arXiv preprint arXiv:1711.10561, 2017

    Pith/arXiv arXiv 2017

  55. [58]

    Karniadakis

    Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019. doi: 10.1016/j.jcp.2018.10.045

    doi:10.1016/j.jcp.2018.10.045 2019

  56. [59]

    Carl Edward Rasmussen and Christopher K. I. Williams.Gaussian Processes for Machine Learning. MIT Press, 2006. URLhttps://gaussianprocess.org/gpml/chapters/RW4.pdf

    2006

  57. [60]

    Williams.Gaussian Processes for Machine Learning

    Carl Edward Rasmussen and Christopher K.I. Williams.Gaussian Processes for Machine Learning. MIT Press, 2006

    2006

  58. [61]

    Fitnets: Hints for thin deep nets

    Adrian Romero, Nicolas Ballas, Samira E Kahou, Cesar Chassang, Carlo Gatta, and Yoshua Bengio. Fitnets: Hints for thin deep nets. InICLR, 2015

    2015

  59. [62]

    Wiley, 1990

    Walter Rudin.Fourier Analysis on Groups. Wiley, 1990

    1990

  60. [63]

    Dynamic programming algorithm optimization for spoken word recog- nition.IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1):43–49, 1978

    Hiroaki Sakoe and Seibi Chiba. Dynamic programming algorithm optimization for spoken word recog- nition.IEEE Transactions on Acoustics, Speech, and Signal Processing, 26(1):43–49, 1978

    1978

  61. [64]

    Dynamic time warping algorithm review

    Pavel Senin. Dynamic time warping algorithm review. Technical Report, University of Hawai‘i at M¯ anoa,

  62. [65]

    Discovering the fourier transform: A tutorial on circulant matrices, circular con- volution, and the dft

    Smarker. Discovering the fourier transform: A tutorial on circulant matrices, circular con- volution, and the dft. GitHub gist, 2019. URLhttps://gist.github.com/Smarker/ 1e6fb0ebfd43778b535cc049868589f3. Online tutorial note

    2019

  63. [66]

    Understanding gaussian process regression using the fourier transform, 2000

    Peter Sollich. Understanding gaussian process regression using the fourier transform, 2000. URLhttps: //homepages.inf.ed.ac.uk/ckiw/postscript/sheff3.pdf. Lecture notes

    2000

  64. [67]

    Fourier-spectral methods for navier–stokes equations in 2d

    Mashbat Suzuki. Fourier-spectral methods for navier–stokes equations in 2d. Course notes/technical report, McGill University (MATH 595), 2014. URLhttps://www.math.mcgill.ca/gantumur/ math595f14/NSMashbat.pdf. December 17, 2014

    2014

  65. [68]

    Physics- based deep learning

    Nils Thuerey, Philipp Holl, Maximilian Mueller, Patrick Schnell, Felix Trost, and Kiwon Um. Physics- based deep learning. arXiv preprint, 2021. URLhttps://arxiv.org/abs/2109.05237

    Pith/arXiv arXiv 2021

  66. [69]

    Trefethen.Spectral Methods in MATLAB

    Lloyd N. Trefethen.Spectral Methods in MATLAB. SIAM, 2000

    2000

  67. [70]

    Physics-informed neural network with adaptive mesh refinement sampling

    Jakub Trynda. Physics-informed neural network with adaptive mesh refinement sampling. SSRN preprint 5148446, 2024

    2024

  68. [71]

    Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers

    Kiwon Um, Robert Brand, Yun (Raymond) Fei, Philipp Holl, and Nils Thuerey. Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers. InNeurIPS, 2020. https://proceedings.neurips.cc/paper/2020/file/43e4e6a6f341e00671e123714de019a8-Paper.pdf

    2020

  69. [72]

    Generalized harmonic analysis.Acta Mathematica, 1930

    Norbert Wiener. Generalized harmonic analysis.Acta Mathematica, 1930. Wiener–Khinchin link ACF↔PSD

    1930

  70. [73]

    Andrew Gordon Wilson and Ryan P. Adams. Gaussian process kernels for pattern discovery and ex- trapolation, 2013. URLhttps://arxiv.org/abs/1302.4245

    Pith/arXiv arXiv 2013

  71. [74]

    Andrew Gordon Wilson and Ryan P. Adams. Gaussian process kernels for pattern discovery and ex- trapolation. InProceedings of the 30th International Conference on Machine Learning (ICML). PMLR,

  72. [75]

    Andrew Gordon Wilson and Eric P. Xing. Advanced gaussian processes (lecture slides). CMU 10-708 Spring 2015, 2015. URLhttps://www.cs.cmu.edu/ ~epxing/Class/10708-15/slides/andrewgp2. pdf

    2015

  73. [76]

    Wood and Grace Chan

    Andrew T.A. Wood and Grace Chan. Simulation of stationary gaussian vector fields.Statistics and Computing, 1994

    1994

  74. [78]

    Chenxi Wu, Min Zhu, Qinyang Tan, Yadhu Kartha, and Lu Lu. A comprehensive study of non- adaptive and residual-based adaptive sampling for physics-informed neural networks.Computer Meth- ods in Applied Mechanics and Engineering, 403:115671, 2023. doi: 10.1016/j.cma.2022.115671. URL https://arxiv.org/abs/2207.10289. Accessed: 2025-09-27

    arXiv 2023

  75. [79]

    On the generalization properties of adversarial training

    Yue Xing, Qifan Song, and Guang Cheng. On the generalization properties of adversarial training. In Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS), pages 505–513, 2021. URLhttps://proceedings.mlr.press/v130/xing21b.html

    2021

  76. [80]

    Karniadakis

    Jeremy Yu, Lu Lu, Xuhui Meng, and George E. Karniadakis. Gradient-enhanced physics-informed neural networks for forward and inverse pde problems.Computer Methods in Applied Mechanics and Engineering, 393:114823, 2022. doi: 10.1016/j.cma.2022.114823

    arXiv 2022

  77. [81]

    Karniadakis

    Dongkun Zhang, Li Guo, and George E. Karniadakis. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems.Journal of Computational Physics, 397:108850, 2019. doi: 10.1016/j.jcp.2019.07.048

    doi:10.1016/j.jcp.2019.07.048 2019

  78. [82]

    Adversarial training: A survey.arXiv preprint arXiv:2410.15042, 2024

    Mengnan Zhao, Lihe Zhang, Jingwen Ye, Huchuan Lu, Baocai Yin, and Xinchao Wang. Adversarial training: A survey.arXiv preprint arXiv:2410.15042, 2024. URLhttps://arxiv.org/abs/2410.15042

    Pith/arXiv arXiv 2024

  79. [83]

    Improving generalization of adversarial training via robust critical fine-tuning

    Kai Zhu et al. Improving generalization of adversarial training via robust critical fine-tuning. InICCV 2023, 2023. URLhttps://openaccess.thecvf.com/content/ICCV2023/papers/Zhu_Improving_ Generalization_of_Adversarial_Training_via_Robust_Critical_Fine-Tuning_ICCV_2023_ paper.pdf. 24 Appendix: Pseudospectral Methods and Spectral GRF Generation A 1D Burgers...

    2023

  80. [86]

    For FFT libraries using real-to-complex transforms (e.g

    Build wavenumber arraysk x, ky andk 2. For FFT libraries using real-to-complex transforms (e.g. RFFT), store the half-spectrum indexing accordingly (truncate the redundant half along the last di- mension). Setk 2(0,0)←1 for safe division, and later force bψ(0,0) = 0

Showing first 80 references.