{"id":"c90e9d56-f03d-4130-a3a5-0802357785ca","arxiv_id":"2505.12020","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"GeoMaNO adapts Mamba state-space models with geometric corrections and 2D scans to achieve reported state-of-the-art accuracy and efficiency on Darcy flow and Navier-Stokes benchmark problems.","lead":"GeoMaNO is a neural network architecture that solves PDEs by combining Mamba's fast state-space sequence modeling with corrections designed to preserve grid geometry. On standard Darcy flow and Navier-Stokes benchmarks it reports large accuracy and speed gains over prior models like Transolver.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Headline SOTA numbers appear to be the best-of-22 ablation configurations selected on the test set, with no validation split described, so the reported 58.9% and 36.8% gains may be inflated by selection bias.","rationale":"I read the paper in good faith and found the proposed architecture plausible: the geometric correction term addresses a genuine structural concern in multi-directional scans, and the ablation results consistently show the correction helping on both benchmarks. However, the central empirical claim—that GeoMaNO improves the previous SOTA by 58.9% and 36.8%—rests entirely on the reported test errors, and the experimental write-up does not describe any validation split. The best rows of the ablation table are exactly the numbers in the headline table, which is strong evidence that the model was selected on the test set. This is a real methodological flaw because the reported error is the minimum over 22 configurations, and the comparison against single-configuration baselines is not statistically matched. That said, the concern does not reverse the qualitative conclusion: even the worst vanilla GeoMaNO variants in Table 3 (e.g., 0.04190 on Navier-Stokes and 0.00380 on Darcy) are still substantially better than Transolver's 0.0900 and 0.0057, so the architecture likely does beat the baseline. The problem is that the exact magnitude of the improvement, the headline numbers, and the efficiency claims (one of which has a clear internal inconsistency) are not rigorously supported. The reader's CONDITIONAL verdict is appropriate: the paper needs code release, a clear model-selection protocol with a validation split, and corrected efficiency tables before the SOTA claim can be accepted. My stress-test therefore leaves the verdict unchanged.","tokens_in":20059,"tokens_out":14703,"duration_ms":137140,"concrete_test":"Hold out a validation split from the 200 test samples (e.g., 160 for validation, 40 held out for testing), select the best GeoMaNO configuration from the Table 3 ablations using only the validation split, and then report the relative L2 error of that single configuration on the held-out 40 samples for both Darcy and Navier-Stokes. Compare these numbers against Transolver's error computed under the identical train/validation/test split. If the held-out error remains within a small margin (e.g., 5% relative) of the reported 0.0036 and 0.0370, the selection-bias concern is mitigated; if the held-out error is substantially higher, the headline SOTA claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper reports GeoMaNO's headline accuracy as 0.0370 relative L2 on Navier-Stokes and 0.0036 on Darcy (Table 1). These numbers exactly match the best rows of the ablation table (Table 3): for Darcy, the best row is '2D, None, 0011' with 0.00362; for Navier-Stokes, it is 'Vanilla, Yes, Learnable' with 0.03701. The experimental setup in Sec. A.1 states only that 1000 samples are used for training and 200 for testing; no validation split is mentioned anywhere in the paper. The ablations in Table 3 therefore appear to be evaluated on the same 200-sample test set, and the best configuration is then reported as the final model in Table 1. This makes the headline result the minimum over 22 configurations (12 Darcy + 10 Navier-Stokes), which is a biased estimate of the selected model's true performance: selecting the best of many correlated trials on the test set systematically lowers the reported error compared with a single pre-registered configuration. The baselines are reported as single fixed configurations from their original papers, so the comparison is not statistically matched. A secondary issue is that the appendix Table 7(b) lists Transolver's memory as 5.21 MB for Navier-Stokes, contradicting the 91.85 MB in Table 2(a); this suggests at least one error in the efficiency tables. The load-bearing concern is that the claimed SOTA gains are not derived from a protocol that separates model selection from evaluation, so the magnitude of the reported improvement is not rigorously