{"id":"35096fb4-b8a5-4a13-8639-9029725f6da2","arxiv_id":"2505.08631","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Operator learning surrogates map cardiac stimulus locations to activation and repolarization times on synthetic 2D and 3D heart meshes, with relative errors from 5e-4 to 7e-2 and large speedups over Monodomain simulation.","lead":"This paper trains two machine learning models, one based on Fourier transforms and one on kernel regression, to map the location of an electrical stimulus in heart tissue to activation and repolarization times. On synthetic 2D and 3D heart-like domains, the trained surrogates reproduce simulation outputs in milliseconds, with the kernel method more accurate and the Fourier method faster at prediction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"KOL's IQ kernel depends on stimuli only through their centroid, so reported accuracy may not generalize to shape-varying stimuli; the operator-learning claim is overbroad.","rationale":"The paper's internally reported errors are internally consistent, and the KOL-vs-FNO comparison is honestly reported for the tested datasets. The weakest point is not the Monodomain discretization or the FNO implementation; it is the gap between 'held-out samples from the same random-stimulus generator' and 'accurate approximations of the target map' asserted in Section 4. In KOL this gap is concrete and demonstrable from the equations: the IQ and RBF kernels map each input to a scalar centroid, so the 'operator' KOL learns is effectively a function of a low-dimensional summary statistic. The reader's concern about clinical representativeness is correct but can be sharpened: even inside the synthetic setup, stimulus shape is a free dimension that the kernel intentionally discards. If the random pulses in Section 3.1 are compact clusters of similar size, centroid may be almost sufficient, and the reported 1e-3 to 1e-2 errors are real for that class. But that is not the general operator claimed. A same-centroid, different-shape test is cheap and decisive. I also flag the 'below 1%' sentence because it is contradicted by the paper's own Table 2 and Table 3; this supports CONDITIONAL rather than ACCEPT, but does not warrant REJECT since the data-generation and evaluation protocol is explicit enough to be checked. Agreement with the reader is partial: the reader stopped at 'real clinical data differ,' while the stronger, more actionable defect is that KOL's kernel cannot distinguish stimuli that the paper's own framework treats as distinct inputs.","tokens_in":26346,"tokens_out":7310,"duration_ms":78888,"concrete_test":"On the Section 3.4 ventricle mesh, generate 100 test stimuli matched pairwise by centroid to training stimuli but with different shapes (compact ball, thin line, two separated clusters, thin annulus) and evaluate the trained KOL (iq4) and FNO models against a fresh Monodomain solve. Because KOL's IQ kernel assigns identical weights to all stimuli with the same centroid, any variation in true activation/repolarization times across shape-matched stimuli will appear directly as KOL error. Compare relative L2 errors (Eq. 17) to FNO and to a shape-aware KOL variant (e.g., adding second moments or radius to the kernel). If KOL error jumps above FNO or above 5%, the claimed accuracy and KOL superiority are artifacts of the training stimulus distribution; if it stays near 1%, the centroid summary is sufficient and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the trained maps approximate the true stimulus-to-time operator on new samples, not merely on the randomly drawn pulse class. Section 2.4 defines the IQ kernel (the only KOL kernel used in the 3D experiments) as S(A1,A2)=1/(sigma1*||c_A1-c_A2||+sigma2), where c_A is the centroid of the non-zero entries of the discrete input. Consequently, in Eq. (14) KOL's prediction is invariant to every geometric property of the stimulus except its centroid: two activation regions with the same centroid but different shape, size, or connectivity produce identical activation and repolarization maps. The reported held-out errors therefore certify accuracy only for the specific random-stimulus family generated in Section 3.1; they do not support the Section 4 claim of accurate clinical integration, where pacing protocols, scar, and tissue heterogeneity produce shape-varying stimuli. The RBF kernel has the same centroid-only defect. This is a boundary-of-validity problem, not an inconsistency in the 2D numerics, and it is compounded by a factual overstatement: the conclusion's 'errors generally below 1%' conflicts with Table 2 (FNO activation 3.27e-2 in 3D slab) and Table 3 (KOL activation 1.22e-2, FNO 7.15e-2 on the ventricle).