{"id":"c9ab6695-8dea-4a5b-bc89-414518b22daa","arxiv_id":"2608.09071","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A two-step probabilistic DeepONet learns uncertainty in a low-dimensional coefficient space and maps it to output fields, producing structured, non-diagonal predictive covariance with single-pass inference.","lead":"This paper presents a new neural operator method, two-step MV-DeepONet, that predicts whole output fields and their uncertainty as a package, including how errors at different points in the field move together. The approach gives a lightweight way to capture non-diagonal uncertainty structure in engineering problems like hypersonic reentry, where previous pointwise methods assumed errors at different locations were independent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The diagonal coefficient-covariance assumption in Eq. (27) is not implied by the rotation in Eq.","rationale":"The reader's weakest assumption is precisely that the conditional coefficient covariance Lambda_c,theta(u) is diagonal in the rotated space, and that the rotation in Eq. (22) only diagonalizes the unconditional empirical covariance. I agree that this is the most load-bearing modeling premise: the central architectural claim (non-diagonal output covariance through a shared basis) is mathematically sound, but the quality of the recovered covariance depends on whether the true conditional coefficient covariance is diagonal in the fixed rotated basis. The paper's error analysis acknowledges this only implicitly through eta_B_app, which is neither estimated nor experimentally separated. The numerical experiments report total covariance recovery, which can be dominated by the mean-covariance term, so they do not stress the diagonal assumption. This is an addressable concern: a synthetic or stochastic test with known input-dependent conditional coefficient correlations would reveal whether the restriction is harmful in practice. Since the reader already made this the basis of a CONDITIONAL verdict, my read does not change the verdict; the paper should either relax the assumption (e.g., predict a full low-rank coefficient covariance) or add evidence that the diagonal assumption is adequate for the target applications.","tokens_in":46341,"tokens_out":9835,"duration_ms":95459,"concrete_test":"Construct a synthetic operator with known input-dependent coefficient covariance, e.g., s(u) = Phi c(u) with c(u) ~ N(0, Sigma_c(u)) where Sigma_c(u) has off-diagonal entries that vary with u (for instance, Sigma_c(u) = D(u) + rho(u) 11^T for a subset of modes). Train two-step MV-DeepONet on samples from this model. For several fixed inputs u, compute the model's predicted conditional covariance from Eq. (30) and compare it with the true conditional covariance Phi Sigma_c(u) Phi^T mapped to the output space. If the off-diagonal correlation patterns deviate systematically (e.g., relative error in off-diagonal entries exceeds the total-covariance recovery error), the diagonal-coefficient assumption is the limiting factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that two-step MV-DeepONet induces a generally non-diagonal conditional predictive covariance in the physical output space via Eq. (30), Sigma_s(u) = eQ* Lambda_c,theta(u) (eQ*)^T. This construction is algebraically correct, but its fidelity depends on the modeling assumption in Eq. (27) that the coefficient vector c(u) has diagonal conditional covariance Lambda_c,theta(u) in the rotated basis eQ*. The rotation in Eq. (22) is obtained by diagonalizing the unconditional empirical coefficient covariance bB_Q over the whole training set: bB_Q = bV_Q bLambda_Q bV_Q^T. There is no reason that, for a fixed input u, the true conditional covariance Cov(c(u)|u) should also be diagonal in this same, input-independent coordinate system. The model's conditional predictive covariance is therefore restricted to the low-dimensional set {eQ* diag(sigma^2) eQ*^T : sigma^2 in R^p}. If the true conditional coefficient covariance has non-negligible off-diagonal entries, or if its eigenvectors vary with u, Eq. (30) cannot represent the actual cross-location conditional dependence. The error decomposition in Eq. (65) folds this into the approximation term eta_B_app, but the paper neither estimates eta_B_app nor provides an experiment that isolates conditional covariance recovery for fixed inputs. The numerical studies evaluate total predictive covariance (Eq. 37), where the mean-covariance term bB_mu can dominate, so they do not test the diagonal-conditional assumption. This is a genuine modeling restriction: the method replaces pointwise conditional independence with a fixed-basis diagonal-coefficient independence, which is more expressive but still a structural prior that may be violated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two-step MV-DeepONet, a probabilistic operator-learning model that combines the two-step DeepONet training strategy of Lee and Shin with Gaussian modeling in a low-dimensional rotated coefficient space. The key algebraic device is Eq. (30), where a diagonal input-dependent coefficient covariance is mapped