established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes GeoMaNO, a neural operator built on a Mamba-style state-space model augmented with a geometric correction designed to dampen duplicate hidden states caused by four-way cross-scans. The architecture is applied to the standard THUML PDE benchmarks: Darcy flow (2D elliptic) and Navier-Stokes (3D parabolic/hyperbolic). The authors claim state-of-the-art accuracy, improving on the previous best Transolver by 58.9% on Navier-Stokes and 36.8% on Darcy flow, and also improved efficiency (training time, inference time, memory). The paper includes architectural ablations, scalability studies, error bars for the main results, and an additional ImageNet-100 experiment demonstrating generalization of the geometric correction module. The mathematical preliminaries and the kernel-integral/SSM equivalence are credited to prior work, while the geometric correction and 2D SSM representation are presented as the main novelties.","tokens_in":20421,"tokens_out":12130,"duration_ms":109283,"significance":"If the empirical claims hold, GeoMaNO would be a noteworthy contribution: it combines linear-complexity selective SSMs with a geometrically motivated correction, addressing a recognized limitation of Transformer-based neural operators on regular grids. The paper ships error bars for its main numbers and ablation tables over a reasonably sized configuration space, and the ImageNet-100 transfer experiment is a useful sanity check for the geometric-correction mechanism. However, the load-bearing evidence for the headline improvements is currently jeopardized by an experimental protocol that appears to select the best of many ablation configurations on the same test set, by an internal contradiction in the efficiency tables, and by an inconsistency between the mathematical formulation and the provided implementation pseudo-code. These issues must be resolved before the claimed state-of-the-art results can be accepted.","major_comments":[{"comment":"The manuscript never describes a validation split; Sec. A.1 states only that 1000 samples are used for training and 200 for testing. The headline numbers in Table 1 (0.0370 for Navier-Stokes, 0.0036 for Darcy) exactly coincide with the best rows of the ablation table (Table 3): 'Vanilla, Yes, Learnable' for N-S and '2D, None, 0011' for Darcy. If, as the text suggests, all ablation configurations were evaluated on the same 200 test samples, then the reported 'state-of-the-art' gains are the result of test-set selection over 22 configurations, which systematically biases the reported errors downward relative to a pre-specified model. Please rerun the experiments with a proper validation split, select the configuration on validation, and report only the test-set performance of the selected model, or otherwise document that model selection is performed without access to the test labels.","section":"Sec. A.1, Table 1, Table 3"},{"comment":"The memory consumption for Transolver on the Navier-Stokes task is reported as 91.85 MB in Table 2(a) but as 5.21 MB in Table 7(b) of the appendix. The two tables cannot both be correct. Since efficiency is one of the paper's core claims, this internal contradiction makes the memory comparison unreliable: if the 5.21 MB figure is the true one, then GeoMaNO's 70.44 MB is substantially larger and the claimed 23.3% memory saving is false; if the 91.85 MB figure is correct, the appendix needs a correction. The authors must identify the error and provide consistent numbers for all efficiency metrics.","section":"Table 2(a) and Table 7(b)"},{"comment":"The geometric correction in the main equations subtracts R_s B_s x from the output (Eqs. 9 and 10), while Algorithm 2 computes y += Cs*(hs - Rs*x), omitting the multiplication by B_s. Since the correction term is the central theoretical novelty, this discrepancy between the formal definition and the implementation pseudo-code is load-bearing. The authors need to clarify which form is actually implemented and used in the experiments, and align the mathematical formulation with the algorithmic description.","section":"Eq. (9), Eq. (10), and Algorithm 2 line 24"},{"comment":"The accuracy comparison in Table 1 reuses baseline numbers from their original papers (e.g., the Transolver error 0.0900 is taken from the Transolver paper) rather than rerunning them under the same training and evaluation protocol as GeoMaNO. This is a fairness concern, especially when juxtaposed with the same-environment efficiency comparison in Table 2, which does run Transolver locally. Please rerun at least the strongest baselines (Transolver, ONO, FactFormer, LSM) under the same environment, data split, and evaluation metric, and