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REVIEW 3 major objections 5 minor 40 references

Deep Operator Networks for Bayesian Parameter Estimation in PDEs

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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%.

desk verdict New architecture, but the Bayesian claim is unsupported: the implemented loss is not the derived ELBO. read the letter →

arxiv 2501.10684 v2 pith:S3W3NOAT submitted 2025-01-18 cs.LG cs.CEstat.ML

Amogh Raj , Carol Eunice Gudumotou , Sakol Bun , Keerthana Srinivasa , Arash Sarshar This is my paper · ORCID
classification cs.LGcs.CEstat.ML
keywords Physics-InformedNeuralNetworksDeepONetoperatorlearningBayesianinferencevariationaluncertaintyquantificationpartialdifferentialequationsparameterestimation
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

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 $0.999 \pm 5.4 \times 10^{-4}$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Formalized claims in Lean

  1. Claim #1: 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 u

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Sec. 2.4, Eqs. (17)-(18e), footnote 1] 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.'
  2. [Sec. 2.3, Eqs. (16a)-(16b)] 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.
  3. [Sec. 3.6 and Fig. 15] 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.
minor comments (5)
  1. [Sec. 3.2, Fig. 2 caption] The caption refers to 'ODD dataset' where it should say 'OOD dataset.'
  2. [Sec. 3.4, Fig. 5 caption] The caption says 'eq. (21) eq. (21)' for the 1D heat equation, which is a duplicated reference.
  3. [Sec. 3.6, Fig. 15] 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.
  4. [Sec. 3.2, Table 2] 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.
  5. [Sec. 2.3, paragraph on data uncertainty] 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.

Circularity Check

2 steps flagged · score 6.0 of 10

The Bayesian posterior claim is not derived: the KL term is replaced by a heuristic log-variance loss, so the reported parameter distributions are pushforwards of a fixed Gaussian through the fitted trunk network.

  1. self definitional [Section 2.4, Eq. (17)-(18e) and footnote 1]
    "The loss component in eq. (18e) was found empirically to be simpler than evaluating the exact KL-divergence. The intuition behind it is that the log of the variance of the predicted solution needs to be of moderate magnitude."

    Eq. (12) derives an ELBO whose optimization includes the analytic KL divergence D_KL(q(λ)||p0(λ)). Eq. (17) defines the actual training loss with L_STD = |log σ_y^2| in place of that KL term. The optimized objective is therefore not the ELBO, so q(λ) is not trained to approximate p(λ|D). The claimed variational inference reduces, by the paper's own equations and footnote, to fitting a heuristic log-variance penalty; the 'posterior' is whatever the trunk network produces, not the result of a Bayesian update.

  2. fitted input called prediction [Section 2.3, Eqs. (16a)-(16b); Section 3.5, Table 3]
    "λ ∼ N(0,I), ˜h = τ(λ)⊙h. ... The posterior distribution q(λ) over the latent parameters can be empirically estimated using the trunk network to understand the model’s confidence in the predicted parameter."

    Because λ is always sampled from the fixed distribution N(0,I), the 'posterior' histogram is, by construction, the pushforward of the fixed Gaussian through the trained trunk network τ(λ); no likelihood update or KL regularization enters. The reported parameter estimates (e.g., k = 0.999 ± 5.4e-4 in Table 3) are the moments of this fitted modulation, so the uncertainty and parameter-recovery claims are readings of the fitted network rather than predictions from an inferred posterior.

full rationale

The forward PDE solving and regression experiments are self-contained and benchmarked independently, so the paper is not wholly circular. However, the central Bayesian UQ claim is not derived: Eq. (12) derives an ELBO with a KL term, but Eq. (17)/(18e) replaces it with L_STD = |log σ_y^2|, as footnote 1 admits. Since the optimized objective is not the ELBO, the trunk-network output τ(λ) for λ∼N(0,I) is not an approximate posterior p(λ|D); the histograms and coverage intervals are properties of the fitted stochastic perturbation. This makes the 'posterior distributions' and the reported parameter uncertainties reduce, by the paper's own equations, to the learned modulation rather than to Bayesian inference. The 3D eigenvalue section also contains an internal inconsistency (the text gives the eigenvalue λ_11 = 20.19, while Fig. 15a is said to converge to λ≈4.493, which is the Bessel zero, not the eigenvalue), further undermining the parameter-estimation claim, though this is a correctness issue rather than circularity. No load-bearing self-citation chain was found; the cited B-PINN and DeepONet works are external.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The listed free parameters are the ones the reported parameter estimates and uncertainty bands depend on: the loss weights, the learned data noise, and the residual noise. The axioms are the standard Bayesian and operator approximation assumptions plus the ad hoc replacement of the KL term. No new physical entities are introduced.

