{"id":"6af2bd10-7435-44dd-9704-65f34aceb823","arxiv_id":"2509.09611","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"ReBaNO combines reduced basis greedy selection with PINN activations to learn parametric PDE maps without high-fidelity training data.","lead":"This paper proposes ReBaNO, an operator learning method that builds a small basis of pre-trained physics-informed neural network solutions, selected greedily, and combines them linearly to solve new PDE instances. It claims better generalization on out-of-distribution inputs and exact discretization invariance than standard neural operators like FNO and DeepONet.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Discretization-invariance test is confounded: ReBaNO may receive a continuous/super-resolved input f while FNO/CNO receive only s-point samples, making 'strict discretization invariance' an artifact rather than a learned property.","rationale":"The reader's verdict conditionally accepts the paper but flags the residual-loss proxy as the weakest assumption. I agree that the lack of an a posteriori error bound is important, but the more immediately load-bearing issue for the paper's headline is the discretization-invariance claim. The paper asserts that ReBaNO is the only operator learning algorithm achieving strict discretization invariance, which is a central and distinctive contribution. That assertion is based on a single Poisson experiment whose methodology is not described with enough precision to rule out an unfair comparison. Specifically, ReBaNO's online solver needs to evaluate the PDE residual, which requires the input function f at arbitrary points, whereas FNO and CNO receive only a finite set of point samples. If ReBaNO is given a continuous representation or a fixed high-resolution evaluation of f, then its flat error across resolutions is by construction and cannot be credited as a learned discretization-invariant property. This concern is concrete and testable: repeat the experiment with ReBaNO restricted to the same s-point samples and an explicit interpolation rule. The reader did note that the claim rests on a single Poisson experiment, which is related, but did not identify the asymmetric input representation as the root methodological issue. Hence 'partial'. The verdict remains conditional because the paper could address this with a revised experiment and appropriate caveats; the core reduced-basis-plus-PINN idea may still be sound, but the current central claim is not substantiated as written.","tokens_in":21373,"tokens_out":4652,"duration_ms":57144,"concrete_test":"Rerun the discretization-invariance experiment (Fig. 5 Right) giving ReBaNO exactly the same information as FNO and CNO: the discrete values of f on the s-point grid only, with a fixed, clearly specified interpolation rule (e.g., piecewise linear) to evaluate f at the collocation points in Eq. (15). If ReBaNO's error then varies significantly with s (especially increasing at small s), the 'strict discretization invariance' claim collapses; if it remains flat under this strictly comparable input representation, the claim is supported. Additionally, report the same experiment for at least one of the Darcy or Navier-Stokes benchmarks.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's headline claim that ReBaNO is 'the only operator learning algorithm achieving strict discretization invariance' rests on the experiment in Section 5.1 (Figure 5 Right) and the associated text. In that experiment, data-driven models (FNO, CNO) are trained on inputs discretized on a 128-point grid and then tested on inputs sampled with s=32,50,...,1024 points. ReBaNO is said to show flat error across all s. But ReBaNO's online phase does not consume a fixed-size discrete input vector; it minimizes the physics loss L_p (Eq. 14), which requires evaluating the source term f at the collocation points C_R (Eq. 15). The paper never specifies how f is represented to ReBaNO when the test input is only given on an s-point grid. If, as is plausible, ReBaNO evaluates f on its own fixed collocation grid (possibly the same 128-point training grid) or uses a continuous Gaussian-process realization, whereas FNO and CNO only see the s discrete values, then the comparison is asymmetric. ReBaNO's flat error curve would reflect that it effectively sees a higher-resolution or continuous version of f at all test resolutions, not that it has learned a discretization-invariant operator from point samples. This would make the uniqueness claim vacuous. The paper also provides no analysis of how interpolation of f from the s samples to C_R affects ReBaNO's error, nor does it test other PDEs for this invariance. Thus the central 'strict discretization invariance' claim is not supported by the evidence as presented.