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents an operator-learning study of the map from an applied current stimulus to cardiac activation and repolarization times, comparing Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL) on synthetic 2D domains, a 3D slab, and an unstructured left-ventricle mesh. Training data are Monodomain simulations with random 1 ms stimuli, and the outputs are activation and repolarization time fields. The authors report that KOL generally achieves lower test errors and shorter training times than FNO, while FNO offers faster inference, and they claim both methods are robust and accurate, with errors generally below 1%, and promising for clinical integration. The central goal is a surrogate for the stimulus-to-time operator, including the repolarization map for which no Eikonal-style PDE is known.","tokens_in":1836,"tokens_out":1896,"duration_ms":44404,"significance":"If the claimed accuracy held across the intended input class, the paper would provide a practically useful surrogate for cardiac activation and repolarization times at a fraction of the Monodomain cost, and the repolarization result would go beyond what Eikonal-type models offer. The study has several strengths: a systematic 2D sensitivity analysis over KOL kernel families and FNO learning-rate policies, a nontrivial 3D unstructured-mesh experiment with about 35k degrees of freedom, and a transparent comparison of training time, memory, and inference time. The KOL formulation is deterministic and exact within its kernel-regression framework. However, the paper's headline claims overstate what the reported numbers and the chosen kernels support, as detailed in the major comments.","major_comments":[{"comment":"The statement 'errors generally below 1%' is contradicted by the paper's own tables. In Table 2 (3D slab), FNO activation error is 3.27e-2 for acti(2000) and KOL activation error is 1.82e-2 for acti(2000). In Table 3 (ventricle), KOL activation error is 1.22e-2 and FNO activation error is 7.15e-2, while FNO repolarization error is 4.86e-2. Additionally, Figure 13 shows KOL activation test medians around 0.027 with 1.5% of test samples above 10% relative L2 error. The claim needs to be restricted to specific datasets and to a statement like 'errors range from below 1% in the 2D and some 3D repolarization cases to several percent in activation cases.'","section":"Section 4 (Conclusions) and Abstract"},{"comment":"The IQ kernel used for KOL is defined as S(A1,A2)=1/(sigma1*||c_A1-c_A2||+sigma2), where c_A is the centroid of the non-zero entries of the discrete input. Consequently, Eq. (14) makes KOL's prediction invariant to every property of the stimulus except its centroid. Two stimuli with the same centroid but different shape, size, or connectivity produce identical activation and repolarization maps. The RBF kernel has the same centroid-only defect. The held-out test sets in Section 3.1 are random stimuli from the same generative family, so the reported L2 errors certify accuracy only for that specific family, not for the broader class of pacing protocols, scarred tissue, or shape-varying stimuli invoked in the clinical-integration claim in Section 4. The authors should explicitly state this boundary of validity, or use a kernel that encodes shape information, before claiming clinical relevance.","section":"Section 2.4 (KOL kernels), especially the IQ kernel definition"},{"comment":"The abstract's statement that both FNO and KOL are 'robust to hyperparameter choices' is not supported by the sensitivity analysis. In Tables A2 and A3, KOL with NTK or RBF kernels gives errors around 1e-1, three orders of magnitude worse than the IQ kernels, while the IQ kernels themselves vary by up to a factor of about 1.5 (for example, iq1 vs iq5 for repo(2000) in Table A3). In Table A4, FNO test errors on the ventricle range from 7.15e-2 to 1.44e-1 as width changes from 32 to 2, and depth changes from L=1 to L=3 produce differences of 7.15e-2 versus 8.81e-2. These are meaningful sensitivities, not robustness.","section":"Section 3.2 and Tables A2, A3, A4"},{"comment":"The comparison of KOL and FNO would be considerably stronger if the test-error distributions were discussed in the conclusion, since the mean relative L2 errors hide substantial outliers. For the 3D slab acti(2000) dataset, Figure 13 reports 1.5% of KOL test data above 10% error, while Figure 12 reports 3.5% of FNO test data above 4%. For the ventricle activation data, Table 3 reports KOL at 1.22e-2 and FNO at 7.15e-2; those are point estimates with no uncertainty interval for KOL. The paper should either report median errors and outlier proportions in the main text or temper the 'generally below 1%' claim, since a reader cannot assess the reliability of the surrogate from the mean alone.","section":"Sections 3.3 and 3.4, Tables 2 and 3"}],"minor_comments":[{"comment":"The phrase 'we compare the performances of FNO e KOL' contains a typo: 'e' should be 'and'.","section":"Section 1, paragraph 2"},{"comment":"The quantity Np in the definition of the discrete L2 relative error is not defined. It should be stated whether Np is the number of test samples or the number of spatial points, because the current notation is ambiguous.","section":"Eq. (17), Section 3.1"},{"comment":"The caption reads '3D unstructed ventricle'; this should be 'unstructured'.","section":"Figure 4 caption"},{"comment":"The Ten Tusscher model is spelled twice as 'Ten Tuss-cher' in the text; the standard spelling is 'Ten Tusscher'.","section":"Section 3.1"},{"comment":"The inference-time comparison would be easier to interpret