through a learned orthonormal output basis to produce a generally non-diagonal conditional predictive covariance in the physical output space. The authors also provide a Frobenius-norm error decomposition for total predictive covariance recovery, and they validate the method on reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems, comparing against Prob-DeepONet in terms of generalization, uncertainty bands, covariance recovery, and calibrated prediction intervals.","tokens_in":1686,"tokens_out":3976,"duration_ms":128340,"significance":"If the central claim holds, the paper offers a lightweight alternative to full covariance parameterization in probabilistic operator learning: it obtains structured conditional covariance while retaining single-pass inference. The construction in Eqs. (22)-(30) is algebraically correct, and the paper is unusually explicit about the decomposition of covariance recovery error into truncation, subspace, statistical, and coefficient-regression terms via Eckart-Young and Davis-Kahan arguments. The empirical study is broad, including three PDE benchmarks and a CFD-based aerothermal problem with a validated solver. The paper also makes a fair comparison setup: covariance recovery is evaluated against held-out empirical covariances, and the branch network minimizes a Gaussian NLL rather than the Frobenius covariance error directly, so the central mechanism is not circular. The main weakness is that the claimed advantage over Prob-DeepONet concerns the conditional predictive covariance, while the numerical evaluation targets the total predictive covariance, for which no ground-truth conditional object exists in the deterministic benchmarks.","major_comments":[{"comment":"The diagonal form of the conditional coefficient covariance is an input assumption, not a consequence of the rotation in Eq. (22). The rotation diagonalizes the unconditional empirical coefficient covariance over the training set; for a fixed input u it does not diagonalize the conditional covariance of the coefficient residuals, and the eigenvectors of that conditional covariance may vary with u. The model class in Eq. (30) is therefore restricted to matrices of the form eQ* diag(sigma^2) eQ*^T. If the true conditional coefficient covariance has non-negligible off-diagonal entries, the predicted off-diagonal output correlations are misspecified. Since the misspecification is absorbed into the unquantified term eta_B_app in Eq. (65), the analysis does not control the error caused by the central modeling assumption. A concrete diagnostic would be to estimate the conditional covariance of the coefficient residuals in fixed input bins, or to test on a synthetic problem with known non-diagonal conditional coefficient covariance.","section":"Section 2.4.2, Eq. (27)"},{"comment":"The numerical evaluation compares the total predictive covariance bSigma_2step of Eq. (37) with the empirical total covariance of the test outputs, not the conditional covariance bSigma_s(u) of Eq. (30). Because the PDE and CFD maps in these benchmarks are deterministic, there is no ground-truth conditional covariance of s(u)|u against which to validate Eq. (30); the non-diagonality of the reference covariance arises from variation of the conditional means across inputs, i.e., the first term in the law of total covariance, Eq. (14). The reported Frobenius errors and correlation maps therefore cannot substantiate the paper's claim that the method recovers off-diagonal conditional dependence. I recommend adding either a benchmark with repeated output realizations for the same input (for example noisy observations or stochastic PDE outputs), or a held-out residual analysis that decomposes the total covariance error into the contributions of bB_mu and bB_sigma, together with a correlation-map comparison against Prob-DeepONet's total covariance.","section":"Section 3, Eq. (70) and Sections 3.1.2-3.4.2"},{"comment":"The presentation calls Eq. (69) an upper bound, but eta_B_app and eta_B_opt are defined as the deviations of the trained model from a best-in-class model and are never estimated or dominated by computable quantities. The recovered error hierarchy T1 << T2, T3 << T2 reported in Sections 3.1.2, 3.2.2, and 3.3.2 is inferred from the gap between the oracle curve and the final model curve; that gap is precisely the unquantified model-dependent error. The theory would be substantially stronger if the approximation term were bounded or estimated, or if the misspecification in Eq. (27) were isolated in a separate term that the experiments explicitly control.","section":"Section 2.4.4, Eqs. (65)-(69)"}],"minor_comments":[{"comment":"The prediction equations describe the centered output only; the paper should state explicitly that the reported mean predictions in Figures 4, 9, 14, and 21 are obtained by adding the training mean from Eq. (19) to eQ* times the predicted coefficient mean.","section":"Eqs. (32) and (41)"},{"comment":"The argument that the covariance is nonzero excludes only the case where all basis functions vanish simultaneously at one of the two locations; a complementary pattern in which some