{"id":"9bfc3c8f-f320-474e-aba8-26cca0bd97b9","arxiv_id":"2505.11766","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new neural operator architecture, SKNO, models embedding evolution along an added auxiliary dimension with Fourier kernels and reports state-of-the-art errors on PDE benchmarks.","lead":"This paper adds an extra 'auxiliary' dimension to neural operators, turning how the model's internal embeddings change into part of the learned operator. The new SKNO model reports lower prediction errors than FNO and Transolver on over ten PDE benchmarks, at similar training cost.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 1's evidence for the d+1 mechanism is confounded: SKNO's linear block includes an untruncated local x-differential operator absent from FNO/Transolver, so the 1-mode heat/advection gap is not attributable to the p-evolution.","rationale":"The reader's weakest assumption concerned whether Eq. 13 faithfully represents embedding dynamics for nonlinear benchmarks. My stress-test found a different, more direct concern: the flagship controlled experiment in Table 1 does not isolate the auxiliary-dimension evolution because the SKNO linear block additionally contains an untruncated local differential operator on x, while the FNO and Transolver blocks are band-limited. This is an internal control flaw rather than a theoretical representability gap. The full-model comparisons in Tables 2, 13, and 15 are less affected by this particular confound, so I do not recommend changing the CONDITIONAL verdict: the paper's empirical claim remains plausible but is not fully supported without code, seeds, and a matched ablation that removes the local x-branch from the comparison. My recommendation is therefore UNCHANGED rather than REJECT, conditional on the authors addressing the confound and releasing reproducible code.","tokens_in":28002,"tokens_out":14372,"duration_ms":162891,"concrete_test":"Re-run the Table 1 heat and advection linear-block setups under a matched budget: (a) SKNO with the local differential operator disabled; (b) FNO and Transolver augmented with the same local differential operator and roughly matched parameter count; (c) all with 1 mode/4 gridp; (d) all with full modes. If SKNO in (a) degrades to O(1) error, or FNO/Transolver in (b) drop to the 1e-2 level, the advantage attributed to the p-dimension evolution is confounded by the untruncated local x-branch. Additionally, run at least five seeds for SKNO, FNO, and Transolver on Burgers, Darcy, and Navier-Stokes to check whether the full-model margins in Tables 2 and 13 survive seed variability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. 3.1 states that SKNO always appends one local propagator that computes finite-difference derivatives on x via Eq. 15 using an untruncated convolution in physical space. In Table 1, SKNO is listed as '(1 mode, 4 gridp)' and Table 7 shows a single layer for the heat/advection linear-block experiments. With L=1, the SKNO linear block therefore includes both a truncated spectral x-branch and a full-resolution local differential x-branch, whereas FNO with 1 mode and Transolver with 4 slices are effectively band-limited in x. The heat and advection solution operators are x-convolutions, so the local differential branch alone can supply high-frequency x content that the truncated FNO/Transolver blocks cannot. Consequently, the roughly 340x gap between SKNO (2.637e-3) and FNO (9.305e-1) on heat, and the analogous advection gap, need not come from the F_p^{-1} A F_p + b evolution along the auxiliary dimension at all. This does not disprove the full-model results, but it removes the paper's cleanest controlled evidence for the central mechanistic claim. The absence of code, seeds, and error bars further leaves the consistent-SOTA claim over Tables 2, 13, and 15 unverified against seed variability.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a d+1-dimensional reformulation of neural operators, in which the usual d-dimensional embedding evolution is lifted to a (d+1)-dimensional domain by introducing an auxiliary dimension p, motivated by the Schrödingerisation method for PDEs. The resulting architecture, the Schrödingerised Kernel Neural Operator (SKNO), uses Fourier-based operators in both the physical domain and the auxiliary dimension, together with residual connections and a final local differential propagator. The authors claim consistent state-of-the-art relative L2 errors across more than ten benchmarks (1D heat, advection, Burgers; 2D Darcy, Gray-Scott, shallow-water, Navier-Stokes, stress/strain; 3D compressible