if the hardware and software versions (e.g., PyTorch version, CPU/GPU details) for both the training and inference runs were reported in a single table, since the paper currently mixes an Apple M1 Pro laptop for prediction with different machines for training and for the Monodomain reference.","section":"Section 3.4 and Tables 2 and 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid empirical benchmark of FNO versus KOL for a specific synthetic stimulus family, and the 3D unstructured-mesh experiment is valuable. My recommendation of major revision is driven by the disconnect between the paper's claims and its own numbers: the 'errors generally below 1%' and 'robust to hyperparameters' statements are not supported, and the centroid-only kernels used for KOL impose a major restriction on the generality of the learned operator. I would encourage the editor to ask for an explicit scope statement and, if possible, code or data availability, because the synthetic dataset is the main evidence for the method's practical relevance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a solid, workmanlike benchmark paper: the first head-to-head of FNO and KOL for mapping an applied stimulus to activation and repolarization times, including a realistic 3D ventricle mesh. The experiments are carefully done, the errors are internally consistent, and the finding that KOL is far faster to train and often more accurate than FNO is well supported. Learning repolarization times, where no Eikonal-style model exists, is a genuinely useful datapoint for the community. The authors also test several kernels and clearly report memory and inference times.\n\nBut the paper's framing oversells itself in ways that matter. The abstract claims errors 'generally below 1%' – contradicted by their own Table 3 (KOL activation 1.22e-2, FNO 7.15e-2 on the ventricle) and Table 2 (FNO activation 3.27e-2). The robustness claim is also too strong: FNO's error goes from 1.44e-1 to 7.15e-2 as width increases, and NTK/RBF kernels for KOL give errors around 1e-1 while IQ gives 1e-3. So both architectures are sensitive to hyperparameters; only the IQ-fixed choices are stable.\n\nThe more serious issue, which the stress-test caught, is that the KOL kernels used for the 3D runs (IQ and RBF) depend on the stimulus only through its centroid. That means KOL's predictions are invariant to shape, size, or connectivity of the stimulation region. The reported test errors therefore only certify accuracy for the family of random stimuli in the training distribution, not the operator mapping from arbitrary stimuli to activation/repolarization times. The paper claims clinical integration potential, but clinical pacing protocols, scars, and heterogeneities produce shape-varying stimuli. The authors never acknowledge this limitation. For the FNO, the NUDFT approach does not have this defect, but its errors on the ventricle are markedly worse.\n\nThere is also no code or data release, which makes replication harder. The math and citation pattern look sound; the issue is the mismatch between claims and evidence.\n\nBottom line: as a benchmark for a specific, well-defined stimulus family, the paper deserves a serious referee and could be accepted after major revision. The authors need to either (a) restrict their claims to the tested stimulus class, (b) use KOL kernels that actually depend on the full input geometry, or (c) add experiments with shape-varying stimuli to show the centroid-only kernel still works. I'd bring it to a reading group, and I'd cite it, but only with the boundary-of-validity caveat.\n\nRegards,","headline":"Useful benchmark of FNO vs KOL for cardiac activation/repolarization surrogates, but the abstract overstates accuracy and robustness and the KOL kernel's centroid-only dependence limits the claim.","tokens_in":27147,"tokens_out":2295,"would_cite":true,"duration_ms":21727,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65M60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that Fourier Neural Operators and Kernel Operator Learning can learn the stimulus-to-activation and stimulus-to-repolarization time maps directly from Monodomain simulation data, making fast cardiac timing surrogates…","keywords":["operator learning","cardiac electrophysiology","activation time","repolarization time","Fourier Neural Operator","Kernel Operator Learning","Monodomain model","Eikonal model"],"falsifier":"Run the trained KOL model on two held-out stimuli that share the same centroid but have very different shapes, for example a point cluster versus a thin stripe. Because the IQ kernel used for KOL depends only on the centroid of the stimulus support, the model will return nearly identical activation and repolarization maps for the two inputs, while a fresh Monodomain solve will show different timing distributions; that discrepancy would falsify the claim that the learned operator is the full stimulus-to-timing map rather than a centroid-only map.","tokens_in":26140,"feed_emoji":"🫀","tokens_out":11792,"duration_ms":104529,"temperature":0.7,"pith_summary":"This paper aims to establish that operator learning can replace the expensive PDE step in