modes vanish at y_i and other modes vanish at y_j is still logically possible. A generic-density or non-vanishing-product assumption would make the argument complete.","section":"Section 2.4.3, Eq. (31)"},{"comment":"The paper does not report the network widths, depths, learning rates, optimizer settings, or the ensemble aggregation rule for the eight-member ensembles shown in the training-loss figures; because the numerical results appear to depend on these ensembles, the experiments are difficult to reproduce without this information.","section":"Section 3, ensemble usage"},{"comment":"The axis labels use a pound sign where a multiplication sign is intended; the labels should read 'Epochs (x 10^3)' or similar.","section":"Figures 2, 3, 7, 8, 12, 13, 18, 19"},{"comment":"The 'mean error' over out-of-distribution correlation lengths is an unweighted average over an irregular grid of ell values; the averaging convention should be stated.","section":"Section 3.1.1, Table 2"},{"comment":"The aerothermal example varies only the freestream Mach number, and Appendix B shows that the effective stochastic dimension is one; the paper should acknowledge that this case is not a test of high-dimensional input-field uncertainty propagation in the same sense as the PDE benchmarks.","section":"Section 3.4 and Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The core algebraic construction is sound and the manuscript is readable. My main concern is the mismatch between the advertised conditional-covariance contribution and the total-covariance evaluation; I would like to see a conditional-covariance diagnostic or a stochastic benchmark before publication. Please also verify that the 2026-dated references cited in the manuscript exist, since several appear very recent for a 2026 arXiv submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the paper does something real, and the central construction is not circular. The two-step basis learning from Lee and Shin is combined with Prob-DeepONet's Gaussian NLL formulation, but the probabilistic model is placed in the rotated coefficient space, and Eq. (30) genuinely produces a generally non-diagonal conditional predictive covariance in the physical output space with single-pass inference. That is a useful extension, not a repackaging of two existing methods. The error decomposition in Section 2.4.4 is standard, but competently assembled, and the numerical study is broad: reaction-diffusion, Burgers, Darcy, and a hypersonic blunt-body problem. The multi-anchor correlation maps are the right kind of evidence, and the CFD validation against the MacLean/DPLR data is a serious touch.\n\nThe main fragility is exactly the one the stress-test note identifies. The rotation in Eq. (22) diagonalizes the empirical unconditional coefficient covariance over the training set, not the conditional covariance for a fixed input. The model's conditional covariance is therefore restricted to matrices of the form eQ* diag(sigma_c^2) eQ*^T. That is a structural prior, not an identity. It is more expressive than Prob-DeepONet's diagonal conditional covariance, but it can still be misspecified if the true conditional dependence in the coefficient space is not diagonal in that fixed basis. The paper does not isolate this experimentally: the reported total covariance in Eq. (37) includes a mean-covariance term that can dominate, and the correlation maps are evaluated over held-out test outputs, so they do not directly test fixed-input conditional dependence. I would ask for an experiment that compares predicted conditional covariance to a Monte Carlo conditional estimate at fixed inputs, or at least a clear statement that this is the modeling assumption being made.\n\nA second, smaller soft spot: the O_P(sqrt(M/K)) Frobenius concentration step in Eq. (69) is invoked loosely. Standard spectral-norm concentration does not automatically give that Frobenius rate in high dimensions, and the paper even shows the estimate is loose in its own aerothermal example. This does not damage the main argument, but the statement should be softened or replaced with an explicit caveat.\n\nMissing code and data, plus no error bars on reported metrics, are addressable but real omissions for reproducibility.\n\nWho this is for: anyone working on lightweight UQ for neural operators, especially with field-valued outputs and random input fields. It deserves a serious referee, and with the conditional-covariance experiment and the concentration caveat addressed, it would be a solid contribution.","headline":"A genuinely useful extension of Prob-DeepONet: the paper moves Gaussian NLL training into a rotated low-dimensional coefficient space and thereby induces non-diagonal conditional covariance in the output field, but the fixed-basis diagonal-coefficient assumption is a structural prior that the experiments never directly test.","tokens_in":47217,"tokens_out":1970,"would_cite":true,"duration_ms":21698,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A two-step probabilistic DeepONet models uncertainty in a low-dimensional coefficient space and recovers cross-location output covariance in a