report their errors alongside GeoMaNO so that the claimed 58.9% and 36.8% gains are statistically matched.","section":"Sec. 5.2 and Sec. A.4"}],"minor_comments":[{"comment":"There is a typo: 'empolyed' should be 'employed'.","section":"Sec. 1, last paragraph"},{"comment":"Eq. (16) is missing the factor v(y) in the integrand; the correct kernel integral should be \\int C e^{Ax} B e^{-Ay} v(y) dy, which is what Eq. (17) later uses. Please fix the typo.","section":"Appendix B, Eq. (16)"},{"comment":"In the derivation of h'(x), the lower integration limit changes from -infinity to 0 without explanation; the limit should remain -infinity for the conclusion h'(x) = A h(x) + B v(x) to hold.","section":"Appendix B, Eq. (18)"},{"comment":"The main-text model configuration does not explicitly state which positional-encoding and geometric-correction settings are used for the final GeoMaNO model on each benchmark; these are only inferable from Table 3. Please state the final configuration in the main text for reproducibility.","section":"Sec. 5.1"},{"comment":"The tensor shapes in the 'Require' block are inconsistent with the usage in the loop: cAt is declared as (N,ED) but is used with spatial subscripts [kh,kw], and cBt is declared as (B,H,W) without the dstate dimension. Please clarify the exact shapes and broadcasting semantics.","section":"Algorithm 2"},{"comment":"The Darcy loss is reported as LrL2 + 0.1Lg, but the spatial gradient regularizer Lg is not defined in the main text; please define it where the training configuration is introduced.","section":"Table 4"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the apparent test-set selection of ablation configurations without a validation split; if the authors cannot provide a clean model-selection protocol, the headline accuracy claims should be substantially downweighted. The memory-table contradiction and the Eq./Algorithm inconsistency also suggest that the supplementary materials were not carefully checked. I would not reject outright, because the core idea is plausible and the issues appear fixable with additional experiments and corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: worth reading for the geometric-correction idea, but don't quote the SOTA numbers until the model-selection protocol is clarified.\n\nWhat's actually new: the paper augments a Mamba-based neural operator with (1) a 2D state-space recursion taken from their own 2DMamba, and (2) a learnable or fixed correction term Rs that subtracts duplicate Bx contributions from the four cross-scan directions. That's a sensible way to address redundancy in multi-way scans, and the ablations show consistent small gains from the correction. The idea is plausibly useful beyond PDEs—they show a small ImageNet-100 bump when plugging GeoMamba-SSM into VMamba. Credit where due: the architecture is clearly described, the benchmarks are standard, and they report error bars for their own runs.\n\nThe soft spot is load-bearing. The Table 1 headline results—0.0370 on Navier-Stokes and 0.0036 on Darcy—are exactly the best rows of the Table 3 ablation grid (22 configurations). The paper nowhere mentions a validation split; Sec A.1 says only 1000 train / 200 test. So the reported SOTA numbers look like the minimum over 22 configurations evaluated on the same test set used for the final table. That systematically inflates the apparent gain, and the baselines are fixed configurations from prior papers, so the comparison isn't matched. The 58.9% and 36.8% claims are not trustworthy as stated.\n\nThere's also a minor but telling error in the efficiency tables: appendix Table 7(b) puts Transolver's memory at 5.21 MB for Navier-Stokes, while Table 2(a) says 91.85 MB. Either one is wrong, which doesn't inspire confidence in the memory claims. Code is not provided, which makes the whole thing hard to verify.\n\nThe central architectural idea—dampening duplicate hidden states from cross-scans—is reasonable and deserves a serious look. But the paper's main evidence for superiority is compromised by the selection problem. If the authors can show a proper validation split and matched baselines, the results may still hold; if not, the gains could shrink substantially.\n\nWho this is for: people working on Mamba-based operators or efficient neural PDE solvers. It should go to peer review, but it needs a major revision on the experimental methodology before the performance claims can be taken at face value.