Navier-Stokes and Rayleigh-Taylor; ERA5 wind), with training time comparable to FNO, and additionally report resolution-invariance, super-resolution, and zero-shot temporal generalization experiments, as well as ablations of the lifting/recovering operators and of individual architectural components.","tokens_in":28322,"tokens_out":8653,"duration_ms":84672,"significance":"If the claims hold, the auxiliary-dimension operator design provides a simple and computationally efficient alternative to brute-force embedding scaling in neural operators, and the d+1 perspective could be a useful organizing principle for future operator architectures. The empirical scope is broad and the ablation study (Table 3) is informative. The manuscript also makes a concrete theoretical claim about efficient energy capture (Theorem D.3) and proposes an entanglement-entropy diagnostic. However, the conclusions are currently weakened by the absence of repeated-seed statistics, by a controlled linear-block comparison that does not isolate the p-evolution mechanism, and by an unproved inclusion at the heart of the main theoretical result. The core idea is plausible and the full-model results are suggestive, but the evidence as presented is not yet at the level required to establish the central mechanistic and SOTA claims.","major_comments":[{"comment":"The Table 1 comparison is confounded and therefore does not provide the claimed controlled evidence for the auxiliary-dimension evolution. With L=1, SKNO has zero global spectral propagators and consists only of the final local propagator, which is the untruncated physical-space differential kernel of Eq. (15), in addition to the W_l residual/p-evolution block. The FNO (1 mode) and Transolver (1-4 slices) columns are band-limited in x. Consequently, the ~340x gap on heat and the analogous advection gap could arise entirely from the untruncated local differential branch, which is absent from the FNO and Transolver blocks. To support the statement in Sec. 4.2 that SKNO \"removes this bottleneck\" along p, please report the heat and advection errors for (i) SKNO with the F_p^{-1} A F_p + b module ablated (keeping the local differential branch), and (ii) SKNO and baselines with L>=2 so that all models include a global spectral/attention branch. Without these controls, Table 1 cannot be cited as evidence for the d+1 mechanism.","section":"Sec. 3.1, Eq. (15); Table 1; Table 7 (1D Heat and 1D Advection rows)"},{"comment":"The key inclusion H_FNO ⊂ H_SKNO is asserted without proof or construction. The proof of Eq. (42) and the entire claim that SKNO captures energy more efficiently than FNO depend on this inclusion, yet no argument is given for why every FNO linear map is realizable by SKNO under the same width budget, nor why the supremum over H_SKNO dominates. Since the paper presents Theorem D.3 as a formal proof, please either provide an explicit parameter assignment (including how the p-evolution module and local propagator are set to reproduce an arbitrary FNO layer) or explicitly label the statement as a conjecture and support it empirically. The sentence \"It is straightforward to extend the above arguments to function spaces\" is not a substitute for this missing justification.","section":"Appendix D, Theorem D.3, paragraph following Eq. (41)"},{"comment":"The central SOTA claim is not statistically supported. All tables report single runs without error bars, seeds, or code, and several margins over the best baseline are very small (e.g., Burgers 5.475e-4 vs 6.479e-4 in Table 2; shallow-water 0.0027 vs 0.0030 in Table 13; stress 0.0472 vs 0.0481 in Table 17). In addition, the main comparisons are not strictly budget-matched: SKNO uses 4+1 layers (four global propagators plus one local propagator) on Darcy, Navier-Stokes, and Rayleigh-Taylor, whereas FNO uses 4 layers (Table 7), and the local propagator adds an untruncated module. Please report mean and standard deviation over at least three seeds, and release the code and data so that the reported single-run numbers can be independently verified.","section":"Sec. 4.1-4.2, Tables 2, 7, 13, 15"}],"minor_comments":[{"comment":"The title contains a spacing error: \"ind+ 1Dimensions\" should read \"in d+1 Dimensions\".","section":"Title"},{"comment":"In the definition of D_mu, the index range \"mu_{-Np/2},...,mu_{Np/2}\" is incomplete; it should specify the ordering of the DFT frequencies, e.g., \"mu_{-Np/2},...,mu_{Np/2-1}\" for an even grid.","section":"Sec. 2.1, Eq. (6)"},{"comment":"The phrase \"a group of Schrödinger / convection equations\" is grammatically awkward; consider