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes ReBaNO, a physics-informed operator learning method that builds a one-hidden-layer network whose activation functions are full PINN solutions selected greedily from a training input set. Online, for a new input, only the linear coefficients are tuned by minimizing the PDE residual. The authors claim that ReBaNO is data-lean, has minimal online computational cost, is mathematically rigorous due to the greedy basis construction, shrinks the generalization gap relative to PCA-Net, DeepONet, FNO, CNO, and PINO, and is the only operator learner with strict discretization invariance. These claims are tested on 1D Poisson, 2D Darcy flow, and 2D Navier-Stokes problems, with in-distribution and out-of-distribution tasks and a resolution-transfer experiment.","tokens_in":21659,"tokens_out":4237,"duration_ms":47183,"significance":"If the central claims held, ReBaNO would be a genuinely useful contribution: a physics-driven operator learner with adaptive basis construction, no paired training data, and strong generalization. The paper has tangible strengths: the code is available, the method is clearly positioned within the encoder-decoder framework, and the greedy residual-based selection is a sensible adaptation of reduced-basis ideas to PINN-based surrogates. The three benchmark problems are standard and the comparisons include both data-driven and physics-informed baselines. However, several headline claims are not supported by the paper's own numerical evidence, and the most distinctive claim, strict discretization invariance, rests on a comparison that appears to be confounded. The underlying algorithmic idea remains interesting, but the manuscript needs substantial revision before the conclusions can be accepted.","major_comments":[{"comment":"The 'strict discretization invariance' claim is confounded. ReBaNO's online phase minimizes the physics loss Eq. (14), which requires f evaluated on the collocation set C_R (Eq. 15). The paper never states how f is represented to ReBaNO when the test input is provided only on a varying s-point grid. If ReBaNO evaluates f on its own fixed or continuous functional representation while FNO and CNO only receive the s discrete samples, the flat error curve in Figure 5 Right is an artifact of asymmetric information, not a learned discretization-invariant operator. At minimum, the authors must specify the exact representation and interpolation procedure for f in the resolution test, and rerun the comparison under identical input information. The claim that ReBaNO is 'the only operator learning algorithm achieving strict discretization invariance' is not established by the current experiment, an","section":"Section 5.1, Figure 5 Right"},{"comment":"The abstract's claim of 'minimal computational cost online' is contradicted by Table 3. For Poisson, ReBaNO inference is 0.399 s per case versus 1.235 ms for FNO and 3.698 ms for PINO; for Darcy flow it is 1.940 s versus 4.222 ms for CNO; for Navier-Stokes it is 14.092 s versus 4.919 ms for CNO. ReBaNO is three to four orders of magnitude slower in online inference. The paper even reports 5000 online fine-tuning epochs for Navier-Stokes. This is not 'minimal computational cost' under any standard reading, and it materially affects the efficiency contribution claimed in the introduction and conclusion.","section":"Abstract and Table 3"},{"comment":"The claim that ReBaNO 'significantly outperforms' baselines in shrinking the generalization gap is not supported by Table 2. For Poisson, the OOD mean error ratio of ReBaNO is 6.975, essentially identical to FNO's 6.976 and worse than PINO's 4.090; the absolute OOD mean error of ReBaNO is 0.023 versus PINO's 0.007. The text states 'PINO achieves the smallest ratio in the OOD test while ReBaNO yields the smallest errors,' which is contradicted by the table. For Navier-Stokes, ReBaNO's absolute errors (0.036 train, 0.072 OOD) are far larger than FNO's (0.002, 0.013) and CNO's (0.004, 0.044). The only metric on which ReBaNO can be said to 'shrink the gap' is the ratio e_test/e_train, and even there the advantage is not uniform. The paper must either soften the superiority claim or define precisely what 'outperforms' means and defend it against the absolute-error numbers.","section":"Section 5.1 and Table 2"},{"comment":"The abstract and contribution list call the greedy algorithm 'mathematically rigorous,' but no a posteriori error bound or certified residual-to-error estimator is provided. Algorithm 1 selects basis inputs by minimizing L_p (Eq. 14), yet there is no proof that a small physics loss controls the true approximation error in the relevant function norm; the paper itself defers convergence analysis to future work. The reasoning is not circular, and the greedy selection is a legitimate heuristic, but without a bound the adjective 'rigorous' is an overstatement. The authors should either supply a residual-based error estimate for the ReBaNO surrogate or replace 'rigorous' with language appropriate to an empirical greedy strategy.","section":"Section 4.3, Algorithm 1, and Section 6"}],"minor_comments":[{"comment":"The VPINN/RVPINN formulations cite two references as '[?]' instead of actual citations; these should be filled in. The notation L_b and L_i is introduced but not consistently used later.","section":"Section 2.2"},{"comment":"There is a typo in 'strict discretizations invariance' (should be 'discretization invariance'). The informal tone of 'ReBaNo' vs. 'ReBaNO' should be unified.","section":"Section 1"},{"comment":"The inference time column is labeled 'infer time/N test'; it would help to state explicitly whether the reported numbers are per test case or averaged over the 200/1000 test cases. The parameter counts like '8+901×8' are opaque and deserve a brief explanation.","section":"Table 3"},{"comment":"The ablation study in Figure 5 Left compares greedy with random selection but reports only the largest loss, not the mean or median, and does not show the corresponding test errors. A boxplot or additional statistics would make the advantage of the greedy strategy more convincing.","section":"Section 5.1"},{"comment":"The Darcy flow experiment uses RVPINN as the high-fidelity solver, whereas the Poisson and Navier-Stokes experiments use plain PINNs. The paper does not discuss whether this choice affects the comparison or the internal consistency of the ReBaNO framework.