{"id":"bc851433-c243-46ef-9ff2-bce3e1f2dd82","arxiv_id":"2501.10684","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A DeepONet-PINN hybrid with latent perturbation is proposed for PDE parameter estimation, but the practical loss is not the derived variational objective.","lead":"This paper proposes DeepBayONet, a network that combines DeepONet, physics-informed losses, and a random latent perturbation to estimate unknown PDE parameters along with uncertainty. It is tested on regression, heat, reaction-diffusion, and eigenvalue problems, but the implemented loss drops the derived Bayesian term, so the advertised Bayesian posterior is not actually trained.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The training objective is not the derived ELBO: the KL divergence is replaced by a heuristic log-variance penalty (Eq. 18e), so the reported posteriors are not Bayesian posteriors.","rationale":"The reader's identified weakest assumption is exactly the load-bearing flaw: Eq. (14)-(16) and Eq. (18e) do not implement the variational inference objective derived in Eq. (12). The paper itself admits in footnote 1 that the KL term was replaced by a heuristic. Since the central novelty is 'Bayesian training implemented through variational inference,' this mismatch undermines the main claim. The 3D eigenvalue contradiction (target 20.19 vs reported 4.493) is an additional red flag but secondary; it may be a unit or notation error, whereas the loss-function mismatch is structural. I read the paper in good faith: the exposition is clear, the code is publicly available, and the regression experiment reports means over ten runs, which is commendable. However, the theoretical foundation for the posterior interpretation is missing, and the single-run PDE experiments cannot compensate for that. The reader's REJECT verdict with moderate confidence is appropriate; I would not adjust it. A conditional acceptance would require the authors to either train the true ELBO and show the posteriors are unchanged, or reframe the method as a heuristic uncertainty-quantification scheme without Bayesian claims. Neither is present in the manuscript.","tokens_in":13857,"tokens_out":3876,"duration_ms":41811,"concrete_test":"Implement the exact ELBO from Eq. (12) with the analytical KL from Eq. (13) for the one-dimensional function approximation in Section 3.3, keeping the same architecture and data. Train both the exact-ELBO model and the paper's heuristic-loss model, then compare the resulting posterior over omega to the histogram in Figure 4 and evaluate the frequentist coverage of 95% credible intervals on held-out synthetic datasets generated with known omega. If the exact-ELBO posterior differs substantially from the reported histogram, or if coverage is far from 95%, the heuristic loss is not approximating variational inference and the Bayesian UQ claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that DeepBayONet performs Bayesian parameter estimation with variational inference, yielding calibrated uncertainty quantification. Section 2.1 derives an ELBO (Eq. 12) whose maximization includes the analytical KL divergence (Eq. 13). However, Section 2.4 defines the actual training loss (Eq. 17) with L_STD = log sigma_y^2 (Eq. 18e), and footnote 1 explicitly states this 'was found empirically to be simpler than evaluating the exact KL-divergence.' The optimized objective is therefore not the ELBO, so the posterior is not trained by variational inference. Moreover, Eq. (16a) samples lambda from N(0,I) as a fixed distribution; the trunk network tau(lambda) is a deterministic transformation, so the histograms in Figures 4, 7, 11, and 15 are pushforwards of a fixed standard normal through a learned network, not samples from p(lambda | D). Nothing in the loss anchors these distributions to the prior or to a proper likelihood beyond the data MSE term. This is not merely a cosmetic mismatch: the abstract and Section 2.3 promise 'comprehensive uncertainty quantification for both aleatoric and epistemic uncertainties,' but the epistemic component is a heuristic perturbation of hidden activations, not a Bayesian posterior. The internal contradiction in the 3D eigenvalue experiment (text gives ground truth lambda_11 = 20.19, while Figure 15a reports lambda ≈ 4.493) further indicates that the reported parameter estimates are not reliably tied to the stated target.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes DeepBayONet, a DeepONet-based architecture with physics-informed loss terms for forward PDE solving and parameter estimation. The central claim is that Bayesian training is implemented through variational inference, yielding posterior distributions over PDE parameters and quantifying both aleatoric and epistemic uncertainties. The authors derive an ELBO in Section 2.1, describe a trunk network that maps samples from a standard normal to an approximate posterior, and present experiments on regression, 1D function approximation, the 1D heat equation, a 2D reaction-diffusion equation, and a 3D eigenvalue problem. Comparisons against SNN, BNN, MCDO, DENN, and B-PINN variants are reported, with DeepBayONet claiming higher 95% CI coverage and accurate parameter