{"id":"d72d81d7-53dd-4bde-9452-f9a3c65d2d4e","arxiv_id":"2608.09764","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"MoNo uses entropy-regularized optimal transport to build balanced, stable latent-space projections in a multiscale neural operator, achieving lower relative L2 errors and GFLOPs than prior neural operators.","lead":"MoNo is a neural operator that replaces the usual learned projection into latent tokens with an optimal-transport assignment that keeps token usage balanced across scales. It reports accuracy and efficiency gains over prior neural operators on several PDE benchmarks, including large airfoil flow cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"AirfRANS SOTA claim is confounded by the modified sampling protocol: MoNo trains on all surface points per batch while borrowed baseline numbers used fully random sampling, so the large surface-error improvements may reflect data access, not CoTAP.","rationale":"I read the paper in good faith: the CoTAP construction is coherent, the balanced-marginal idea is well-motivated, the ablations show consistent gains from both CoTAP and multiscale modeling, and the custom Sinkhorn kernel is a plausible engineering contribution. The central claim, however, is empirical SOTA, so the evaluation protocol is load-bearing. The reader's weakest-assumption analysis identifies the strongest genuine vulnerability: the AirfRANS results, which contain the largest headline margins, were produced under a training protocol different from the one used to generate the borrowed baseline numbers. Appendix F explicitly documents that MoNo includes all surface points in every sampled case and uses batch size 4, while previous works used fully random sampling with batch size 1. Since the largest improvements are in surface and lift-coefficient errors, retraining baselines under the new protocol could materially shrink or reverse the claimed advantage. I checked the theoretical appendices as independent support: Theorems 1 and 2 are consistency statements showing that the architecture is an integral operator and a function-to-function map; they are correct but do not provide falsifiable predictive content that would rescue the empirical claim. Efficiency comparisons are plausible but also depend on implementation choices and were not independently reproduced. Missing comparisons to a few cited recent works (e.g., SPiDER-Solver, WOFormer) are secondary. The protocol confound is the single most load-bearing concern, and the concrete retraining test would settle it. If the test shows the baselines remain unchanged under the modified protocol, the SOTA claim survives; if not, the verdict should shift toward caution. Either way, the current CONDITIONAL verdict is appropriate, so I recommend no change.","tokens_in":26270,"tokens_out":7080,"duration_ms":67647,"concrete_test":"Retrain LinearNO, Transolver, and Transolver++ on AirfRANS in-place using exactly the Appendix F protocol: every sampled case contains all surface points plus randomly sampled volume points (32k total), batch size 4, relative L2 loss on volume and surface with equal weights, OneCycle schedule with 100 warmup epochs, learning rate 2e-4, and the same total optimization steps as MoNo. Report Vol., Surf., CL, and rho_L averaged over at least 3 seeds including seed 0. If LinearNO's surface error falls below roughly 0.002 or CL below roughly 0.042, the claimed AirfRANS advantage is substantially protocol-induced; if errors stay near 0.0077 and 0.0491, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that MoNo \"outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency\" (abstract). The most decisive evidence is AirfRANS (Table 2), where MoNo reports surface error 0.0013 versus LinearNO's 0.0077 and lift-coefficient error 0.0415 versus 0.0491. This is exactly where the protocol confound bites. Appendix F states: \"we find that the fully random sampling strategy with a batch size of 1 adopted in previous works causes severe training instability. To address this issue, each sampled case contains all surface points together with randomly sampled volume points... We further set the batch size to 4.\" The Table 2 baselines are borrowed from prior papers, so they were trained with fully random 32k-point sampling and batch size 1. Surface quantities (Surf., CL) are precisely what are most affected: under fully random sampling, surface points are a tiny fraction of each batch and are frequently absent, so surface supervision is weak. Giving MoNo guaranteed surface coverage in every training batch is an advantage independent of CoTAP. Under the modified protocol, LinearNO and Transolver might close most of the 83% surface gap and the CL gap, reducing the \"outperforms SOTA\" claim to a modest margin on a single benchmark. The comparison is further weakened by single-seed runs (seed 0) and by importing standard-benchmark baseline numbers from prior work without reruns. This does not invalidate the method, but it is a real, testable confound on the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MoNo, a multiscale Transformer-based neural operator for PDEs on