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's underlying idea is worth pursuing, and the code availability is a clear plus. However, the headline claims are broader than the evidence. The discretization-invariance experiment, in particular, may be structurally unfair to the baselines; that is a correctness-level concern that must be addressed before the manuscript can be published. The efficiency claim is also quantitatively contradicted by the authors' own Table 3. I would recommend a major revision focusing on a fair, fully specified resolution-transfer experiment, a recalibrated set of claims, and a more precise treatment of the greedy algorithm's guarantees."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"ReBaNO is worth knowing about. The core idea—greedy selection of a few full-order PINN solutions to form a reduced basis, then per-instance online coefficient solving by minimizing the PDE residual—is a sensible and genuinely new combination. It is data-lean (no paired training data for the operator itself), the greedy selection is a legitimate adaptive procedure rather than circular fitting, and the method clearly outperforms PCA-Net and DeepONet on the three benchmarks. The ablation showing greedy selection beats random selection is convincing.\n\nThe soft spots are real, though, and some are load-bearing. The abstract promises \"minimal computational cost online,\" but Table 3 tells a different story: ReBaNO inference is 0.399 s vs 1.235 ms for FNO on Poisson, and 14.092 s vs 4.919 ms on Navier-Stokes. For a surrogate meant to replace expensive simulations, that is not a minor detail. The \"mathematically rigorous greedy algorithm\" is also oversold: the paper itself defers convergence analysis to future work, and there is no a posteriori error bound, unlike classical RBM. The \"only algorithm achieving strict discretization invariance\" claim rests entirely on a single Poisson experiment, and the stress-test confound holds: ReBaNO's online phase evaluates f on its own collocation grid (presumably the 128-point training grid), while FNO and CNO receive only the s discrete samples. That is not a fair comparison, and the paper does not specify how f is represented to ReBaNO at other resolutions. The OOD generalization ratio on Poisson is also essentially tied with FNO (6.975 vs 6.976), undercutting \"significantly outperforms\" on that benchmark. No error bars are reported anywhere, and two references are missing placeholders (VPINN, RVPINN).\n\nNone of this kills the method. The architecture is publishable and could be useful for problems where per-instance online cost is acceptable and data are scarce. But the framing needs to be recalibrated: the computational cost should be reported honestly, the discretization-invariance experiment needs a level playing field (and more than one PDE), and the uniqueness claims should be softened. This deserves a serious referee—the core idea is novel and the numerical evidence, though overstated, is substantial. I would send it to review with the expectation of major revision.","headline":"A genuinely new reduced-basis-plus-PINN operator learner with attractive data-lean properties, but the headline claims outrun the evidence: online cost is high, the discretization-invariance test is confounded, and several uniqueness claims lack support.","tokens_in":22240,"tokens_out":1853,"would_cite":true,"duration_ms":20590,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"ReBaNO builds a neural operator from a handful of PINN solutions, closing generalization gaps and achieving true mesh invariance.","keywords":["reduced basis method","neural operator","physics-informed neural network","discretization invariance","generalization gap","greedy algorithm","out-of-distribution generalization","knowledge distillation"],"falsifier":"Run ReBaNO on an advection-dominated problem with a boundary layer; if the greedy indicator selects basis inputs that do not resolve the layer, and online solutions have small PDE residual but large L2 error compared to a high-fidelity solver, the central claim would be falsified. Alternatively, find a PDE where the residual minimum is non-unique and the chosen coefficients produce poor predictions despite small residual.","tokens_in":21167,"feed_emoji":"🧮","tokens_out":3323,"duration_ms":33624,"temperature":0.7,"pith_summary":"This paper claims that the generalization failures and mesh dependence of neural operators can be avoided by constructing the operator as a linear combination of a small number of full physics-informed neural network (PINN) solutions selected greedily. The resulting method, ReBaNO, needs no paired training data offline, only a few full-order PINN solves, and its online prediction is a one-layer network whose hidden neurons are pre-trained PINNs. On Poisson, Darcy flow, and Navier-Stokes benchmarks, the authors report that ReBaNO outperforms PCA-Net, DeepONet, FNO, and CNO in