recovery.","tokens_in":14177,"tokens_out":2453,"duration_ms":26522,"significance":"If the claims were supported, the method would be a useful contribution: it combines operator learning with physics-informed losses, provides a unified architecture for forward and inverse problems, and ships reproducible code. However, as written, the central Bayesian claim is not supported by the paper's own equations. The optimized loss is not the derived ELBO, and the reported parameter distributions are not posterior distributions conditioned on the data. The paper is therefore not yet a reliable basis for Bayesian UQ in PDE surrogate modeling.","major_comments":[{"comment":"The training objective implemented in the paper is not the ELBO derived in Eq. (12). The ELBO contains the KL divergence term of Eq. (13), but the actual loss in Eq. (17) replaces this term with L_STD = log sigma_y^2 (Eq. 18e), and footnote 1 states this was chosen empirically as simpler than evaluating the exact KL divergence. Consequently, the optimized objective is not a variational lower bound, and the q(lambda) represented by the trunk network is not trained as a variational posterior. This directly undermines the abstract's claim that 'Bayesian training is implemented through variational inference.'","section":"Sec. 2.4, Eqs. (17)-(18e), footnote 1"},{"comment":"The 'posterior' histograms in Figures 4, 7, 11, and 15 are not samples from p(lambda | D). Equation (16a) draws lambda from a fixed standard normal independent of the data, and Eq. (16b) computes a deterministic transformation tau(lambda) through the trunk network. The resulting distribution over tau(lambda) is a pushforward of the fixed prior N(0, I) through a learned map; it is not conditioned on observations through Bayes' rule. Nothing in the loss anchors this distribution to the posterior, so the reported uncertainty intervals are not Bayesian credible intervals.","section":"Sec. 2.3, Eqs. (16a)-(16b)"},{"comment":"The 3D eigenvalue experiment contains an internal inconsistency: the text states the ground-truth eigenvalue is lambda_11 = 4.49341^2 ≈ 20.19 (dipole mode), while Figure 15a is described as showing convergence to 'lambda ≈ 4.493.' If 4.493 is the first zero of the spherical Bessel function, it would correspond to a different quantity. As written, the reader cannot tell which eigenvalue was actually estimated, and this undermines the claim of accurate parameter recovery in the 3D example.","section":"Sec. 3.6 and Fig. 15"}],"minor_comments":[{"comment":"The caption refers to 'ODD dataset' where it should say 'OOD dataset.'","section":"Sec. 3.2, Fig. 2 caption"},{"comment":"The caption says 'eq. (21) eq. (21)' for the 1D heat equation, which is a duplicated reference.","section":"Sec. 3.4, Fig. 5 caption"},{"comment":"Figure 15 labels the panels as '(a) 3D Axis Plot' and '(b) 3D Histogram,' but the text refers to Figure 15a as the histogram; the panel labels and the textual references should be aligned.","section":"Sec. 3.6, Fig. 15"},{"comment":"The BNN baseline reports a total 95% CI coverage of 13.30%, which is unusually low for a Bayesian method; the authors should clarify the baseline configuration and diagnostic checks to ensure the comparison is not affected by a poorly tuned prior or variational posterior.","section":"Sec. 3.2, Table 2"},{"comment":"The paper claims sigma_y^2 can be learned as an input-dependent function, but the loss terms in Eqs. (18d) and (18e) use a single scalar sigma_y^2. Either the implementation should be described consistently or the claim of input-dependent aleatoric uncertainty should be moderated.","section":"Sec. 2.3, paragraph on data uncertainty"}],"recommendation":"reject","confidential_remarks":"The core Bayesian claim is refuted by the manuscript's own equations and footnote, not by a difference in interpretive taste. This is not a case where a small local fix would suffice; the method would need to be reframed as a non-Bayesian stochastic regularization approach and the uncertainty comparisons would need to be redone. The eigenvalue inconsistency in Section 3.6 further suggests that some numerical results may not have been cross-checked against the stated ground truths."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. The DeepBayONet architecture—a trunk network mapping standard normal samples to multiplicative perturbations of branch features—is genuinely new relative to B-DeepONet and B-PINNs. And the regression experiment is properly executed, with ten runs and 89% total 95% CI coverage versus 63% for the best baseline. But the central methodological claim does not survive contact with the loss function.\n\nThe paper is clearly written and the authors are transparent about their choices. Code is on GitHub, and footnote 1 explicitly admits that the KL divergence in the derived ELBO was replaced by the heuristic penalty L_STD = log σ_y^2. That admission is honest, but it is also the paper's fatal flaw. Section 2.1 derives variational inference; Section 2.4 optimizes something else. The histograms in Figures 4, 7, 11, and 15 are therefore pushforwards of a fixed N(0, I) through the trunk network, not posterior samples. Nothing in the training objective ties these distributions to the prior or to a proper likelihood in the Bayesian sense. Calling them posteriors is the load-bearing claim, and it is not supported.