cardiac electrophysiology for two clinically relevant output maps: the activation time and the repolarization time of every point in the tissue, given the region where a brief current stimulus was applied. The trained surrogate would matter because today the standard route is to solve the Monodomain reaction-diffusion model for each new stimulus, costing minutes on GPU-accelerated solvers, while the learned map evaluates in milliseconds. The paper also targets a map that has no known reduced PDE formulation: repolarization time has no Eikonal-style counterpart, so learning is not just accelerating an existing fast model but approximating a map that previously had no cheap surrogate at all. Using Monodomain-generated training data on 2D, 3D, and unstructured left-ventricle geometries, the paper reports that both Fourier Neural Operators and Kernel Operator Learning achieve useful held-out accuracy, with Kernel Operator Learning the more accurate and cheaper to train of the two.","feed_headline":"Operator surrogates map stimulation to cardiac timing in milliseconds","feed_subtitle":"Kernel Operator Learning beats Fourier Neural Operators in accuracy and training time on cardiac tissue maps.","key_machinery":"The machinery is a pair of learned surrogate maps $\\widehat{\\mathcal{G}}$ approximating $\\mathcal{G}: I_{\\text{app}} \\mapsto (t_{\\text{act}}, t_{\\text{rep}})$. For Fourier Neural Operators, the key identity is the convolution theorem used in equations (11)-(13): each integral kernel layer is applied in Fourier space as $\\mathcal{F}^{-1}(R_\\theta \\cdot \\mathcal{F}(a_t))$, with the weight tensor $R_\\theta$ parameterizing the kernel. For Kernel Operator Learning, the key object is the closed-form kernel-regression predictor of equation (14), $\\bar{\\mathcal{G}}(a)(x)=K(x,X)K(X,X)^{-1}\\sum_j S(\\phi(a),A_j)\\alpha_j$, where $S$ is an inverse-quadratic kernel built from the centroid of the stimulated region; training consists of solving one symmetric positive-definite linear system by Cholesky factorization rather than iterative optimization. These surrogates do the work of replacing repeated full-order Monodomain solves with a single function evaluation.","core_discovery":"The central claim is that the operator $\\mathcal{G}$ sending a 1 ms applied-current stimulus to the spatial distribution of activation time, and the analogous operator for repolarization time, can be learned from Monodomain simulation data by both Fourier Neural Operators and Kernel Operator Learning, and that the learned surrogates are accurate and vastly cheaper to evaluate than the full PDE model. The paper studies this on a 2D square, a 3D slab, and a realistic unstructured left-ventricle mesh of about 35k degrees of freedom, using Rogers-McCulloch or Ten Tusscher ionic models to produce ground truth. It reports that Kernel Operator Learning outperforms Fourier Neural Operators in accuracy and training time on all test cases, while Fourier Neural Operators remain competitive and have faster per-sample inference; both remove the need to re-solve the Monodomain equations for each new stimulus. The paper highlights that the repolarization-time map has no known Eikonal-style reduced PDE, so the learned surrogate approximates a map for which no cheap classical model exists.","pith_inferences":["Because Kernel Operator Learning's IQ kernels collapse each stimulus to its centroid, the method is unlikely to transfer to multi-site pacing, extended scar-adjacent wavefronts, or stimuli whose shape matters; a natural test is adding same-centroid, different-shape stimuli to the training distribution and measuring the drop in accuracy.","The reported cost comparison compares a single surrogate inference with one full Monodomain solve; including training-data generation and kernel selection, the surrogate pays off only when many evaluations are needed, so the efficiency claim is an amortized one.","Since repolarization time has no Eikonal-style reduced model, the learned operator could be used in reverse, for example inferring the stimulus region from observed repolarization maps, an inverse problem the paper does not address.","The same operator-learning pipeline should extend to tissue-conductivity or fiber-orientation inputs, not just stimulus location; if it does, it would give clinicians a single surrogate spanning multiple patient-specific parameters."],"forward_implications":["A clinician or researcher can obtain activation and repolarization time maps for a new stimulus location in tens of milliseconds on a laptop rather than minutes of GPU-accelerated Monodomain simulation.","Repolarization time, which has no known Eikonal-style reduced PDE, becomes accessible through a trained surrogate built from Monodomain data alone.","Kernel Operator Learning with a physics-informed inverse-quadratic kernel is the more accurate and faster-to-train of the two methods on these tasks, including on an unstructured 35k-node ventricle mesh.","Fourier Neural Operators, with a non-uniform Fourier transform modification, can be applied to unstructured cardiac meshes, giving an alternative when many fast predictions are needed.","Both methods are