single forward pass, with an error bound tied to four identifiable factors.","keywords":["uncertainty propagation","operator learning","DeepONet","probabilistic surrogate","covariance recovery","two-step training","random input fields"],"falsifier":"For a fixed test input $u$, draw many solver evaluations (or use a reference surrogate), center the outputs, project them onto the learned rotated basis $\\widetilde{Q}^*$, and estimate the conditional coefficient covariance $C = \\mathrm{Cov}(c \\mid u)$; if the off-diagonal entries of $C$ are not negligible compared with its diagonal entries, then Eq. (30) will miss output correlations and the claimed recovery fails even with a perfect trunk subspace.","tokens_in":46086,"feed_emoji":"📐","tokens_out":8453,"duration_ms":69076,"temperature":0.7,"pith_summary":"The paper aims to establish that a probabilistic operator surrogate can represent structured, cross-location predictive covariance without parameterizing a full high-dimensional covariance matrix. The proposed two-step MV-DeepONet performs Gaussian modeling in a low-dimensional, rotated coefficient space and maps the resulting diagonal coefficient covariance back through a shared orthonormal basis, producing a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. If the claim is right, uncertainty propagation for PDE-driven engineering systems gains a lightweight surrogate whose uncertainty bands and correlation patterns reflect the physical coupling of the output field rather than isolated pointwise variance. The paper also derives a Frobenius-norm error decomposition and tests the method on reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems.","feed_headline":"Coefficient-space Gaussian yields non-diagonal output covariance","feed_subtitle":"Modeling uncertainty in a latent coefficient space captures cross-location correlations in a single forward pass.","key_machinery":"The load-bearing object is the covariance propagation identity $\\widehat{\\Sigma}_s(u) = \\widetilde{Q}^* \\widehat{\\Lambda}_{c,\\theta}(u)(\\widetilde{Q}^*)^\\top$, which converts a diagonal Gaussian assumption on the rotated modal coefficients into structured output covariance through the shared orthonormal trunk basis $\\widetilde{Q}^* \\in \\mathbb{R}^{M \\times p}$ with $p \\ll M$. The machinery also includes the two-step training schedule (trunk basis learning first, branch coefficient regression second, with QR orthogonalization and empirical coefficient-covariance diagonalization), and the error analysis built on the Eckart-Young-Mirsky truncation, a spectral-gap bound relating trunk reconstruction risk to subspace distance, and a Davis-Kahan-type finite-sample bound for the empirical eigenspace. Together these components support the claim that covariance recovery is governed by low-rank compressibility, subspace quality, sample size, and branch covariance estimation.","core_discovery":"The central claim is that probabilistic modeling in a rotated low-dimensional coefficient space, followed by linear mapping through a shared trunk basis, induces a generally non-diagonal conditional predictive covariance in the physical output space: $\\widehat{\\Sigma}_s(u) = \\widetilde{Q}^* \\widehat{\\Lambda}_{c,\\theta}(u)(\\widetilde{Q}^*)^\\top$, where $\\widehat{\\Lambda}_{c,\\theta}(u)$ is diagonal. The shared basis is the orthonormal trunk basis after QR orthogonalization and an empirical-covariance eigendecomposition, so each modal coefficient fluctuation reaches multiple output locations and creates cross-location dependence without explicit covariance parameterization. This relaxes the pointwise conditional-independence assumption of Prob-DeepONet while keeping single-pass inference, and the paper argues that the total predictive covariance decomposes by the law of total covariance into a mean-induced part and this structured conditional part. The theoretical analysis further claims that the Frobenius-norm recovery error is bounded by four terms: low-rank truncation of the true output covariance, trunk subspace projection error, finite-sample statistical error, and branch coefficient-covariance estimation error.","pith_inferences":["The rotation in the paper diagonalizes the unconditional empirical coefficient covariance over training data; whether the conditional coefficient covariance for a fixed input is also diagonal is a modeling assumption the paper does not prove, and non-diagonal conditional coefficients would change the recovered output correlations.","A natural extension is to relax the diagonal Gaussian coefficient model to non-Gaussian or correlated latent distributions, since the covariance propagation identity would still map latent dependence into output covariance; the paper lists this direction as future work.","The framework suggests a general design pattern for operator surrogates: choose any low-dimensional latent distribution, propagate it through a learned linear basis, and obtain an output covariance with structure inherited from the basis, so the method's expressiveness is tied to the quality of the trunk subspace.","The rank-one aerothermal case illustrates that low-rank