free parameters (3)
  • Loss weights w_Interior, w_IC, w_BC, w_STD, w_Data = heat: wIC=3, wData=6, wBC=1, wInterior=1, wSTD=1; reaction-diffusion: wInterior=20000, wData=60000, wBC=100, wSTD=20…
    Hand-tuned via grid search (Sections 2.4, 3.4, 3.5). These weights determine the balance between data, PDE residual, BC, IC, and STD terms in Eq. 17, so the parameter estimates depend strongly on them.
  • Observation noise variance σ_y^2 = learned, values not reported
    Appears in the denominators of Eqs. 18b to 18d and in L_STD Eq. 18e; it is treated as learnable (Section 2.3) and directly sets the width of the reported aleatoric uncertainty.
  • Residual noise variance σ_R^2 = constant or learned, not specified in experiments
    Used in the interior residual term Eq. 18a; the paper does not report its value or how it was set.
assumptions (5)
  • domain assumption The joint likelihood factorizes as a product over data points of the data likelihood and the PDE residual likelihood (Eq. 4).
    The residual is evaluated at collocation points but indexed like the data; independence is assumed without discussion.
  • domain assumption The PDE in Eq. 1 is well posed and y(t,x;λ) exists uniquely with the stated initial and boundary conditions.
    The problem definition requires this but does not state regularity conditions.
  • standard math Both p0(λ) and q(λ) are Gaussian with σ0^2 = 1, enabling the closed-form KL in Eq. 13.
    This is the textbook Gaussian KL result, but the trunk network's pushforward distribution is not guaranteed to be Gaussian.
  • standard math DeepONet can approximate the solution operator M_θ(t,x,λ) as a product of branch and trunk outputs (Eq. 14).
    Borrowed from the universal approximation theorem for operators, cited as [30,33].
  • ad hoc to paper L_STD = log σ_y^2 is an acceptable replacement for the exact KL divergence (Eq. 18e, footnote 1).
    This replacement is load-bearing because nothing else constrains the posterior over λ; the paper admits it is empirical.

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Cite this review

Pith. "Pith review of Deep Operator Networks for Bayesian Parameter Estimation in PDEs." pith.science (2026). https://pith.science/paper/S3W3NOAT

@misc{pith2026250110684,
  author       = {Pith},
  title        = {Pith review of: Deep Operator Networks for Bayesian Parameter Estimation in PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3W3NOAT}},
  note         = {Machine review of arXiv:2501.10684}
}
read the original abstract

We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven learning with physical constraints, our method achieves robust and accurate solutions across diverse scenarios. Bayesian training is implemented through variational inference, allowing for comprehensive uncertainty quantification for both aleatoric and epistemic uncertainties. This ensures reliable predictions and parameter estimates even in noisy conditions or when some of the physical equations governing the problem are missing. The framework demonstrates its efficacy in solving forward and inverse problems, including the 1D unsteady heat equation and 2D reaction-diffusion equations, as well as regression tasks with sparse, noisy observations. This approach provides a computationally efficient and generalizable method for addressing uncertainty quantification in PDE surrogate modeling.

Figures

Figures reproduced from arXiv: 2501.10684 by the authors.

Figure 1
Figure 1. DeepBayONet feedforward architecture with trunk and branch networks. The trunk network learns the PDE parameters, while the branch network predicts the forward solution of the problem. 2.2. Model architecture We used a deep operator network (DeepONet) [33, 32] to parametrize Mθ(t, x; λ). This architecture consists of two components: the branch net￾work and the trunk network. The branch network approximates the forwa… view at source ↗
Figure 2
Figure 2. (a) DeepBayONet without trunk (SNN). (b) Monte Carlo Dropout (MCDO). [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. DeepBayONet learned solution for the regression problem in eq. (20). (a) σ = 0.1. (b) σ = 0.01 (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Posterior of the parameter ω learned by DeepBayONet from eq. (20). where D and α are unknown parameters. When D = 1 and α = 1, the equation has the exact solution: y(t, x) = e −t sin(πx). A DeepBayONet model is trained to approximate this solution while learn￾ing the p…
Figure 5
Figure 5. Figure 5: (a) Loss history plot for 1D Heat equation in eq. (21) eq. (21). (b) Loss history [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: A sample of the predicted solution (left) and the exact solution (right) for the [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Histograms showing the distributions of Parameter [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Error and PDE residuals for the DeepBayONet learned solution of the unsteady heat eq. (21). (a) Forward solution absolute error compared to exact solution using the mean of the parameter posterior distribution. (b) PDE residual using the primary mode of the parameter p…
Figure 9
Figure 9. Figure 9: Training samples, exact u and f, and sensor positions for the 2D Reaction￾diffusion equation in eq. (22c). Comparing the training loss history, as shown in fig. 5, we observe that the reaction-diffusion losses in fig. 5b have magnitudes of difference, while in fig. 5a …
Figure 10
Figure 10. Figure 10: PDE residuals for the 2D Reaction-diffusion equation in eq. (22c). [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Posterior distribution of the parameter k for the 2D Reaction-diffusion equation in eq. (22c). error in u, not just local residuals, which makes accurate estimation of both quantities challenging. We solve this problem using DeepBayONet by approximating the eigen￾mode…
Figure 12
Figure 12. Figure 12: Predicted mean solution, absolute error, and uncertainty for the 2D Reaction [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Predicted mean solution, absolute error, and uncertainty for the 2D Reaction [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Combined results for the 3D Helmholtz experiment. [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Axis and Histogram plot for the 3D Helmholtz equation in eq. (23) [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

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Pith tools

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