rephrasing to \"a family of uncoupled Schrödinger equations (after Fourier transform in p) and a family of uncoupled convection equations (after Fourier transform in x)\".","section":"Fig. 1 caption"},{"comment":"The sentence \"As shown in Fig. 5 (Li et al., 2021) which illustrates the spectral decay of truncation modes\" appears to cite the reference for its own figure; if Fig. 5 is the paper's own figure, the citation should be removed or moved to the text.","section":"Sec. 4.2 and Appendix B, paragraph on spectral decay"},{"comment":"Several equations have unmatched parentheses (e.g., \"Grad(bV^(SKNO)_{L-1})_{ij} =\" in Eq. (47) is missing a closing bracket). Please carefully proofread the display math.","section":"Appendix D, Theorem D.4, Eqs. (46)-(53)"},{"comment":"The reference to \"ANO (Lanthaler et al., 2025)\" seems inconsistent with the content of the cited paper (which is about universality of nonlinear operators); please verify that the reference supports the averaging-operator claim.","section":"Appendix B, \"Averaging Neural Operator on Dp\""}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious architectural contribution but the current evidence does not yet support the central mechanistic claim. The Table 1 confound is the most important issue; the authors should be asked to provide the proposed ablations before the claims about the d+1 evolution are accepted. The missing proof of H_FNO ⊂ H_SKNO is also a genuine gap in a stated theorem. I would not reject the paper on these grounds, since the full-model results and ablations suggest the architecture is valuable, but the revision must address the confound and the statistical verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on arXiv:2505.11766. The genuinely new thing is the d+1 dimensional reformulation of neural operators: instead of evolving embeddings with matrix multiplications, the authors add an auxiliary p dimension, lift the input to v(x,p), evolve with Fourier kernels in x and p jointly, and recover by integrating over p. This is not a routine FNO variant. The Schrödingerisation inspiration is real, and the paper does a service by showing that a single architecture, trained with standard losses, can beat FNO and Transolver across a dozen benchmarks, with quite consistent compute advantages. The ablation on Darcy, the super-resolution tests, and the zero-shot temporal experiments are all to the paper's credit. The authors also explore lifting and recovering choices, which is useful.\n\nHowever, the soft spots are substantial. First, the cleanest experiment meant to isolate the p-evolution advantage, Table 1 on heat and advection, is confounded. With L=1, SKNO still includes the untruncated local differential operator on x, while FNO and Transolver in that comparison are truncated in x. The local operator alone can supply the high-frequency content needed for those x-convolutions, so the 340x gap does not establish that the Fourier evolution along p does the work. The stress-test note is correct on the paper as written. This does not kill the full-model results, but it removes the best controlled evidence for the central mechanism.\n\nSecond, no code or data are shipped, and every reported number appears to be a single run without error bars. For a claim of consistent SOTA across more than ten benchmarks, seed variability matters. This is correctable, but as is, I cannot independently verify the headline numbers.\n\nThird, the theoretical appendix is not ready. Theorem D.3 asserts H_FNO ⊂ H_SKNO without proof, and that inclusion is not obviously true, since the SKNO parameterization with the differential operator is not a superset of all FNO realizations at fixed width. The other appendix theorems are informal or purely computational. The placeholder references and incomplete visualization captions should have been caught before submission.\n\nOn the positive side, the Schrödingerisation motivation is not used to fit constants—it is a design inspiration, and the paper does not overclaim it. The circularity burden is low. The empirical breadth is real, and the architecture is worth understanding.