\n\nThere is also an internal numerical contradiction in the 3D eigenvalue experiment: the text states λ11 = 4.49341² ≈ 20.19, while Figure 15a reports convergence to λ ≈ 4.493. That is a factor-of-twenty gap in a headline result, not a minor typo. It makes me doubt the reliability of the reported parameter estimates elsewhere.\n\nWhat the paper does well: the regression study is careful, the comparison against SNN, BNN, MCDO, and DENN is reasonable, and the authors acknowledge limitations (manual hyperparameter tuning, no learned solution operator). The abstract also promises robustness when some physical equations are missing, but no missing-equation experiment appears in the text—another gap between claim and evidence.\n\nWho is this for? A reader interested in new architectures for uncertainty quantification in operator learning might find the latent perturbation idea worth exploring. But as a Bayesian parameter estimation paper, it fails on its own terms. The architecture could be repositioned as a heuristic UQ method with a clearly stated non-Bayesian motivation.\n\nRecommendation: this deserves a serious referee, but with an expectation of heavy revision. The core flaw is fixable—either implement the actual KL term or reframe the method as an approximate, non-Bayesian UQ scheme. As it stands, I would not accept it in its current form.","headline":"New architecture, but the Bayesian claim is unsupported: the implemented loss is not the derived ELBO.","tokens_in":14751,"tokens_out":2765,"would_cite":false,"duration_ms":28276,"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":"DeepBayONet claims to solve PDEs, estimate unknown parameters, and quantify both aleatoric and epistemic uncertainty in a single training run, with regression coverage of 89.05%.","keywords":["Physics-Informed Neural Networks","DeepONet","operator learning","Bayesian inference","variational inference","uncertainty quantification","partial differential equations","parameter estimation"],"falsifier":"Generate many synthetic datasets from a PDE with known parameter $\\lambda_{\\mathrm{true}}$, train DeepBayONet on each, and measure how often the trunk's 95% credible interval for $\\lambda$ contains the true value. If the empirical coverage is far from 95%—especially if it is systematically overconfident—the 'posterior' is not calibrated and the Bayesian claim does not hold.","tokens_in":13614,"feed_emoji":"🎯","tokens_out":8617,"duration_ms":80008,"temperature":0.7,"pith_summary":"The paper aims to establish that one neural architecture, DeepBayONet, can solve forward PDE problems, recover unknown parameters, and report uncertainty for both data noise and model uncertainty in a single training run. It does this by combining the DeepONet operator-learning structure with physics-informed losses and a variational-inference training scheme in which the trunk network turns standard-normal draws into an approximate posterior over the PDE parameters. If the claim holds, practitioners would get point estimates and uncertainty bands for parameters from sparse, noisy observations without running expensive MCMC or separate inversion steps. The headline quantitative results are 89.05% coverage of 95% confidence intervals in the regression benchmark versus 63.45% for the best baseline, and recovery of the reaction rate $k = 1$ in a 2D reaction-diffusion problem as $0.999 \\pm 5.4 \\times 10^{-4}$.","feed_headline":"DeepBayONet recovers PDE parameters with 89% CI coverage","feed_subtitle":"A physics-informed operator network yields parameter estimates and error bars from sparse noisy data.","key_machinery":"The engine is the deep operator network with latent modulation: $M_\\theta(t,x,\\lambda) = \\sigma\\big(W_0\\,(b(t,x)\\odot \\tau(\\lambda)) + b_0\\big)$, where the branch network $b$ encodes the spatiotemporal input, the trunk network $\\tau$ maps independent standard-normal draws into a distribution over the unknown parameter $\\lambda$, and the element-wise product $\\odot$ couples the two before the output layer. This coupling lets uncertainty in the parameter flow into the forward solution, and the variational loss (data likelihood plus PDE-residual likelihood plus a KL-type term) trains $\\tau$ to be an approximate posterior over $\\lambda$. The simplification that replaces the exact KL divergence with $L_{\\mathrm{STD}} = \\log \\sigma_y^2$ is what makes the training practical, and it is the component on which the Bayesian interpretation rests.","core_discovery":"The central claim is that modulating a DeepONet's branch activations by a trunk-generated latent variable $\\lambda \\sim \\mathcal{N}(0, I)$ and training with an ELBO-style loss yields an approximate posterior over the PDE parameters rather than a point estimate. The paper argues that this posterior carries both data uncertainty, through a learned observation-noise variance $\\sigma_y^2$, and model uncertainty, through the spread of the trunk's output. It demonstrates the claim on a regression benchmark, a 1D unsteady heat equation, a 2D reaction-diffusion