{"id":"166ef881-78d3-4943-b7a2-3dbc1c2843cd","arxiv_id":"2412.16883","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A CNN surrogate for the forward model is used inside pCN-MCMC to accelerate Bayesian inversion for EIT, DOT, and QPAT, with accuracy that sometimes falls below the FEM baseline.","lead":"This paper replaces the slow finite-element forward solver inside Markov Chain Monte Carlo with a convolutional neural network for three imaging problems. The method speeds up sampling but its accuracy claims are only partly supported by the paper's own tables.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that MCMC-Net 'performs similar ... with significant speedup' is not supported because the trained surrogates' held-out forward errors are never measured, and QPAT results show the surrogate posterior is biased; Theorem 3's guarantee is conditional on an L2(µ) error that is never…","rationale":"The reader's weakest assumption is that the theoretical results are conditional on a small surrogate forward error that is never measured; this is exactly the point on which the paper's central claim turns. The QPAT table provides direct evidence that the surrogate posterior is biased relative to FEM, so the missing error measurement is not a formality. I deliberately did not make the proof gap in Theorem 2 the primary concern: the notation around Eq. (35) is sloppy, but the surrounding modulus-of-continuity argument on the compact set Z can be repaired, and the same is true for the bounded-y condition in Theorem 3 if one fixes the data. The unmeasured surrogate error, by contrast, is a gap between theory and the actual trained networks that no amount of proof repair can close without new experiments. The proposed check is deliberately minimal: one held-out error evaluation over the training prior, plus the same error over the posterior samples actually used in the QPAT reconstructions, would settle whether the Hellinger theorem applies to the deployed surrogates. Since the reader's verdict was already REJECT and this concern supports that verdict, no adjustment is needed.","tokens_in":26739,"tokens_out":10961,"duration_ms":107689,"concrete_test":"Compute the held-out forward error of the trained QPAT CNN from Sec. 5.3 on 10,000 parameter samples drawn from the prior used for training (star-shaped and circular profiles), evaluating ||G(γ)-Gθ(γ)||² with the FEM solver as reference. Report the resulting L2(µ) error, and also the error restricted to the MCMC-Net posterior samples behind Figure 7. If either error is too large for the Theorem 3 bound to force a small Hellinger distance (e.g., below 0.05), the theoretical guarantee does not cover the reported QPAT reconstructions and the central claim fails for that problem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the transfer of Theorems 2-3 to the experiments. Theorem 2 proves only existence of an encoder/approximator pair with ||G-Gθ||_{L2(µ)} ≤ ϵ; Theorem 3 then gives Hellinger convergence of the surrogate posterior. No number in the paper measures the held-out L2(µ) forward error of the trained CNNs, and the conclusion (Sec. 6) explicitly defers training and generalization error bounds to future work. An existence result does not certify the specific networks used in Sec. 5. The QPAT numbers in Table 7 are the signature of a nonzero surrogate bias: at 2% noise, MAE is 0.001708 vs 0.001329 and MSE is 0.000156 vs 0.000101 for FEM, and the speedup is only 1.6x. EIT with three anomalies (Table 4) also shows larger errors for MCMC-Net (MAE 0.350951 vs 0.282551). Without quantifying ||G-Gθ|| on the prior or on the posterior region visited by the chain, the theoretical guarantee cannot be invoked, and the central claim is unsupported. A secondary gap is that Theorem 3's proof uses boundedness of y in Eq. (44) without stating it as a hypothesis, but the primary issue is the missing measurement of the surrogate error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MCMC-Net, a method for accelerating MCMC-based Bayesian inverse problems by replacing the FEM forward solver with a CNN surrogate in the likelihood evaluation. The authors present a universal approximation theorem for the neural operator surrogate, a Hellinger-distance convergence result for the surrogate posterior, and numerical experiments on EIT, DOT, and QPAT comparing reconstruction accuracy and speed against a FEM-based MCMC baseline. The central empirical claim is that MCMC-Net performs similarly to the classical likelihood counterpart while providing significant speedups.","tokens_in":27070,"tokens_out":7243,"duration_ms":60796,"significance":"If fully supported, the paper would address a major practical bottleneck in Bayesian inverse problems—the repeated evaluation of expensive PDE-based forward models inside MCMC. The strengths are the clean theoretical setup, the breadth of numerical experiments across three imaging modalities, the sensitivity and robustness analyses, and the inclusion of out-of-distribution tests. However, the central empirical claim is contradicted by the paper's own tables in several scenarios, and the theoretical convergence guarantee is not connected to the trained networks because the surrogate forward error is never measured. The significance of the contribution is therefore currently limited.","major_comments":[{"comment":"The