{"id":"f75c2c2f-4f05-499e-9b52-7f0d677bd0e3","arxiv_id":"2507.18813","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Scale-consistency training, which enforces agreement between global and rescaled sub-domain predictions, enables neural PDE solvers to extrapolate to unseen scale parameters such as Reynolds number or wavenumber.","lead":"This paper introduces a scale-consistency loss that trains neural network solvers for partial differential equations to agree with their own predictions on rescaled sub-domains, plus a scale-informed neural operator architecture. The approach is designed to let models trained at one scale (e.g., Reynolds number 1000) generalize to unseen scales without retraining.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The self-referential super-domain loss can be minimized by degenerate operators, so Theorem B.2's hypotheses are not shown to hold for the trained models; the theoretical guarantee is not connected to the empirical results.","rationale":"The reader's conditional verdict is appropriate. The exact-scale-consistency theorem is mathematically plausible for the idealized Darcy setting, and the empirical tables show genuine gains in several regimes. However, the most load-bearing point is not the micro-scale PDE-family limitation the authors acknowledge; it is the disconnect between Theorem B.2's hypotheses and the training objective. Algorithm 2's super-domain term is self-referential: it compares Ψ to itself across scales, so a degenerate extension (e.g., zero on unseen scales with matching zero-boundary behavior at the training scale) can drive this term to zero without approximating the true operator. The theorem assumes exact or near-exact scale consistency, boundary matching, and a supremum-level error bound on all near-constant inputs; none of these are verified in the experiments, and the training loss is only a finite-sample, weighted surrogate. This concern is close to the reader's second fragile premise about degenerate operators and unverified theorem conditions. I additionally flag a concrete normalization gap in the proof of Theorem B.2: boundary values inserted into Ψ during the argument are the model's own outputs and need not satisfy the ||g||≤1 constraint defining Dδ. These issues do not necessarily invalidate the paper's empirical contribution or its idealized theorem, but they mean the paper currently overclaims that the theorem 'guarantees' the observed generalization. The right decision remains conditional acceptance with a request for a direct check of the self-consistency error on unseen scales, error bars, and code/data release.","tokens_in":25004,"tokens_out":13504,"duration_ms":150337,"concrete_test":"Using the trained SINO+scale model from the Navier-Stokes experiment (training Re=1000), evaluate the Algorithm 2 consistency loss with λ=10 on the held-out Re=10000 test set, and simultaneously measure the relative L2 error against the ground-truth Re=10000 solutions. If the consistency loss at Re=10000 is small (comparable to its value at Re=1000) while the prediction error at Re=10000 remains large or is not better than the no-scale-consistency baseline, then the self-referential loss is being satisfied by a self-consistent but incorrect extension, and the theorem's premise is not met. If, instead, the consistency loss at Re=10000 is large, then the paper must demonstrate how Theorem B.2 applies to a model that is not approximately scale-consistent on the extrapolation scale.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem B.2 is the paper's main theoretical bridge: if the trained operator Ψ has small supervised error on nearly constant coefficients (Err_Dδ), small boundary error, and small self-consistency error (Err_selfcon), then its out-of-distribution error is also small. The experiments, however, never measure any of these three quantities, and there is a structural reason to doubt that they are small in the way the theorem requires.\n\nThe super-domain loss in Algorithm 2 uses the model's own prediction as the target. Concretely, for a low-scale input (a,g,k0), it forms u = Ψ(a,g,λk0) and then penalizes ||Ψ(a|Ωhat, u|∂Ωhat, k0) − u|Ωhat||. This is a self-consistency condition on Ψ, not a condition tying Ψ to the true operator G. A model that outputs zero on all unseen high-scale inputs, and that also outputs zero at the training scale when given zero boundary data, satisfies this loss exactly while remaining completely wrong at the high scale. The supervised loss at the training scale does not rule this out unless the training set already contains enough near-constant-coefficient inputs with all admissible boundary conditions—but the theorem requires a supremum over the whole set Dδ, not a finite empirical average. The paper does not report the sup-error Err_Dδ, nor does it check that the trained network is close to exactly scale-consistent and exactly boundary-matching on the relevant input class. Thus the central claim that scale-consistency 'provably' enforces the correct operator is not tied to the actual training procedure.