equation, and a 3D eigenvalue problem, with the regression experiment showing 89.05% coverage of 95% confidence intervals overall and 97% out-of-distribution coverage.","pith_inferences":["The trunk's 'posterior' is only as trustworthy as the simplified STD loss; a direct check would be to retrain with the exact KL divergence and compare posterior widths and coverage.","Because the model does not learn a parametrized solution operator (the paper states this explicitly), an untested regime is extrapolation to PDE coefficients or source terms that differ from the training instance.","The multiplicative perturbation of branch features is formally similar to multiplicative noise in dropout, so a fair cost/benefit test would compare against MC-dropout with a learned input-dependent variance on the same benchmarks.","For the eigenvalue problem, the normalization term $(E[u^2]-1)^2$ is the only thing preventing collapse to $u\\equiv 0$; a stressed test would be searching for higher or degenerate eigenvalues where mode separation is harder."],"forward_implications":["A single training run yields both a forward PDE solution and a posterior over unknown parameters, so the method shortcuts the usual two-stage solve-then-invert pipeline.","In the regression benchmark, DeepBayONet achieves 89.05% coverage of 95% confidence intervals overall and 97% out-of-distribution, exceeding every tested baseline.","For the 2D reaction-diffusion problem, the mean recovered reaction rate is $0.999$ with standard deviation $5.4 \\times 10^{-4}$, matching the true $k=1$ to three decimal places.","Withdrawing boundary conditions from the loss widens the posterior and increases residuals at the boundary, indicating the network can express its reduced confidence.","The 3D eigenvalue experiment suggests the method can pick out the first dipole eigenmode and its eigenvalue in a curved domain without using spherical coordinates."],"supporting_citations":[{"why":"Supplies the DeepONet architecture (branch + trunk networks) that DeepBayONet modifies with a latent modulation.","marker":"[32]"},{"why":"Provides the universal approximation theorem for operators that underlies the DeepONet construction.","marker":"[33]"},{"why":"Introduces B-PINNs, the Bayesian physics-informed baseline used for parameter estimation and uncertainty comparison.","marker":"[29]"},{"why":"Motivates the work by extending DeepONet with Bayesian posterior sampling via SGLD; DeepBayONet contrasts with it.","marker":"[31]"},{"why":"Defines the physics-informed neural network loss and the residual-based training that DeepBayONet adopts.","marker":"[7]"},{"why":"Supplies the aleatoric/epistemic uncertainty decomposition that the paper uses to interpret its learned variances.","marker":"[34]"},{"why":"Provides the regression test problem and baseline uncertainty-aware networks (SNN, BNN, MCDO, DENN) against which DeepBayONet is compared.","marker":"[38]"}],"fun_headline_variants":["Bayesian DeepONet yields PDE parameters with 89% CI coverage","DeepONet plus PINN estimates PDE parameters with error bars","Variational Inference in DeepONets for calibrated uncertainty","Operator learning with uncertainty: DeepONet does Bayesian PDEs","89% coverage: Bayesian DeepONet handles sparse noisy data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that drawing $\\lambda \\sim \\mathcal{N}(0,I)$ and multiplying the branch network's hidden activations by $\\tau(\\lambda)$ produces a valid approximate posterior over the unknown PDE parameters, and that replacing the exact KL divergence with $L_{\\mathrm{STD}} = \\log \\sigma_y^2$ leaves the variational Bayesian interpretation intact.","fun_headline_variants_meta":{"raw":{"variants":["Bayesian DeepONet yields PDE parameters with 89% CI coverage","DeepONet plus PINN estimates PDE parameters with error bars","Variational Inference in DeepONets for calibrated uncertainty","Operator learning with uncertainty: DeepONet does Bayesian PDEs","89% coverage: Bayesian DeepONet handles sparse noisy data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000426,"raw_usage":{"total_tokens":2121,"prompt_tokens":825,"completion_tokens":1296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":1210}},"tokens_in":441,"tokens_out":1296,"duration_ms":9905,"temperature":1.0,"reasoning_tokens":1210,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:02:19.700840+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate many synthetic datasets from a PDE with known parameter $\\lambda_{\\mathrm{true}}$, train DeepBayONet on each, and measure how often the trunk's 95% credible interval for $\\lambda$ contains the true value. If the empirical coverage is far from 95%—especially if it is systematically overconfident—the 'posterior' is not calibrated and the Bayesian claim does not hold.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates the work by extending DeepONet with Bayesian posterior sampling via SGLD; DeepBayONet contrasts with it."},{"cited_title":"URLhttps://is.gd/ZvwCFr","cited_arxiv_id":null,"evidence_quote":"Provides the regression test problem and baseline uncertainty-aware networks (SNN, BNN, MCDO, DENN) against which DeepBayONet is compared."}],"review_version":1}