robust to hyperparameter choices over the tested ranges, with Kernel Operator Learning deterministic and Fourier Neural Operators showing small variation across random initializations."],"supporting_citations":[{"why":"Defines the Fourier Neural Operator architecture whose Fourier-parameterized kernel convolutions are the FNO surrogate used here.","marker":"[23]"},{"why":"Provides the Neural Operator framework of lifting, kernel-integral layers, and projection on which the FNO implementation rests.","marker":"[22]"},{"why":"Establishes Kernel Operator Learning as a competitive operator-learning method and supplies the kernel-regression formulation.","marker":"[4]"},{"why":"Provides the closed-form KOL formulation with RKHS kernels that the paper adapts to cardiac activation and repolarization maps.","marker":"[44]"},{"why":"Supplies the Rogers-McCulloch ionic model used to generate the 2D Monodomain training labels.","marker":"[31]"},{"why":"Supplies the Ten Tusscher ionic model used to generate the 3D Monodomain training labels.","marker":"[36]"},{"why":"Provides the Monodomain and Eikonal model background, parameter choices, and the activation-time framework the surrogates are compared with.","marker":"[9]"},{"why":"Supplies the non-uniform Fourier transform used to adapt FNO to the unstructured ventricle mesh.","marker":"[24]"},{"why":"Provides the full-order PDE solver used as the baseline for the reported computational-cost comparison.","marker":"[3]"}],"fun_headline_variants":["Operator learning predicts cardiac timing without solving PDEs","Kernel operators beat Fourier nets for cardiac timing maps","Learn activation and repolarization maps via operator surrogates","Surrogate operators accelerate cardiac simulations to milliseconds","No PDE model for repolarization: operators learn it directly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Monodomain simulations of randomly placed 1 ms pulses are representative of the stimuli and tissue conditions the surrogate will actually face; if the real operating regime differs, the reported held-out errors no longer measure the surrogate's accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Operator learning predicts cardiac timing without solving PDEs","Kernel operators beat Fourier nets for cardiac timing maps","Learn activation and repolarization maps via operator surrogates","Surrogate operators accelerate cardiac simulations to milliseconds","No PDE model for repolarization: operators learn it directly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000911,"raw_usage":{"total_tokens":3923,"prompt_tokens":962,"completion_tokens":2961,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":2884}},"tokens_in":578,"tokens_out":2961,"duration_ms":21244,"temperature":1.0,"reasoning_tokens":2884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:49:30.206101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the trained KOL model on two held-out stimuli that share the same centroid but have very different shapes, for example a point cluster versus a thin stripe. Because the IQ kernel used for KOL depends only on the centroid of the stimulus support, the model will return nearly identical activation and repolarization maps for the two inputs, while a fresh Monodomain solve will show different timing distributions; that discrepancy would falsify the claim that the learned operator is the full stimulus-to-timing map rather than a centroid-only map.","supporting_citations":[{"cited_title":"Neural operator: Learning maps between function spaces with applications to PDEs.Journal of Machine Learning Research24, no","cited_arxiv_id":null,"evidence_quote":"Provides the Neural Operator framework of lifting, kernel-integral layers, and projection on which the FNO implementation rests."},{"cited_title":"Kernel methods are com- petitive for operator learning.Journal of Computational Physics496 (2024): 112549","cited_arxiv_id":null,"evidence_quote":"Establishes Kernel Operator Learning as a competitive operator-learning method and supplies the kernel-regression formulation."},{"cited_title":"Learning epidemic trajectories through Kernel Operator Learning: from modelling to optimal control.Numerical Mathematics: Theory, Methods and Applications(2025): 104208","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form KOL formulation with RKHS kernels that the paper adapts to cardiac activation and repolarization maps."},{"cited_title":"McCulloch","cited_arxiv_id":null,"evidence_quote":"Supplies the Rogers-McCulloch ionic model used to generate the 2D Monodomain training labels."},{"cited_title":"Panfilov","cited_arxiv_id":null,"evidence_quote":"Supplies the Ten Tusscher ionic model used to generate the 3D Monodomain training labels."},{"cited_title":"Pavarino, and Simone Scacchi","cited_arxiv_id":null,"evidence_quote":"Provides the Monodomain and Eikonal model background, parameter choices, and the activation-time framework the surrogates are compared with."},{"cited_title":"PETSc users manual","cited_arxiv_id":null,"evidence_quote":"Provides the full-order PDE solver used as the baseline for the reported computational-cost comparison."}],"review_version":1}