\n\nThere is also a technical gap in the proof of Theorem B.2 as written. The set Dδ is defined with ||g||_{L∞} ≤ 1, but in the proof the boundary condition fed into Ψ is the model's own output TλΨ(a,g)|∂Ω, whose L∞ norm is not controlled by the assumptions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a scale-consistency framework for training neural operators that can generalize across PDE scales. The authors define a scale-consistency loss based on PDE rescaling and sub-domain/super-domain sampling, provide a theorem claiming that scale consistency plus accuracy on nearly constant coefficient fields recovers the true solution operator (Theorems 3.1 and B.2), and introduce a scale-informed Fourier neural operator (SINO) with frequency embedding and a U-shaped multi-band structure. Experiments on Darcy flow, Burgers, Helmholtz, and Navier-Stokes report zero-shot generalization to unseen scales and a test-time domain-decomposition refinement that reduces Darcy error by about 40%.","tokens_in":25204,"tokens_out":6643,"duration_ms":64915,"significance":"If the theoretical and empirical claims hold, the work is a meaningful step toward scale-robust operator learning: the quantitative bound in Theorem B.2 is a genuine attempt to connect self-consistency to out-of-distribution error, and the proposed dataset and architecture address a real limitation of fixed-scale neural PDE solvers. The empirical direction is well motivated and, for the proposed SINO model, the improvements are consistent and sometimes large, for example on Burgers and Helmholtz. However, the advertised 'provable' statement is weaker than presented, and the experiments do not verify the quantities that appear in the theorem; moreover, several baseline comparisons show that scale-consistency can hurt. These issues currently prevent me from endorsing the central claims as stated.","major_comments":[{"comment":"The theorem as stated in the main text claims that scale-consistency (4) plus accuracy on nearly constant coefficient functions implies Ψ ≡ G. The proof in Appendix B.2, Theorem B.1, requires two additional conditions: translation symmetry and exact boundary satisfaction (conditions 2 and 3 of Theorem B.1). The main text does not state these hypotheses, so the formal result is overstated. Please either state the full hypotheses in Theorem 3.1 or label it as an informal summary that points to the precise Theorem B.2.","section":"Section 3.1, Theorem 3.1; Appendix B.2, Theorem B.1"},{"comment":"The super-domain loss is self-referential: Algorithm 2 penalizes mismatch between Ψ(a|Ω̂, Ψ(a,g,λk)|∂Ω̂, k) and Ψ(a,g,λk)|Ω̂, both of which use the model's own outputs as targets. An operator that outputs zero on the relevant inputs can satisfy this term exactly, so minimizing the training loss does not by itself tie Ψ to G. Theorem B.2 bounds Err_DM in terms of Err_Dδ, Err_boundary, and Err_selfcon, but the experiments neither measure these three sup-errors nor verify that the trained network is approximately scale-consistent, boundary-matching, and accurate on all of Dδ; training on a finite dataset with a weighted loss cannot establish the sup-error hypotheses. As a result, the theoretical guarantee is not connected to the empirical results. Please report these error quantities on validation sets or adapt the theorem to finite-sample or average errors with explicit assumptions.","section":"Section 3.2.1, Algorithm 2; Section B.3, Theorem B.2"},{"comment":"The empirical claim that scale-consistency helps each model extrapolate is not consistently supported. For Burgers, UNO+scale is worse than UNO at the unseen ν=1/1000 (8.720 vs 8.624) and at the training ν=1/400 (1.387 vs 1.235), while UNet+scale is worse both in training and at ν=1/1000 (30.747 vs 26.481). For Navier-Stokes, UNet+scale deteriorates dramatically at the training Re=1000 (13.802 vs 0.809) and at Re=2000. The '34% average reduction' in the abstract needs to specify the aggregation across models and scales and should be accompanied by multiple seeds, error bars, and a discussion of cases where the method hurts.","section":"Tables 1 and 2"}],"minor_comments":[{"comment":"There is a typo: 'scale consistenct framework' should be 'scale-consistency framework'.","section":"Section 2, Related Work"},{"comment":"The sentence 'As shown in Table 6, a minimum resolution of 32 per each dimension of space and time works the best' refers to a table that does not clearly present the minimum-size ablation; Table 6 reports the 2+1 dimensional Navier-Stokes results and has an unclear 'Aug. size min' column.","section":"Appendix D.1"},{"comment":"Equation (6) and Algorithm 1 