general geometries. The central component is CoTAP, which constructs cross-scale assignments between adjacent latent spaces by solving an entropy-regularized optimal transport problem with uniform marginals, thereby producing balanced bidirectional projection matrices. The authors argue that this avoids token imbalance and collapse that arise with softmax-based projections, and enables stable deep multiscale hierarchies. The method is evaluated on six standard benchmarks plus AirfRANS, with claims of improved accuracy and computational efficiency; a fused Sinkhorn implementation is described, and theoretical appendices provide integral-operator and neural-operator interpretations.","tokens_in":26591,"tokens_out":7280,"duration_ms":60199,"significance":"If the empirical claims hold under controlled comparison, the paper makes a useful contribution: it identifies a concrete failure mode of softmax latent-space construction (token imbalance and collapse), offers an OT-based alternative with uniform marginal constraints, and demonstrates a practical fused Sinkhorn implementation. The multiscale hierarchy is a natural and potentially efficient design. The ablations in Table 6 support the qualitative importance of CoTAP and multiscale modeling, and the visualizations in Figures 2, 4, and 5 provide intuitive evidence of more balanced assignment patterns. The main caveat is that the headline 'outperforms SOTA' claim, especially on AirfRANS, is currently supported by a comparison protocol that differs between MoNo and the baselines. The theoretical representation as an integral operator is a useful framing, though the stated theorems are stronger than the assumptions used in their proofs.","major_comments":[{"comment":"The AirfRANS comparison in Table 2 is confounded by a training-protocol mismatch. The authors state that 'the fully random sampling strategy with a batch size of 1 adopted in previous works causes severe training instability' and therefore use 'all surface points together with randomly sampled volume points' with a batch size of 4. The baseline numbers in Table 2 are taken from prior papers that used the fully random protocol. Because surface points are a tiny fraction of the 32,000 sampled points under random sampling, MoNo's guaranteed surface coverage in every batch provides a supervision advantage that is independent of CoTAP. This directly affects the largest claimed gains: surface error 0.0013 vs. 0.0077 (an 83.1% relative improvement) and lift-coefficient error 0.0415 vs. 0.0491. To support the 'outperforms SOTA' claim, the baselines should be retrained under the same sampling and batching protocol, or a matched protocol should be used, and the resulting numbers reported.","section":"Appendix F, 'Training'"},{"comment":"All experiments are run with a single random seed (seed 0), and no error bars or variance estimates are reported anywhere. Several close margins in Tables 1 and 3 (e.g., Airfoil MoNo 0.0048 vs. LinearNO 0.0049; Darcy 141x141 MoNo 0.0049 vs. LinearNO 0.0053) may not be statistically meaningful without multiple seeds. The authors should report means and standard deviations over at least three to five seeds for the main tables, or at minimum for the closest comparisons. This is essential for evaluating the claimed state-of-the-art accuracy.","section":"Appendix F, 'Training'; all experimental sections"},{"comment":"Theorem 1 is stated for any bounded measurable physical state field and any continuous anchor token field, but the proof begins by assuming that the continuous CoTAP coupling 'admits a nonnegative measurable density' with respect to the product measure, and the kernel is defined in terms of that density. The existence and regularity of such a density for arbitrary continuous G0 is not established. Therefore the theorem as stated is stronger than what is proven; it should be restated as a conditional result that assumes the existence of the CoTAP coupling density. The same issue applies to Theorem 2, which assumes such densities at every level. This does not invalidate the empirical method, but it overstates the theoretical contribution.","section":"Appendix B, Theorem 1 and Eq. (8); Appendix D"}],"minor_comments":[{"comment":"The label 'LinerNO' in Figure 1(b) is a typo and should read 'LinearNO'.","section":"Figure 1"},{"comment":"The text states that 'Both MoNo and MoNo-light effectively utilize the additional information provided by higher-resolution observations', but in Table 3 MoNo-light's error at 211x211 (0.0059) is higher than at 141x141 (0.0057). The sentence should be qualified to acknowledge this non-monotonic behavior.","section":"Evaluation at Multiple Resolutions"},{"comment":"The claim that MoNo 'outperforms existing state-of-the-art neural operators' is repeated in the abstract and conclusion without qualification. Given the protocol mismatch on AirfRANS and the lack of multiple seeds, this statement should be softened or explicitly conditioned on the comparisons performed.","section":"Abstract and Conclusion"},{"comment":"The citation style for Transolver++ is inconsistent: Table 2 uses 'Transolver++ (2025)' while