\n\nMy recommendation: this deserves peer review, not desk rejection. The idea is novel and the experimental scope is large enough that a serious referee should sort out whether the advantage holds under seeds and whether the mechanism claim can be repaired. I would bring it to our reading group and would likely cite it as related work, but I would not rely on its numbers until the code and error bars appear.","headline":"A novel d+1 operator architecture with a broad empirical sweep, but the controlled evidence for its central mechanism is confounded and the results lack error bars and code.","tokens_in":28827,"tokens_out":3130,"would_cite":true,"duration_ms":29070,"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":"Neural operators that evolve embeddings along an auxiliary Fourier dimension achieve lower PDE errors with training cost comparable to FNO.","keywords":["neural operators","operator learning","Schr\\\"odingerisation","Fourier neural operator","embedding evolution","auxiliary dimension","kernel integral operator","PDE solving"],"falsifier":"Train SKNO and FNO with matched compute on a benchmark where the auxiliary-dimension coupling is known to be densely nonlocal, e.g., a manufactured operator whose kernel couples all $p$ values with comparable strength rather than acting as a local differential operator. If SKNO's advantage over FNO disappears or reverses, the linear-PDE form in Eq. (13) is the operative limitation.","tokens_in":27765,"feed_emoji":"🌀","tokens_out":9136,"duration_ms":83500,"temperature":0.7,"pith_summary":"Most neural operators evolve a lifted embedding by matrix multiplications, so improving accuracy tends to mean lengthening the embedding at quadratic cost. This paper argues that embedding evolution itself should be treated as an operator over an extra auxiliary dimension p, turning the whole pipeline into a (d+1)-dimensional evolution. The proposed SKNO applies Fourier kernels jointly in the physical and auxiliary domains, implementing the p-evolution as a learnable linear PDE of the form $F_p^{-1}\\tilde{A}F_p + b$. Across more than ten benchmarks, from 1D heat to 3D Rayleigh-Taylor instability, SKNO reports the lowest relative $L_2$ error among the evaluated baselines, with training time comparable to FNO. The paper presents this as an alternative to brute-force embedding scaling.","feed_headline":"Reformulating neural operators in d+1 dimensions cuts PDE errors","feed_subtitle":"SKNO models embedding evolution as a linear PDE on an auxiliary axis, beating FNO and Transolver on 10+ benchmarks.","key_machinery":"The load-bearing mechanism is the auxiliary dimension $p$ and the warped-phase lift $v(x,p)=w^T(p)a(x)$, which turns a $d$-dimensional function evolution into a $(d+1)$-dimensional phase-space evolution. Along $p$, the model learns an operator of the form $F_p^{-1}\\tilde{A}F_p+b$---a Fourier-diagonal differential operator plus bias that discretizes a general linear PDE along $p$ with an Euler step---so that embedding evolution is learned as an operator rather than as a dense matrix. Combined with truncated spectral convolutions and a local differential operator on $x$, this operator-based evolution is what the paper argues gives SKNO its accuracy and resolution stability.","core_discovery":"The paper proposes to bring embedding evolution into the operator formalism by redefining neural operators in $d+1$ dimensions: lift the input $a(x)$ to $v(x,p)=w(p)^T a(x)$, evolve $v$ with kernel integral operators over $D_x \\times D_p$, and recover the output by $u(x)=\\int_{D_p}\\chi(p)v(x,p)\\,dp$. Along the auxiliary dimension, the evolution is parameterized as a learnable Fourier-domain linear PDE, $F_p^{-1}\\tilde{A}F_p+b$, discretized with an Euler step. The resulting Schr\\\"odingerised Kernel Neural Operator (SKNO) is claimed to achieve the lowest relative $L_2$ error among the evaluated baselines on a spectrum of benchmark PDEs, while remaining stable under mixed-resolution training, super-resolution inference, and zero-shot generalization to unseen time horizons.","pith_inferences":["Because the $p$-axis is learned as a linear operator, a natural testable extension is to run SKNO with adaptively chosen grid sizes along $p$ at inference: if the learned operator is genuinely resolution-invariant along $p$, the same weights should give accurate predictions on denser $p$-grids than those used in training, without retraining.","The sparsity and energy-capture results suggest the auxiliary dimension behaves like a learned dictionary; one could test whether SKNO's $p$-space can be pruned, e.g., with a sparse $\\tilde{A}$, to shrink model size while preserving accuracy---an extension not explored in the paper.","If the linear-PDE assumption holds for a given problem family, the spectrum of the learned operator $\\tilde{A}$ may be interpretable as an effective equation discovered for the embedding; comparing these spectra across benchmark families could reveal common structural features across PDEs."],"forward_implications":["SKNO offers a concrete alternative to embedding