abstract's claim that MCMC-Net 'performs similar' to the classical likelihood counterpart is contradicted by the paper's own results. In Table 4, the three-anomaly EIT case gives MAE 0.350951 and MSE 1.403706 for MCMC-Net versus 0.282551 and 1.130100 for MCMC-FEM. In Table 7, MCMC-FEM has lower MAE and MSE at every QPAT noise level (e.g., at 2% noise, MAE 0.001329 vs 0.001708 and MSE 0.000101 vs 0.000156), and the speedup is only about 1.6x (10.62 vs 6.61 minutes). Later text (Sections 5.8 and 6) even asserts that the neural-operator method 'outperforms' the traditional method in error metrics, which is the opposite of Table 7. These discrepancies affect the paper's central claim.","section":"Sections 5.4 and 5.6, Tables 4 and 7"},{"comment":"The theoretical guarantee is an existence result: Theorem 2 states that for every ε there exists some encoder–approximator pair with ||G − Gθ||_{L2(µ)} ≤ ε. It does not certify that the specific CNNs trained in Section 5 have small forward error, and the paper never reports a held-out estimate of ||G − Gθ||_{L2(µ)} over the prior or over the posterior region sampled by the chain. Section 6 explicitly defers training and generalization error bounds to future work. Without this measurement, Theorem 3's Hellinger convergence cannot be invoked for the numerical results, and the QPAT results in Table 7 are consistent with a nontrivial surrogate bias. This is the main methodological gap.","section":"Section 4, Theorems 2–3; Section 5"},{"comment":"The proof of Theorem 3 uses the boundedness of y in Eq. (44) ('is bounded if y is bounded') without stating it as a hypothesis of the theorem. The same bound is used to control I2, so this is a load-bearing omission. The theorem statement should include the assumption that y is bounded, or the proof should be adjusted to avoid this requirement.","section":"Theorem 3, Eq. (44)"},{"comment":"The notation in the proof of Theorem 2 is imprecise: in Eq. (35) the expression ||G(Id − I_P)w|| should read ||G(w) − G(I_P w)||, and the integrals over the sets K and Z are written without explicitly specifying the restrictions of µ and the integrands. The argument can be repaired, but as written it is difficult to follow.","section":"Theorem 2 proof, Eqs. (34)–(35)"}],"minor_comments":[{"comment":"There is a typo: 'the goal will; be' should be 'the goal will be'.","section":"Section 4.1"},{"comment":"The statement 'there exists a finite-dimensional spaces' should be 'there exists a finite-dimensional space'.","section":"Theorem 2 statement"},{"comment":"The caption repeats 'the same number of layers (NL = 4)' for three successive triples of subfigures; the intended differences (e.g., number of neurons) are unclear and should be stated more precisely.","section":"Figure A.11 caption"},{"comment":"The text asserts that MCMC-Net 'outperforms' MCMC-FEM in error metrics for QPAT, which is inconsistent with Table 7 and should be corrected.","section":"Section 5.8"},{"comment":"The acceptance probability is written as min(1, L(q*_Prop.) − L(q_i)) without showing the prior ratio; although pCN with a Gaussian prior can cancel prior terms, the expression as written is incomplete and should be clarified.","section":"Section 5.1"}],"recommendation":"reject","confidential_remarks":"The paper has a solid experimental framework and the theoretical content is workmanlike, but the main empirical claim is contradicted by the authors' own tables, and the link between the theory and the experiments is missing. If the authors add a direct measurement of the surrogate forward error and revise the claims to match the data (for example, reporting QPAT as a case with limited speedup and accuracy loss), a resubmission to a more specialized venue could be considered. The current version does not meet the standard for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth reading if you work on surrogate-accelerated Bayesian inversion, but the central accuracy claim overreaches.\n\nWhat's actually new: not the core idea—replacing the expensive forward solve with a neural surrogate inside MCMC appears in the paper's own references [20]–[23], [29], [33]. The contribution here is the systematic application to three PDE-based imaging problems (EIT, DOT, QPAT) using a simple CNN architecture, a careful sensitivity analysis over network depth/width/hyperparameters, and an out-of-distribution test on non-circular inclusions that the network never saw. The theory is a standard but clean formalization: existence of an encoder-approximator pair with small L2(µ) error, plus Hellinger convergence of the surrogate posterior. That part is fine.\n\nThe soft spots are substantial. The abstract says 'performs similar ... with a significant speedup,' but Table 7 shows the QPAT surrogate is worse at every noise level (MAE 0.0017 vs 0.0013 at 2% noise) and only ~1.6x faster. For three-anomaly EIT, MCMC-Net has MAE 0.351 vs 0.283. So the claim is not uniformly true. More fundamentally, the theory is conditional on ‖G−Gθ‖_{L2(µ)} being small, and that number is never measured for the trained networks. The conclusion explicitly defers training and generalization error bounds to future work. Without that measurement, Theorem 3 does not certify the experiments—it remains an existence result. There's also a minor gap in Theorem 3: boundedness of y is used in Eq (44) but not stated as a hypothesis.