use ground-truth solutions, while Equation (5) and Algorithm 2 are self-supervised; the paper should distinguish the supervised sub-domain loss from the self-supervised super-domain loss more clearly, since both are called scale-consistency loss.","section":"Equations (5)-(6) and Algorithms 1-2"},{"comment":"The caption uses 'RE=5000' while the text uses Re=5000; please unify the notation.","section":"Figure 5"},{"comment":"The claimed '34% error reduction on average of all datasets' is not directly traceable to a single table or aggregation rule; please define the average and identify which models and scales are included.","section":"Abstract and Section 5"}],"recommendation":"major_revision","confidential_remarks":"The core idea is promising, and I do not see a mathematical error in the proof of Theorem B.2 under its stated assumptions. The main problems are the overstatement of Theorem 3.1, the disconnect between the sup-error hypotheses of Theorem B.2 and the quantities measured in the experiments, and the lack of error bars and seed variation for the central empirical claim. These are fixable with a revised theorem statement and additional experiments, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The core idea is a real step forward: instead of treating scaling symmetry as a generic data augmentation, the authors build a loss that compares a solution on a full domain with the solution on a rescaled subdomain, explicitly including boundary conditions and scale parameters. That is the right way to encode rescaling for boundary-value PDEs, and it appears to help zero-shot transfer on Burgers, Darcy, and Navier-Stokes in ways prior Lie-symmetry augmentations did not. The scale-informed FNO architecture is also reasonable, and the test-time domain-decomposition refinement on Darcy gives a clean extra gain.\n\nThe paper is weaker where it matters most: the theory is oversold. Theorem 3.1 in the main text omits assumptions that the appendix needs (translation symmetry, exact boundary matching, exact consistency). Theorem B.2 is an honest quantitative bound in the ideal case, but none of the three quantities it bounds—the near-constant supervised error, the boundary error, or the self-consistency error—is measured in the experiments, so the theorem is not connected to the trained models. The stress-test note is right that Algorithm 2's super-domain loss uses the model's own prediction as a target; a degenerate constant operator can satisfy that loss exactly. The supervised sub-domain term anchors things in practice, but the paper should say plainly that the ideal uniqueness theorem does not directly apply to the actual training objective. I also think the proof has a small real gap: ErrDδ requires the boundary input to have L∞ norm at most 1, but TλΨ(a,g)|∂Ω is not controlled. That is fixable with an extra normalization assumption.\n\nEmpirically, the results support the central direction but not the broadest claim. The gains on Burgers and Darcy are substantial. On Helmholtz the gains appear only below the training wavenumbers; at k=50 the scale-consistent model is worse than one baseline, which cuts against the abstract's \"34% on all datasets\" phrasing. There are no error bars, no seeds, and no code or data release. For a method whose main evidence is empirical, that is a real gap in the current version.\n\nWho is this for? Anyone working on neural operators or scientific ML foundations. The scale-consistency regularizer is worth taking seriously and the failures are about presentation and verification, not about the central mechanism. I would send it to peer review and ask for a revision that fixes the theorem statement, reports the three error terms from Theorem B.2, and releases code and data so the claims can be checked. My own verdict is conditional: the idea is good, the paper as written overclaims.","headline":"A genuinely useful scale-consistency regularizer for neural operators, with a theory that is real but narrower than the abstract and a self-referential super-domain loss that needs to be honestly caveated.","tokens_in":25923,"tokens_out":2554,"would_cite":true,"duration_ms":30920,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A neural PDE solver trained at one scale can generalize to unseen scales if it is trained to respect scale consistency, with a theorem showing that exact scale consistency plus accuracy on near-constant inputs pins down the true solution…","keywords":["neural operators","partial differential equations","scale consistency","self-supervised learning","zero-shot generalization","Fourier neural operator","domain decomposition","Reynolds number extrapolation"],"falsifier":"A concrete check: train a neural operator with the scale-consistency loss on nearly constant coefficient data, measure the three terms in Theorem B.2, and evaluate on far-from-constant