the text and reference list use '(Luo et al. 2025a)'.","section":"Table 2 and Table 12"},{"comment":"The attention kernel alpha is stated to satisfy alpha >= 0 and the integral normalization in Eq. (12) only after it is used in Eq. (13) to define beta. Moving the assumptions forward would make the proof easier to follow.","section":"Appendix B, Eq. (12)-(13)"}],"recommendation":"major_revision","confidential_remarks":"The main risk in this paper is the AirfRANS comparison: the modified sampling protocol gives MoNo guaranteed surface supervision, and the baselines were not retrained under the same protocol. This is fixable, but it is essential that the authors provide a matched-protocol comparison before the SOTA claim is accepted. The single-seed reporting is also a concern across all empirical tables. The theoretical appendices contain a gap between the theorem statements and the assumptions used in the proofs, but that is straightforward to fix by restating the theorems as conditional. The core idea is interesting and the code is promised, so I would not reject the paper, but the empirical evidence needs strengthening."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a serious architecture paper with a real, if incremental, idea, but the headline SOTA claim is not yet supported because the AirfRANS comparison is confounded by a training-protocol change. Don't kill it over that; send it for review with the expectation that the authors tighten the evaluation.\n\nWhat's new: CoTAP uses entropy-regularized OT with uniform marginals to build balanced bidirectional projections between latent spaces, replacing softmax token assignment. That is a clean, motivated solution to token collapse, and the paper shows it with clear visualizations and ablations. The multiscale progressive hierarchy built on it is a natural extension, and the fused Sinkhorn implementation is a real engineering contribution. The efficiency numbers are plausible: MoNo-light at 23 GFLOPs on 65k points vs LinearNO's 133 is the kind of win that comes from proper multiscale compression rather than from tricks. If the code is as advertised, that reproducibility is worth real credit.\n\nThe soft spots are in the experimental comparison. First, Appendix F says MoNo trains on batches that always contain all surface points, while the baselines' numbers are imported from papers that used fully random sampling with batch size 1. On AirfRANS, surface error is exactly where MoNo claims the biggest gains. That is a real confound: the improvement may come from data access, not from CoTAP. The authors need to rerun at least LinearNO and Transolver under their protocol. Second, everything is a single seed without error bars, so the margins on standard benchmarks are not robust. Third, the theorems are essentially restatements of the architecture. Theorem 1's kernel is defined as the composition after marginalization; it is not circular, but it is also not a substantive new result. The contribution should be sold as empirical and engineering work, not theoretical.\n\nWho should read it: people working on latent-space neural operators and anyone who cares how evaluation protocols change conclusions. It deserves a serious referee; the reviewer should push on rerunning baselines and multi-seed reporting. I would want to see those before believing the AirfRANS numbers, but the core method is worth engaging with.","headline":"Solid architecture paper with a real AirfRANS confound; deserves serious refereeing, not desk rejection.","tokens_in":27122,"tokens_out":3129,"would_cite":false,"duration_ms":30506,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"MoNo claims that replacing softmax-based latent-token projection with an optimal-transport coupling, CoTAP, yields balanced stable multiscale latent spaces that beat prior neural operators on seven PDE benchmarks at lower computational…","keywords":["neural operator","optimal transport","Sinkhorn iteration","latent token assignment","transformer attention","partial differential equations","multiscale architecture","general geometries"],"falsifier":"Retrain the strongest baseline, LinearNO, on AirfRANS under MoNo's exact training protocol (each sampled case contains all surface points plus randomly sampled volume points, batch size 4, the same number of optimization steps), and recompute the surrounding-field, surface-field, and lift-coefficient errors; if the gaps over MoNo shrink below the claimed 18.2% and 83.1% improvements, the state-of-the-art claim is not robust to the protocol difference.","tokens_in":26029,"feed_emoji":"⚖️","tokens_out":15855,"duration_ms":112012,"temperature":0.7,"pith_summary":"Transformer-based neural operators compress the spatial observations of a partial differential equation into a small set of latent tokens, but the softmax projection used to build those tokens does not guarantee they are used evenly; in a hierarchy of latent spaces the imbalance is inherited and worsens until deeper tokens collapse and carry almost no information. This paper proposes MoNo, a neural operator that replaces softmax projection with CoTAP, a cross-scale