closing the gap between training and test error, especially for out-of-distribution inputs, and is the only method whose error is unchanged when the grid resolution changes.","feed_headline":"ReBaNO neural operator achieves true discretization invariance","feed_subtitle":"Greedily chosen PINN solutions shrink generalization gaps to near 1x while data-driven operators degrade up to 30x.","key_machinery":"The central object is the reduced basis of PINN solutions {u_i^PINN} built by a mathematically rigorous greedy algorithm (Algorithm 1): starting from one random full-order solve, at each step the input with the largest residual loss of the current rank-n surrogate is chosen, and its PINN solution is added as a new hidden neuron. The online solver then minimizes the same residual loss (Eq. 14) over the coefficients c, using precomputed derivatives; for linear PDEs this collapses to a small least-squares problem.","core_discovery":"ReBaNO treats each full-order PINN solution as a neuron and constructs a rank-N surrogate space by a greedy algorithm that repeatedly adds the PINN solution of the input for which the current surrogate's PDE-residual loss is largest. Online, a new input is mapped to coefficients by minimizing the same residual loss over the linear combination; because the basis functions are mesh-free PINNs and all derivatives are precomputed, the prediction is indifferent to discretization. In all three benchmarks, ReBaNO had test/training error ratios near 1 for both in-distribution and out-of-distribution inputs, while data-driven models showed gaps of 1.5x to 30x, and its accuracy remained flat when grid","pith_inferences":["If the residual loss is a trustworthy error indicator (which the paper assumes but does not certify), the same greedy construction could be applied to any PDE family with a stable weak form, including inverse problems or time-dependent controls, giving a physics-driven alternative to transfer learning.","The method's online cost scales with the number of reduced basis functions and the cost of evaluating PINNs; for problems requiring many basis functions, the greedy selection may become expensive, but one could replace full PINNs with cheaper surrogate solvers as activations without losing the framework.","The reported discretization invariance suggests a testable extension: train ReBaNO on one grid and evaluate on an adaptively refined mesh in regions of high gradients; if accuracy stays flat, it would support the claim of true continuous-discrete equivalence rather than mere insensitivity to uniform resolution.","Since the paper defers convergence analysis, an immediate research question is whether the greedy residual indicator yields provable convergence rates like classical reduced basis methods; if not, the method's robustness in adversarial PDEs is an open question."],"forward_implications":["If ReBaNO works as claimed, operator learning no longer requires large labeled training datasets: 8-48 full-order PINN solves replace thousands of input-output pairs.","The online solve is physics-informed per test instance, which is why out-of-distribution generalization gaps shrink compared to purely data-driven operators.","Strict discretization invariance follows because the basis functions are continuous mesh-free functions (PINNs), not grid-dependent features; this is the only method in the comparison that does not degrade under resolution changes.","ReBaNO is data-lean in the sense that it does not need high-fidelity training data for the operator, only for the handful of selected basis solutions."],"fun_headline_variants":["ReBaNO: greedy PINN surrogate beats data-driven operators","Greedy PINN basis yields discretization-invariant operator","ReBaNO shrinks generalization gap to near 1x","Mesh-free neural operator achieves strict discretization invariance","Data-lean operator learning: ReBaNO beats PCA-Net, DeepONet, FNO"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The greedy selection and online coefficient solve both rely on the PDE residual loss being a trustworthy proxy for the actual solution error; the paper provides no certified error bound for this proxy.","fun_headline_variants_meta":{"raw":{"variants":["ReBaNO: greedy PINN surrogate beats data-driven operators","Greedy PINN basis yields discretization-invariant operator","ReBaNO shrinks generalization gap to near 1x","Mesh-free neural operator achieves strict discretization invariance","Data-lean operator learning: ReBaNO beats PCA-Net, DeepONet, FNO"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000138,"raw_usage":{"total_tokens":961,"prompt_tokens":686,"completion_tokens":275,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":184}},"tokens_in":430,"tokens_out":275,"duration_ms":3006,"temperature":1.0,"reasoning_tokens":184,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:44:18.352430+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run ReBaNO on an advection-dominated problem with a boundary layer; if the greedy indicator selects basis inputs that do not resolve the layer, and online solutions have small PDE residual but large L2 error compared to a high-fidelity solver, the central claim would be falsified. Alternatively, find a PDE where the residual minimum is non-unique and the chosen coefficients produce poor predictions despite small residual.","supporting_citations":[],"review_version":1}