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REVIEW 4 major objections 5 minor 83 references

MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.16883 v2 pith:X4L3O7KT submitted 2024-12-22 math.NA cs.NA

Sudeb Majee , Anuj Abhishek , Thilo Strauss , Taufiquar Khan This is my paper · ORCID
classification math.NAcs.NA MSC 62G2035J1562F15
keywords BayesianinverseproblemsMarkovChainMonteCarloneuraloperatorsurrogateconvolutionalnetworkHellingerdistanceelectricalimpedancetomographydiffuseopticalquantitativephotoacoustic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Sections 5.4 and 5.6, Tables 4 and 7] 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.
  2. [Section 4, Theorems 2–3; Section 5] 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.
  3. [Theorem 3, Eq. (44)] 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.
  4. [Theorem 2 proof, Eqs. (34)–(35)] 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.
minor comments (5)
  1. [Section 4.1] There is a typo: 'the goal will; be' should be 'the goal will be'.
  2. [Theorem 2 statement] The statement 'there exists a finite-dimensional spaces' should be 'there exists a finite-dimensional space'.
  3. [Figure A.11 caption] 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.
  4. [Section 5.8] The text asserts that MCMC-Net 'outperforms' MCMC-FEM in error metrics for QPAT, which is inconsistent with Table 7 and should be corrected.
  5. [Section 5.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the surrogate-posterior convergence is a conditional continuity bound in the forward-model error, the CNN is trained on FEM forward data rather than on posterior targets, and the self-citations are background or baseline material rather than load-bearing premises.

full rationale

The derivation chain is not circular. The central theoretical claim, Theorem 3, shows that the Hellinger distance between the surrogate posterior mu_y^theta and the true posterior mu^y is controlled by ||G - G_theta||_{L2(mu)}: Eqs. (40)-(43) bound I1 and I2 by powers of this forward-operator error, so the posterior convergence is a continuity statement with respect to the surrogate's forward accuracy. The quantity being controlled is the same forward error that Theorem 2 asserts can be made small for some encoder/approximator pair; nothing in the theorem defines the posterior or the likelihood in terms of the surrogate posterior, and the proof cites external results (Marzouk and Xiu [79], Iglesias-Lu-Stuart [45], Lanthaler-Mishra-Karniadakis [74], Kovachki-Lanthaler-Mishra [75]). The numerical surrogates are trained on input-output pairs generated by FEM forward solves (Sec. 5.3), not on posterior samples or posterior summary statistics, so the inversion results are not fitted parameters renamed as predictions. The self-citations ([38], [46], [81], [82]) provide background, baselines, or pCN tuning heuristics and do not carry the load-bearing approximation or posterior-convergence argument. The paper's own limitation statements are relevant: Sec. 6 explicitly defers training and generalization error bounds to future work, and Theorem 3's proof uses boundedness of y in Eq. (44) without stating it as a hypothesis; moreover, the paper does not report a held-out ||G - G_theta||_{L2(mu)} measurement for the trained CNNs. These are genuine soundness or rigor gaps between the asymptotic theorem and the particular networks used in the experiments, but a missing measurement or an omitted hypothesis is not a circular reduction. No step of the derivation is equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard Bayesian inversion theory, on a compactness-and-approximation argument for the forward maps, and on an unverified premise that the trained CNN has small L2 error against the FEM forward model. The main hand-chosen quantities are the prior and network hyperparameters; the most fragile assumptions are the boundedness of y in Theorem 3 and the unmeasured surrogate error.