assignment formed as an entropy-regularized optimal transport problem with uniform marginal constraints: every source point must send the same total mass and every latent token must receive the same total mass. The balanced, bidirectional projections that CoTAP derives from one shared transport plan make a four-level progressive multiscale hierarchy stable, which the authors argue is what lets the model learn long-range physical interactions cheaply on arbitrary geometries. On seven PDE benchmarks, including the real-world AirfRANS dataset, the paper reports lower relative L2 errors than existing state-of-the-art neural operators while using far fewer FLOPs at large point counts. If the claim is right, balanced optimal-transport assignment is a practical alternative to softmax projection for building latent spaces in Transformer neural operators.","feed_headline":"Optimal transport fixes token collapse in neural physics solvers","feed_subtitle":"Balanced token assignment across latent scales cuts error on seven physics benchmarks at a fraction of the compute.","key_machinery":"The central object is CoTAP (Cross-scale Optimal Transport Assignment and Projection), the latent-space construction method that carries the whole argument. Between adjacent spaces $\\Omega^{l-1}$ and $\\Omega^{l}$, an MLP projector builds an initial score matrix $S_{\\mathrm{init}}$, and CoTAP solves the entropy-regularized optimal transport problem $\\arg\\max_{S\\in C^{l-1,l}}\\langle S, S_{\\mathrm{init}}\\rangle + \\tau H(S)$ over the transport polytope with uniform marginal constraints, using log-domain Sinkhorn iterations (alternating row and column scalings toward the uniform totals). The solution $S_{\\mathrm{ot}}$ has every row summing to $1/N_{l-1}$ and every column to $1/N_l$, so the encoding projection $S_{\\mathrm{enc}} = N_l S_{\\mathrm{ot}}$ and the decoding projection $S_{\\mathrm{dec}} = N_{l-1} S_{\\mathrm{ot}}$ are balanced bidirectional projections derived from one shared plan: no latent token can be over- or under-assigned, and the two directions transfer consistent mass rather than arbitrary learnable mixing. Reusing the same construction at each of the four latent scales is what makes progressive compression stable, because imbalance is never inherited from a previous level. A Triton-based fused Sinkhorn solver (CoTAP Iteration) with an explicitly derived reverse-mode gradient keeps the OT solve cheap, and it is this solver that carries the efficiency claim at large point counts.","core_discovery":"The paper's central claim is that the instability and imbalance of softmax-based latent projection, rather than the attention mechanism itself, is what limits the accuracy and efficiency of Transformer neural operators, and that this limit can be lifted by constructing latent spaces with optimal transport. MoNo diagnoses a specific failure mode: unconstrained learnable assignment matrices, normalized only row- or column-wise by softmax, concentrate assignment mass on a few latent tokens, and when such projections are stacked into a hierarchy the imbalance is inherited and amplified until deeper latent spaces suffer token collapse. CoTAP replaces that projection with a shared transport plan: an MLP produces an initial score matrix, which is projected onto the transport polytope with uniform row and column marginals via entropy-regularized Sinkhorn iterations, and the normalized plan is rescaled once to give the encoding matrix and once to give the decoding matrix, so both directions preserve the same mass-transfer relation and each latent token receives equal total mass. On top of this, MoNo stacks four progressively compressed latent spaces with token counts halving at each level, and proves that (i) one CoTAP projection followed by one latent self-attention layer is an input-dependent integral kernel operator on the original physical domain, and (ii) the full hierarchy defines a neural operator. Empirically the paper reports the best relative L2 errors on all seven benchmarks, including the AirfRANS surrounding-field, surface-field, and lift-coefficient metrics, and reports 23.46 GFLOPs for MoNo-light at 65,536 points versus 132.83 GFLOPs for LinearNO at a comparable parameter count.","pith_inferences":["A natural extension the paper does not pursue is replacing uniform marginals with physically informed non-uniform ones (for example, allocating more mass to high-gradient or boundary regions); whether that beats uniform marginals would test whether balance itself, rather than entropy regularization, is the source of the gains.","The paper reports that its AirfRANS training protocol differs from the baselines' published one (all surface points plus randomly sampled volume points, batch size 4, versus fully random sampling with batch size 1); whether that difference changes baseline errors is untested, so the claimed margins are the load-bearing unknown.","The resolution-stability of CoTAP assignments suggests the learned coupling behaves like a discretization-invariant transport map; if so, a trained MoNo could be evaluated zero-shot on re-meshed or upsampled versions of the