test fields; if a model achieves near-zero supervised, boundary, and consistency errors yet still shows large out-of-distribution error, the theorem's conclusion fails. Experimentally, one can also directly test whether the left and right sides of identity (4) agree to machine precision on all sampled subdomains for a trained model; if not, the model is in the regime the quantitative bound already leaves uncontrolled.","tokens_in":24632,"feed_emoji":"🌊","tokens_out":6377,"duration_ms":61426,"temperature":0.7,"pith_summary":"The paper tries to establish that PDE solution operators can be learned across scales, not just at a single training scale, by enforcing a scale-consistency constraint during training. The central claim is that if a learned operator obeys the identity that a solution restricted to a subdomain equals the solution of the rescaled PDE on that subdomain, and matches the true operator on nearly constant coefficient fields, then it must be the true solution operator. If true, this makes scale generalization a built-in property rather than an accident of architecture, and their experiments show a model trained only at Re=1000 zero-shot generalizing to Re from 250 to 10,000 with 34% lower average error than baselines. A reader should care because one of the main obstacles to usable ML-based PDE solvers is that they fail outside their training scale; this is a self-supervised regularizer, needing no extra labeled data.","feed_headline":"Scale-consistency loss lifts PDE models from Re 1000 to Re 10,000","feed_subtitle":"A self-supervised loss cuts zero-shot error by 34%, with a theorem linking consistency to correctness.","key_machinery":"The load-bearing object is the scale-consistency identity (4), paired with sub-domain and super-domain sampling that turns it into a loss. Sub-domain sampling restricts input coefficient, boundary, and solution to a random patch, rescales the patch to unit size at the same grid resolution, and penalizes the difference between the model output on the patch and the restricted solution; super-domain sampling does the same when no ground-truth output is available, using the model's own prediction as the reference. The scale-informed neural operator backs this up by embedding the scale parameter (Reynolds number, wavenumber, viscosity) into Fourier-space features so the same network can represent very different frequencies. The theorem's proof machinery is an overlapping partition of unity: on each small subdomain the coefficient is nearly constant, exactness there plus scale consistency forces the operator to be a weak solution on each patch, and summing the weak forms over the partition yields the global solution.","core_discovery":"The paper's core discovery is a self-supervised regularization scheme built on the exact scale-consistency identity (4): for an elliptic PDE, $T_\\lambda G(a,g)$ restricted to the unit domain equals $G(T_\\lambda a, T_\\lambda G(a,g)|_{\\partial\\Omega})$. The authors prove Theorem 3.1 and its quantitative version Theorem B.2, showing that any operator satisfying this identity and matching the true solution operator on nearly constant coefficient fields must coincide with the true operator everywhere, with a bound expressing out-of-distribution error as the sum of supervised error, boundary error, and self-consistency error. Empirically, adding this loss to a scale-informed neural operator (an FNO with scale parameters embedded in Fourier space and a U-shaped multiband architecture) yields zero-shot extrapolation across scales: a 34% average error reduction across Burgers, Darcy, Helmholtz, and Navier-Stokes, including training at Re=1000 and testing at Re from 250 to 10,000.","pith_inferences":["An implication the paper leaves implicit is that scale-consistency turns the scale parameter into a continuous dial: the same checkpoint could serve as the initialization for fine-tuning on a newly observed scale, potentially reducing data requirements for each new operating condition.","Because the consistency identity applies even to PDEs like Helmholtz with no exact scale symmetry (after rescaling the wavenumber), the same machinery may extend to other parameter families, such as material contrasts or forcing frequencies, whenever the parameter transformation under rescaling is known.","A testable extension is to measure the three error terms in Theorem B.2 during training and track whether out-of-distribution error tracks their sum; if the bound is tight in practice, it gives a model-selection criterion for when to trust a scale extrapolation."],"forward_implications":["A model trained with scale-consistency at one scale parameter can be zero-shot evaluated at unseen scales; the paper reports Re 250–10,000 from Re 1000 training, and viscosity or wavenumber ranges beyond training.","Scale-consistency is a self-supervised term: it requires