free parameters (5)
  • Matérn prior length scale ℓ = 0.4 (EIT), 0.2 (DOT)
    Chosen heuristically in Sections 5.1 and 5.3; controls the smoothness and regularization strength of the Gaussian process prior.
  • Matérn prior smoothness υ = 3
    Fixed for all problems; a modeling choice for the Gaussian process prior.
  • pCN proposal step size Δ = Adapted during burn-in to maintain about 25% acceptance
    Tuned per chain to approximate the optimal acceptance rate; affects mixing but not the target posterior.
  • CNN architecture and training hyperparameters = 4 layers, 16-64 neurons, minibatch 8-128, epochs 100-2000, learning rate 0.001
    Chosen by systematic experimentation (Section 5.3, Tables 6, A.8, A.9); the reported inversion results depend on these choices, though sensitivity analysis suggests mild dependence.
  • Level-set and star-shaped prior constants = σ_i, c_i for EIT; κ_i for QPAT
    Set by the problem and prior definitions; not fitted to data.
assumptions (7)
  • standard math Bayes' theorem for infinite-dimensional inverse problems (Theorem 1, from [40])
    Used to define the true posterior and the surrogate posterior in Eq. (3) and Eq. (7).
  • domain assumption The forward maps G for EIT, DOT, and QPAT are continuous and bounded on their parameter spaces
    Invoked in the proof of Theorem 2 via compactness and Lusin's theorem; standard for these level-set and star-shaped formulations, but not proved in this paper.
  • standard math Trigonometric interpolation error decays as ∥(Id−IP)w∥_{L∞} ≲ P^{−ξ(s)} for w in H^s
    Quoted from [75, Theorems 39 and 40] to control the encoder error T1 in Theorem 2.
  • standard math ReLU networks can uniformly approximate continuous functions on compact subsets of R^P
    Quoted from [77, Theorem 2] and [78, Theorem 1] to justify the approximator A.
  • standard math The pCN proposal is reversible with respect to the Gaussian prior, so only the likelihood ratio appears in the acceptance probability
    Used in Section 5.1 without proof; standard result for pCN.
  • ad hoc to paper The observed data y is bounded in Theorem 3
    The proof of Eq. (44) needs ∥y∥<∞, but the theorem statement and the QPAT white-noise model do not guarantee bounded data.
  • domain assumption Point evaluations plus trigonometric interpolation capture enough of the parameter field for the forward map
    The proof of Theorem 2 decomposes the error through DP∘EP; the implemented network has no decoder, so the analysis is conditional on the CNN approximating G∘DP on the encoded space.

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Cite this review

Pith. "Pith review of MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems." pith.science (2026). https://pith.science/paper/X4L3O7KT

@misc{pith2026241216883,
  author       = {Pith},
  title        = {Pith review of: MCMC-Net: Accelerating Markov Chain Monte Carlo with Neural Networks for Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4L3O7KT}},
  note         = {Machine review of arXiv:2412.16883}
}
read the original abstract

In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different PDE-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography. MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance.

Figures

Figures reproduced from arXiv: 2412.16883 by the authors.

Figure 1
Figure 1. Flowchart illustrating the MCMC-Net workflow [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The true map G is approximated by a composition of two maps, encoder E and approximator A. Note that in the Bayesian inverse problem formulated in section 3 for EIT, the space Hs (Ω) := ¯ X is a separable Hilbert space, and H : X → Γ is a level set map from the space X to the space of piecewise constant admissible conductivities. In this case, we denote the forward problem as: y = GEIT(w) + η (30) where GEIT : X → R… view at source ↗
Figure 3
Figure 3. Reconstruction of electrical conductivity using MCMC-FEM and MCMC-Net for EIT (data ob [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: EIT: Convergence of the MCMC method for FEM and CNN. Data obtained at 1 % relative noise. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Trace plots of the posterior samples of level-set functions using the proposed approach for EIT [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Reconstruction of absorption coefficient ( [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Reconstruction of the absorption coefficient using MCMC-FEM and MCMC-Net for QPAT. From [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: This figure shows qualitative results corresponding to the quantitative results in Table 6, arranged [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Comparison of inversion times between MCMC-FEM and MCMC-Net. For each method, 100,000 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]
Figure 8
Figure 8. Figure 8: We also report the inversion time using the MCMC-Net. [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]

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