\n\nCredit where due: the EIT speedup (~30x) and DOT speedup (~9x) are real, and the out-of-distribution generalization results (Appendix D) are a genuine strength. The authors are also transparent about the 100k-sample budget.\n\nI'd send this to peer review, but the reviewers should require: (1) a measured held-out forward error for each trained CNN, (2) a revised claim that doesn't promise similar accuracy for QPAT at 1.6x speedup, and (3) the extra boundedness hypothesis. The work is substantial enough to warrant a revision rather than a desk reject.","headline":"A substantial empirical study of CNN-surrogate MCMC on three imaging problems, but the 'similar accuracy' claim overreaches and the missing forward-error measurement keeps the theory from biting.","tokens_in":27630,"tokens_out":4710,"would_cite":true,"duration_ms":39955,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G20","35J15","62F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"MCMC-Net replaces an expensive forward solver with a CNN surrogate and shows the resulting posterior converges to the true posterior while running tens of times faster.","keywords":["Bayesian inverse problems","Markov Chain Monte Carlo","neural operator surrogate","convolutional neural network","Hellinger distance","electrical impedance tomography","diffuse optical tomography","quantitative photoacoustic tomography"],"falsifier":"Run MCMC-FEM and MCMC-Net on the same held-out phantom, measure the CNN's forward error at the posterior samples, and compute the Hellinger distance between the two posterior distributions; the paper's Theorem 3 predicts this distance tends to zero as the measured forward error does, so a case where the forward error is small but the posteriors differ noticeably would show the theoretical mechanism is not what drives the numerical result.","tokens_in":26505,"feed_emoji":"⚡","tokens_out":6206,"duration_ms":52446,"temperature":0.7,"pith_summary":"This paper proposes MCMC-Net, a method that replaces the expensive forward-model evaluation inside Markov Chain Monte Carlo with a trained convolutional neural network. The authors show that for three PDE-based imaging inverse problems—electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography—the surrogate likelihood produces posterior reconstructions comparable to the classical finite-element likelihood while cutting computation time by roughly an order of magnitude, up to about 30x. The paper also proves two theoretical guarantees: the neural operator can approximate the forward map to arbitrary accuracy in an L2 sense, and as that approximation error goes to zero, the surrogate posterior converges to the true posterior in Hellinger distance. If correct, this provides a principled justification for using fast learned surrogates in Bayesian uncertainty quantification.","feed_headline":"Surrogate CNN speeds MCMC 30x at similar accuracy","feed_subtitle":"By swapping the expensive FEM forward solver for a neural operator, MCMC-Net matches posterior accuracy while running far faster.","key_machinery":"The engine is the composition $G_\\theta = A \\circ E$: an encoder $E$ that samples the parameter field on a grid, and a CNN approximator $A$ that maps those samples to the measurement vector. Theorem 2 shows this composition can approximate the true forward map to any desired $L^2(\\mu)$ accuracy, with the error split into encoding and approximation parts. Theorem 3 then converts that forward error into a Hellinger-distance bound between the true and surrogate posterior measures, using the Gaussian likelihood's exponential structure.","core_discovery":"The central claim is that MCMC-Net's surrogate posterior, built from a CNN replacement $G_\\theta$ of the true forward map $G$, converges to the true posterior in Hellinger distance when the $L^2(\\mu)$-error $\\|G-G_\\theta\\|$ tends to zero, and that in practice this convergence is close enough for reliable reconstructions. The paper derives this by splitting the error into encoding and approximation terms and coupling them to bounds on the likelihood potentials; numerically, MCMC-Net achieves mean absolute error and mean square error values on par with or better than MCMC-FEM for EIT and DOT, and slightly higher errors for QPAT, while reducing inversion times from 125 to 4 minutes for EIT, 34 to 3.6 minutes for DOT, and 10.7 to 6.5 minutes for QPAT.","pith_inferences":["Editorial inference: the theory implies a practical quality check—measure the trained surrogate's forward error on samples drawn from the posterior or its high-probability region, since the Hellinger bound is controlled by that $L^2(\\mu)$ error; the paper does not report this quantity.","Editorial inference: the slight accuracy loss seen for QPAT is consistent with a small but nonzero surrogate forward error; a testable extension would be to train with a loss weighted toward the posterior-relevant region and then re-measure the Hellinger distance.","Editorial inference: the same surrogate-posterior convergence argument should carry over to other MCMC variants that use the same likelihood ratio, such as Metropolis-adjusted Langevin or Hamiltonian Monte Carlo, provided the surrogate error is controlled on the regions those samplers explore.","Editorial inference: an implicit corollary is that reliability is bounded by distribution shift—if test-time parameter fields fall outside the training distribution, the surrogate error can spike and bias the posterior, a risk only partially addressed by the paper's out-of-distribution experiments."],"forward_implications":["The surrogate posterior converges to the true posterior in Hellinger distance as the trained forward surrogate error goes to zero, so Bayesian credible intervals computed with MCMC-Net are asymptotically faithful.","For EIT, DOT, and QPAT, likelihood evaluations with the CNN take minutes instead of hours, making full MCMC-based uncertainty quantification practical for these imaging modalities.","The speedup grows with discretization dimension: MCMC-FEM inversion time rises exponentially with mesh resolution, while MCMC-Net rises only linearly.","Because the proof only relies on the forward error being small in $L^2(\\mu)$, the method extends to any Bayesian inverse problem whose likelihood is expensive and whose forward map admits a continuous approximator."],"supporting_citations":[{"why":"Supplies the infinite-dimensional Bayesian inverse problem framework and posterior formulation used throughout the paper.","marker":"[9]"},{"why":"Provides the preconditioned Crank-Nicolson (pCN) sampler that the MCMC-Net algorithm uses for posterior exploration.","marker":"[11]"},{"why":"Gives the level-set Bayesian formulation for EIT, including boundedness of the forward map used in the approximation proof.","marker":"[42]"},{"why":"Establishes the Bayesian level-set framework and posterior convergence calculations used in the Hellinger-distance proof.","marker":"[45]"},{"why":"Supplies the DOT forward model and the FEM-based MCMC baseline against which MCMC-Net is compared.","marker":"[46]"},{"why":"Provides the QPAT forward model and star-shaped prior parametrization used in the numerical experiments.","marker":"[49]"},{"why":"Supplies the DeepONet error-estimate machinery, including the encoder-approximator decomposition, used in Theorem 2.","marker":"[74]"},{"why":"Provides trigonometric interpolation error bounds used to control the encoding error in the theorem's proof.","marker":"[75]"},{"why":"Gives the likelihood-potential comparison lemma used to bound the first term in the Hellinger-distance proof.","marker":"[79]"}],"fun_headline_variants":["MCMC-Net: Neural operator speeds MCMC 30x, matches accuracy","Neural network surrogate cuts MCMC time 30x for inverse problems","MCMC-Net: CNN surrogate accelerates MCMC 30x, same posterior quality","MCMC-Net: 30x faster MCMC with provable posterior convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the trained network's forward error being small over the parameter regions the sampler actually visits, and on the observed data being bounded; the paper does not measure the network's held-out forward error and explicitly leaves training and generalization error bounds for future work.","fun_headline_variants_meta":{"raw":{"variants":["MCMC-Net: Neural operator speeds MCMC 30x, matches accuracy","Neural network surrogate cuts MCMC time 30x for inverse problems","MCMC-Net: CNN surrogate accelerates MCMC 30x, same posterior quality","MCMC-Net: 30x faster MCMC with provable posterior convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1822,"prompt_tokens":907,"completion_tokens":915,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":829}},"tokens_in":523,"tokens_out":915,"duration_ms":7951,"temperature":1.0,"reasoning_tokens":829,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T06:02:17.545479+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run MCMC-FEM and MCMC-Net on the same held-out phantom, measure the CNN's forward error at the posterior samples, and compute the Hellinger distance between the two posterior distributions; the paper's Theorem 3 predicts this distance tends to zero as the measured forward error does, so a case where the forward error is small but the posteriors differ noticeably would show the theoretical mechanism is not what drives the numerical result.","supporting_citations":[{"cited_title":"Cotter, G","cited_arxiv_id":null,"evidence_quote":"Provides the preconditioned Crank-Nicolson (pCN) sampler that the MCMC-Net algorithm uses for posterior exploration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Bayesian level-set framework and posterior convergence calculations used in the Hellinger-distance proof."},{"cited_title":"Abhishek, T","cited_arxiv_id":null,"evidence_quote":"Supplies the DOT forward model and the FEM-based MCMC baseline against which MCMC-Net is compared."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the QPAT forward model and star-shaped prior parametrization used in the numerical experiments."},{"cited_title":"Lanthaler, S","cited_arxiv_id":null,"evidence_quote":"Supplies the DeepONet error-estimate machinery, including the encoder-approximator decomposition, used in Theorem 2."},{"cited_title":"Kovachki, S","cited_arxiv_id":null,"evidence_quote":"Provides trigonometric interpolation error bounds used to control the encoding error in the theorem's proof."},{"cited_title":"Marzouk, D","cited_arxiv_id":null,"evidence_quote":"Gives the likelihood-potential comparison lemma used to bound the first term in the Hellinger-distance proof."}],"review_version":1}