only samples of inputs, not labeled solutions, so it can augment datasets where high-fidelity solution data is scarce.","The quantitative bound in Theorem B.2 ties out-of-distribution error to three measurable quantities, giving a concrete regularization objective: reduce supervised error, boundary error, and consistency error separately.","Test-time domain decomposition with the pre-trained operator further reduces Darcy flow error by about 40% by iteratively refining local patches with informed boundary initialization.","The architecture itself, with scale embedding plus a multiband U-shape, contributes to accuracy and parameter efficiency, matching FNO accuracy with roughly one tenth of the parameters in the cost-accuracy study."],"supporting_citations":[{"why":"Supplies the Fourier neural operator backbone whose unit-domain Fourier basis automatically rescales inputs, making scale-consistency tractable.","marker":"[9]"},{"why":"Provides the weight-sharing kernel parameterization that the scale-informed operator extends by conditioning on scale and mode features.","marker":"[10]"},{"why":"Supplies the U-shaped operator architecture that the multiband Fourier design adapts.","marker":"[11]"},{"why":"Documents the limited effectiveness of prior scale-symmetry augmentation, motivating the boundary-aware consistency formulation.","marker":"[14]"},{"why":"Provides the symmetry-augmentation baseline (rotation, reflection) that scale-consistency is compared against.","marker":"[13]"},{"why":"Shows how physics-based losses can act as self-supervised training signals, the same role scale-consistency plays here.","marker":"[12]"},{"why":"Provides the neural-operator domain decomposition approach that the test-time iterative refinement algorithm builds on.","marker":"[31]"},{"why":"Another domain-decomposition operator-learning baseline that motivates the overlapping-patch refinement scheme.","marker":"[30]"}],"fun_headline_variants":["Scale-consistency loss enables zero-shot PDE extrapolation","Scale-consistent learning extrapolates PDEs across scales","Scale-consistency lifts PDEs from Re 1000 to Re 10k","Neural PDEs extrapolate Re by 10x via scale-consistency","34% error cut via scale-consistent PDE learning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the system being modeled is governed by the same family of PDEs across all scales, so that rescaling a subdomain produces a valid instance of the same learned problem; the theorem also assumes the trained operator is exactly scale-consistent, exactly boundary-matching, and exactly correct on nearly constant coefficient fields, while training only minimizes those errors.","fun_headline_variants_meta":{"raw":{"variants":["Scale-consistency loss enables zero-shot PDE extrapolation","Scale-consistent learning extrapolates PDEs across scales","Scale-consistency lifts PDEs from Re 1000 to Re 10k","Neural PDEs extrapolate Re by 10x via scale-consistency","34% error cut via scale-consistent PDE learning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000857,"raw_usage":{"total_tokens":3778,"prompt_tokens":1060,"completion_tokens":2718,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":2627}},"tokens_in":676,"tokens_out":2718,"duration_ms":19887,"temperature":1.0,"reasoning_tokens":2627,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:08:15.883907+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: train a neural operator with the scale-consistency loss on nearly constant coefficient data, measure the three terms in Theorem B.2, and evaluate on far-from-constant test fields; if a model achieves near-zero supervised, boundary, and consistency errors yet still shows large out-of-distribution error, the theorem's conclusion fails. Experimentally, one can also directly test whether the left and right sides of identity (4) agree to machine precision on all sampled subdomains for a trained model; if not, the model is in the regime the quantitative bound already leaves uncontrolled.","supporting_citations":[{"cited_title":"Lie point symmetry data augmentation for neural pde solvers","cited_arxiv_id":null,"evidence_quote":"Documents the limited effectiveness of prior scale-symmetry augmentation, motivating the boundary-aware consistency formulation."},{"cited_title":"Physics-informed neural operator for learning partial differential equations","cited_arxiv_id":null,"evidence_quote":"Shows how physics-based losses can act as self-supervised training signals, the same role scale-consistency plays here."},{"cited_title":"Mosaic flows: A transferable deep learning framework for solving pdes on unseen domains.Computer Methods in Applied Mechanics and Engineering, 389:114424, 2022","cited_arxiv_id":null,"evidence_quote":"Another domain-decomposition operator-learning baseline that motivates the overlapping-patch refinement scheme."}],"review_version":1}