same geometry, which the paper does not test.","Because Theorem 1 recasts CoTAP-plus-attention as an input-dependent integral kernel on the original domain, the learned hierarchy can be compared spectrally with Fourier-type operators; examining the kernel's effective rank across scales would show whether the multiscale compression is genuinely exploiting long-range structure rather than just reducing point count."],"forward_implications":["Replacing softmax-based projection with the balanced OT coupling lowers error even at matched model scale: MoNo-light, with a parameter count comparable to prior methods, improves relative L2 error on Airfoil, Plasticity, NS2D, and Elasticity by at least 2.0%, 9.1%, 3.7%, and 16.0% respectively.","The multiscale hierarchy only pays off when assignments are stable: in the ablation, adding progressive multiscale modeling without CoTAP provides only limited improvements, while CoTAP and the hierarchy together cut Elasticity error from 0.0077 to 0.0042.","Balanced assignment converts extra observation points into accuracy: on Darcy, MoNo's error falls steadily as resolution rises from 85×85 to 211×211, whereas Transolver and LinearNO degrade with resolution, and at 211×211 MoNo leads by at least 13.2%.","The same latent construction transfers out of distribution: on AirfRANS OOD Reynolds and OOD angle-of-attack extrapolation, MoNo reports the best surrounding- and surface-field errors and the best lift-coefficient Spearman correlation.","Efficiency is not sacrificed: at 65,536 observation points MoNo-light computes 23.46 GFLOPs versus 132.83 GFLOPs for LinearNO at comparable parameter count, and full MoNo at 97.26 GFLOPs still undercuts Transolver++ (121.17)."],"supporting_citations":[{"why":"LinearNO, the strongest prior baseline: its published benchmark numbers are the main comparison targets MoNo must beat, and its 132.83 GFLOPs at 65,536 points is the efficiency benchmark.","marker":"(Hu et al. 2026)"},{"why":"Transolver, the physics-attention baseline whose AirfRANS evaluation protocol and OOD settings MoNo follows.","marker":"(Wu et al. 2024)"},{"why":"LNO, the latent neural operator whose softmax-based learnable projection CoTAP is designed to replace.","marker":"(Wang and Wang 2024)"},{"why":"FNO, the foundational Fourier neural operator that supplied the Darcy and NS2D benchmark tasks.","marker":"(Li et al. 2020a)"},{"why":"GEO-FNO, which supplied the Airfoil, Pipe, Plasticity, and Elasticity benchmark datasets on structured meshes and point clouds.","marker":"(Li et al. 2023a)"},{"why":"AirfRANS, the real-world CFD dataset used for the main, surface-field, and out-of-distribution comparisons.","marker":"(Bonnet et al. 2022)"},{"why":"The Sinkhorn iteration algorithm that CoTAP uses to solve the entropy-regularized optimal transport problem.","marker":"(Sinkhorn 1967)"},{"why":"The integral-operator representation of neural operators and self-attention that Theorem 1 builds on.","marker":"(Kovachki et al. 2023)"},{"why":"The optimal transport theory background that defines the transport polytope and entropy regularization.","marker":"(Villani et al. 2009)"}],"fun_headline_variants":["Optimal transport stabilizes token assignments in neural operators","MoNo: multiscale optimal transport neural operator for PDEs","Balanced transport assignments prevent token collapse in neural operators","Optimal transport boosts neural operator efficiency and accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the AirfRANS state-of-the-art comparison is fair even though MoNo is trained under a modified sampling scheme (each sampled case contains all surface points plus randomly sampled volume points, with batch size 4) that the published baselines did not use; if the baselines were retrained under the same scheme, their errors could differ and the claimed margins could shrink.","fun_headline_variants_meta":{"raw":{"variants":["Optimal transport stabilizes token assignments in neural operators","MoNo: multiscale optimal transport neural operator for PDEs","Balanced transport assignments prevent token collapse in neural operators","Optimal transport boosts neural operator efficiency and accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000564,"raw_usage":{"total_tokens":2763,"prompt_tokens":1122,"completion_tokens":1641,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":738,"completion_tokens_details":{"reasoning_tokens":1576}},"tokens_in":738,"tokens_out":1641,"duration_ms":10973,"temperature":1.0,"reasoning_tokens":1576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:10:08.544786+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Retrain the strongest baseline, LinearNO, on AirfRANS under MoNo's exact training protocol (each sampled case contains all surface points plus randomly sampled volume points, batch size 4, the same number of optimization steps), and recompute the surrounding-field, surface-field, and lift-coefficient errors; if the gaps over MoNo shrink below the claimed 18.2% and 83.1% improvements, the state-of-the-art claim is not robust to the protocol difference.","supporting_citations":[],"review_version":1}