covariance is not always easier to recover: with a single dominant mode, Frobenius error is highly sensitive to bias in that mode's variance even when the correlation structure is accurate, so metrics should be read together."],"forward_implications":["A diagonal Gaussian in coefficient space becomes a generally non-diagonal conditional predictive covariance in output space via the shared basis, so cross-location dependence is represented without a full $\\mathbb{R}^{M \\times M}$ covariance matrix.","Single-pass inference is retained: the branch network predicts $p$ coefficient means and variances, and the output covariance is obtained by the low-rank product $\\widetilde{Q}^* \\widehat{\\Lambda}_{c,\\theta}(u)(\\widetilde{Q}^*)^\\top$.","The Frobenius error bound implies that fast spectral decay of the true output covariance, a well-learned trunk subspace separated by a spectral gap, and accurate branch covariance regression are the practical conditions for reliable covariance recovery.","On the tested reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems, the method reports lower out-of-distribution mean errors, more structured uncertainty bands, and accurate multi-anchor correlation maps relative to Prob-DeepONet.","Conformal calibration brings interval coverage to nominal levels while the two-step model maintains noticeably narrower intervals than the baseline in the reported experiments."],"supporting_citations":[{"why":"Supplies the two-step training strategy that decouples trunk basis learning from branch coefficient regression and provides the orthogonalized basis used for the covariance construction.","marker":"[45]"},{"why":"Defines Prob-DeepONet, the pointwise Gaussian baseline whose diagonal conditional covariance the paper relaxes.","marker":"[43]"},{"why":"Introduces the DeepONet architecture and the universal approximation setting that the paper extends probabilistically.","marker":"[27]"},{"why":"Provides the Eckart-Young-Mirsky theorem used to identify the low-rank truncation term in the covariance error bound.","marker":"[52]"},{"why":"Supplies the Davis-Kahan-type spectral bound used to control the finite-sample statistical error in the trunk subspace.","marker":"[53]"},{"why":"Gives the covariance concentration rate that yields the finite-sample term in the combined error bound.","marker":"[54]"}],"fun_headline_variants":["Latent coefficient space captures cross-location uncertainty","Non-diagonal output covariance from a diagonal latent model","Single-pass probabilistic operator learning with structured covariance","MV-DeepONet: two-step training for correlated uncertainty"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, after the rotation, the modal coefficients are conditionally independent given the input, so the coefficient covariance $\\widehat{\\Lambda}_{c,\\theta}(u)$ can be taken as diagonal; the rotation diagonalizes only the empirical covariance averaged over training samples, not the conditional covariance at each fixed input.","fun_headline_variants_meta":{"raw":{"variants":["Latent coefficient space captures cross-location uncertainty","Non-diagonal output covariance from a diagonal latent model","Single-pass probabilistic operator learning with structured covariance","MV-DeepONet: two-step training for correlated uncertainty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1641,"prompt_tokens":1051,"completion_tokens":590,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":667,"tokens_out":590,"duration_ms":5926,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:11:57.962413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed test input $u$, draw many solver evaluations (or use a reference surrogate), center the outputs, project them onto the learned rotated basis $\\widetilde{Q}^*$, and estimate the conditional coefficient covariance $C = \\mathrm{Cov}(c \\mid u)$; if the off-diagonal entries of $C$ are not negligible compared with its diagonal entries, then Eq. (30) will miss output correlations and the claimed recovery fails even with a perfect trunk subspace.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-step training strategy that decouples trunk basis learning from branch coefficient regression and provides the orthogonalized basis used for the covariance construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Prob-DeepONet, the pointwise Gaussian baseline whose diagonal conditional covariance the paper relaxes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the DeepONet architecture and the universal approximation setting that the paper extends probabilistically."},{"cited_title":"Eckart, G","cited_arxiv_id":null,"evidence_quote":"Provides the Eckart-Young-Mirsky theorem used to identify the low-rank truncation term in the covariance error bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Davis-Kahan-type spectral bound used to control the finite-sample statistical error in the trunk subspace."},{"cited_title":"Koltchinskii, K","cited_arxiv_id":null,"evidence_quote":"Gives the covariance concentration rate that yields the finite-sample term in the combined error bound."}],"review_version":1}