widening: instead of increasing embedding width to raise capacity, the model adds an auxiliary dimension governed by a learnable linear PDE, which the paper argues is cheaper while improving accuracy.","The $d+1$ formulation makes the auxiliary dimension a first-class part of the operator, aligning the architecture with Schr\\\"odingerisation; the paper credits this alignment for SKNO's ability to capture sharp features such as material interfaces and vorticity.","Resolution-invariance experiments show SKNO degrades less than FNO when tested on grids far beyond the training resolution, directly supporting applications like super-resolution forecasting from a single trained model.","Ablations attribute a substantial part of the gains to the Fourier-domain operator and bias along $p$; removing either raises error, indicating the linear-PDE inductive bias is load-bearing for the reported performance.","Zero-shot temporal generalization on 2D Navier-Stokes means the operator learned on seen time steps remains accurate on unseen future steps, improving autoregressive rollout usability."],"supporting_citations":[{"why":"Supplies the Schr\\\"odingerisation transformation that motivates the $d+1$ dimensional lift of the embedding.","marker":"(Jin et al., 2024)"},{"why":"Provides the Fourier spectral convolution operator and the FNO baseline architecture that SKNO extends and is compared against.","marker":"(Li et al., 2021)"},{"why":"Provides the Transolver physics-attention baseline and the MLP-Linear lifting/recovering design used in comparisons.","marker":"(Wu et al., 2024a)"},{"why":"Gives the universal approximation and error-bound results leveraged to justify stacking linear operators and to prove Theorem D.1.","marker":"(Kovachki et al., 2021)"},{"why":"Supplies the localized integral/differential kernel implementation used for SKNO's local propagator.","marker":"(Liu-Schiaffini et al., 2024)"},{"why":"Supplies PDEBench datasets (advection, shallow-water, 3D compressible Navier-Stokes) used in the benchmark suite.","marker":"(Takamoto et al., 2022)"},{"why":"Provides the 2D Gray-Scott reaction-diffusion data generation used as a benchmark.","marker":"(Kassa\\u00ef Koupa\\u00ef et al., 2024)"},{"why":"Supplies the 3D Rayleigh-Taylor instability dataset used as the most challenging benchmark.","marker":"(Ohana et al., 2024)"}],"fun_headline_variants":["d+1 neural operators: embedding evolution beats 10+ PDE benchmarks","Adding a dimension to neural operators improves PDE accuracy","SKNO: embedding evolution in extra dimension wins on PDEs","Neural operators with auxiliary dimension outperform on PDEs","From d to d+1: neural operators model evolution, cut errors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that the evolution of the lifted signal along the auxiliary dimension is governed by a linear PDE of the form Eq. (13), and that a single learnable Fourier-domain operator plus bias with a fixed Euler step captures it faithfully; if a problem's embedding dynamics resist that representation, the theoretical motivation and the claimed advantage of SKNO would not follow.","fun_headline_variants_meta":{"raw":{"variants":["d+1 neural operators: embedding evolution beats 10+ PDE benchmarks","Adding a dimension to neural operators improves PDE accuracy","SKNO: embedding evolution in extra dimension wins on PDEs","Neural operators with auxiliary dimension outperform on PDEs","From d to d+1: neural operators model evolution, cut errors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000484,"raw_usage":{"total_tokens":2396,"prompt_tokens":957,"completion_tokens":1439,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":1354}},"tokens_in":573,"tokens_out":1439,"duration_ms":12915,"temperature":1.0,"reasoning_tokens":1354,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:49:22.499449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train SKNO and FNO with matched compute on a benchmark where the auxiliary-dimension coupling is known to be densely nonlocal, e.g., a manufactured operator whose kernel couples all $p$ values with comparable strength rather than acting as a local differential operator. If SKNO's advantage over FNO disappears or reverses, the linear-PDE form in Eq. (13) is the operative limitation.","supporting_citations":[{"cited_title":"01\") denotes the output after theL0 and before theσ0 betweenv0 andv1, whereas a single-digit index (e.g.,","cited_arxiv_id":null,"evidence_quote":"Gives the universal approximation and error-bound results leveraged to justify stacking linear operators and to prove Theorem D.1."}],"review_version":1}