\n\nBest.","headline":"GeoMaNO has a clever architectural idea and strong-looking results, but the headline SOTA numbers appear to be best-of-22 ablations selected on the test set, so the gains are probably inflated.","tokens_in":20930,"tokens_out":2625,"would_cite":false,"duration_ms":25932,"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":"A geometric correction to the state-space scan lets a Mamba-based neural operator cut PDE solution error by up to 58.9 percent.","keywords":["neural operator","PDE surrogate","state-space model","Mamba","geometric correction","cross-scan","Darcy flow","Navier-Stokes"],"falsifier":"Re-run the comparison with a held-out validation set: select the geometric-correction configuration by validation error, then report test error. If the chosen GeoMaNO does not beat the previous best model by roughly 58.9% on Navier-Stokes and 36.8% on Darcy flow, the reported state-of-the-art claim fails.","tokens_in":19858,"feed_emoji":"🌊","tokens_out":7746,"duration_ms":72470,"temperature":0.7,"pith_summary":"This paper sets out to build a neural operator—a learned mapping from PDE parameters to solutions—that is both accurate and efficient on regular grids. It claims that Mamba-style state-space models, which process sequences in linear time, can outperform the Transformer-based operators that currently dominate the field, provided the geometry of the grid is respected. To that end it introduces GeoMaNO, whose GeoMamba-SSM module adds a learnable or fixed geometric correction that suppresses duplicate hidden states produced by the four-way cross-scan, and, for 2D problems, replaces the one-dimensional scan with a two-dimensional state-space recurrence. On the standard Darcy flow and Navier-Stokes benchmarks the paper reports relative L2 errors of 0.0036 and 0.0370, beating the previous state of the art by 36.8% and 58.9%, with lower inference time and GPU memory. If these numbers hold, fast and cheap surrogate PDE solvers become significantly more accurate on the grid-based problems that dominate engineering practice.","feed_headline":"State-space PDE solver beats transformer rivals by 58.9%","feed_subtitle":"Geometric scan correction lowers Navier-Stokes and Darcy flow error while cutting inference time and memory.","key_machinery":"The load-bearing object is the GeoMamba-SSM module, a selective state-space model whose output subtracts an input-adaptive correction term, $-R_sB_sx$, from the hidden-state readout in each scan direction. The subtraction is the mechanism that removes the fourfold duplication of $Bx$ created when a grid is scanned in four directions and then merged. For two-dimensional PDEs the module uses a two-dimensional recurrence that keeps vertical neighbors adjacent in the hidden state, making the decay of each contribution a Manhattan distance instead of a flattened row-major distance. The ablation study attributes the accuracy gain to these two geometric choices, with the best fixed correction patterns chosen per benchmark.","core_discovery":"The central claim is that the geometric form of the state-space scan, not just the backbone choice, is what unlocks accuracy on regular-grid PDEs. GeoMamba-SSM writes the output as $y^{(d)}[t]=\\sum_s(C_s^{(d)}[t]h_s^{(d)}[t]-R_s^{(d)}[t]B_s^{(d)}[t]x^{(d)}[t])$, so the duplicated contribution $Bx$ that appears four times under the four-way cross-scan is dampened by a correction coefficient $R_s$ that can be fixed or learned. For two-dimensional problems it further replaces the flattened 1D recursion with the 2D recurrence $g_s[i,j]=A_s[i,j]g_s[i,j-1]+B_s[i,j]x[i,j]$ and $h_s[i,j]=A_s[i,j]h_s[i-1,j]+g_s[i,j]$, which makes the hidden-state decay depend on Manhattan distance rather than row-major linear distance. With these two mechanisms GeoMaNO reports relative L2 errors of 0.0370 on Navier-Stokes and 0.0036 on Darcy flow, improvements of 58.9% and 36.8% over the prior best model, alongside reduced training and inference time and memory.","pith_inferences":["Because the paper shows GeoMamba-SSM also improves ImageNet-100 accuracy over vanilla Mamba-SSM inside a vision backbone, the duplicate-state correction is likely transferable to non-PDE multi-scan state-space models; this is an inference beyond the paper's PDE claim.","The correction coefficient is only a per-scan constant or a scalar learned coefficient; an input-dependent $R_s$ could dampen duplicates where they actually occur, a testable variant the paper does not explore.","Extending the 2D recurrence to three dimensions, which the paper names as future work, is the direct test of whether the same geometric principle carries to volumetric and time-dependent PDEs."],"forward_implications":["On the two benchmarks tested, a state-space backbone with geometric correction outperforms the previous best Transformer-based operator in both accuracy and efficiency, suggesting linear-complexity backbones can replace quadratic attention for grid PDEs.","The large error reductions imply that the four-way cross-scan's hidden-state duplication was a real source of error, not a cosmetic artifact, and other multi-scan Mamba models may carry the same overhead.","The 2D recurrence preserves spatial adjacency for 2D PDEs, but the paper reports it fails on the 3D Navier-Stokes domain, so dimension-matched scan geometry is needed rather than one universal scan.","The hardware-aware tiling described in the appendix applies the correction at the thread level, so the geometric fix does not add memory traffic; GPU memory drops substantially on Darcy flow.","If the reported gains are reproducible, PDE surrogate modeling on regular grids can move from quadratic-complexity attention to linear-complexity scanning without sacrificing accuracy."],"supporting_citations":[{"why":"Supplies the previous state-of-the-art baseline and the standard training configuration that GeoMaNO follows.","marker":"[3]"},{"why":"Provides the selective state-space model and hardware-aware scan that GeoMaNO builds on.","marker":"[4]"},{"why":"Supplies the two-dimensional state-space recurrence used for 2D PDEs.","marker":"[10]"},{"why":"Establishes the Darcy flow and Navier-Stokes benchmark setup and the relative L2 metric.","marker":"[13]"},{"why":"Provides the four-way cross-scan pattern whose duplicate hidden states GeoMamba-SSM corrects.","marker":"[8]"},{"why":"Supplies the derivation connecting Mamba-SSM to neural operator kernel integrals, which the paper reuses.","marker":"[6]"},{"why":"Provides the standard PDE solving benchmark used for evaluation.","marker":"[11]"}],"fun_headline_variants":["GeoMaNO beats transformer PDE solvers by 58.9% with geometric Mamba","Geometric Mamba neural operator cuts PDE error 58.9% below transformers","PDE solver with geometric Mamba: 58.9% better than transformer rivals","GeoMaNO: state-space PDE model outperforms transformer rivals by 58.9%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported gains depend on choosing the best geometric-correction configuration from the ablation table while using the same test set for the final numbers, and the paper describes no separate validation split for that selection.","fun_headline_variants_meta":{"raw":{"variants":["GeoMaNO beats transformer PDE solvers by 58.9% with geometric Mamba","Geometric Mamba neural operator cuts PDE error 58.9% below transformers","PDE solver with geometric Mamba: 58.9% better than transformer rivals","GeoMaNO: state-space PDE model outperforms transformer rivals by 58.9%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000812,"raw_usage":{"total_tokens":3561,"prompt_tokens":947,"completion_tokens":2614,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":2522}},"tokens_in":563,"tokens_out":2614,"duration_ms":17553,"temperature":1.0,"reasoning_tokens":2522,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:42:51.157346+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the comparison with a held-out validation set: select the geometric-correction configuration by validation error, then report test error. If the chosen GeoMaNO does not beat the previous best model by roughly 58.9% on Navier-Stokes and 36.8% on Darcy flow, the reported state-of-the-art claim fails.","supporting_citations":[{"cited_title":"Transolver: a fast transformer solver for pdes on general geometries,","cited_arxiv_id":null,"evidence_quote":"Supplies the previous state-of-the-art baseline and the standard training configuration that GeoMaNO follows."},{"cited_title":"Mamba: Linear-time sequence modeling with selective state spaces,","cited_arxiv_id":null,"evidence_quote":"Provides the selective state-space model and hardware-aware scan that GeoMaNO builds on."},{"cited_title":"2DMamba: Efficient state space model for image representation with applications on giga-pixel whole slide image classification,","cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional state-space recurrence used for 2D PDEs."},{"cited_title":"Fourier Neural Operator for Parametric Partial Differential Equations,","cited_arxiv_id":null,"evidence_quote":"Establishes the Darcy flow and Navier-Stokes benchmark setup and the relative L2 metric."},{"cited_title":"Vmamba: Visual state space model,","cited_arxiv_id":null,"evidence_quote":"Provides the four-way cross-scan pattern whose duplicate hidden states GeoMamba-SSM corrects."},{"cited_title":"Alias-free mamba neural operator,","cited_arxiv_id":null,"evidence_quote":"Supplies the derivation connecting Mamba-SSM to neural operator kernel integrals, which the paper reuses."},{"cited_title":"PDE solving standard benchmark","cited_arxiv_id":null,"evidence_quote":"Provides the standard PDE solving benchmark used for evaluation."}],"review_version":1}