Distribution free uncertainty quantification in neuroscience-inspired deep operators

Shailesh Garg; Souvik Chakraborty

REVIEW 4 major objections 8 minor 47 references

Distribution free uncertainty quantification in neuroscience-inspired deep operators

T0 review · 4 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that randomized-prior ensembles plus split conformal calibration yield per-location 95% prediction bands for wavelet and spiking neural operators across four PDE benchmarks, missing at only one grid location in the…

desk verdict A useful, mostly sound conformal-wrapper paper for spiking neural operators whose headline coverage claim is not verifiable as written because the width multiplier z is never specified. read the letter →

arxiv 2412.09369 v1 pith:PPZLD46C submitted 2024-12-12 stat.ML cs.LG

Shailesh Garg , Souvik Chakraborty This is my paper · ORCID

classification stat.MLcs.LG MSC 62G1568T07

keywords conformalpredictionuncertaintyquantificationneuraloperatorspikingnetworkwaveletrandomizedpriorlearningsuper-resolution

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 introduces a distribution-free uncertainty-quantification wrapper, the Conformalized Randomized Prior Operator (CRP-O), that attaches calibrated prediction bands to wavelet and spiking neural operators. Randomized-prior ensembles provide a per-location mean and standard deviation; split conformal prediction then rescales this spread by a location-dependent quantile so that the bands meet a target coverage of 95% in the experiments. Across four PDE benchmarks (Burgers, Darcy on rectangular and triangular domains, and seismic Helmholtz wave propagation), the conformalized operators CRP-WNO and CRP-VSWNO achieve the target coverage at every grid location except one, whereas quantile-based baselines undercover at many locations. A Gaussian-process extension transfers the calibrated quantile from the training grid to finer super-resolution grids without retraining. The paper argues that a good initial uncertainty estimate is what lets conformal calibration succeed with a meager calibration set.

What carries the argument

The load-bearing object is the randomized-prior operator ensemble: $n_c$ copies of the base operator (vanilla WNO or VSWNO) are trained on the same data with different initializations, each augmented by a frozen prior network whose parameters enter the loss as $\lambda L_{\mathrm{prior}}(\phi)$. The ensemble spread supplies a heuristic uncertainty $s(u_t)$ alongside the mean $\mu(u_t)$; the normalized residual $|y-\mu|/s$ is the conformal score. Split conformal prediction turns that score into a per-location multiplier $q$ (one quantile per element of the output grid), and the final band is $\mu \pm z q s$, with $z$ a user-set width parameter that the paper never assigns a value. For super-resolution, a Gaussian process regression model treats $q$ as a function of grid location and predicts it on the finer grid, so no retraining or recalibration is needed. VSWNO is the spiking variant of the wavelet neural operator, replacing continuous activations with variable spiking neurons (VSNs) that emit sparse, event-driven outputs; its uncertainty is the paper's main target.

What would settle it

Recompute per-location coverage with $z$ set explicitly to 1 in Algorithm 1, using the same trained RP operators, calibration sets, and test sets; if Examples I-III no longer achieve at least 95% coverage at all grid locations, the paper's central claim depends on the unspecified multiplier rather than on randomized priors plus split conformal calibration.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that randomized-prior (RP) operators give a sufficiently informative initial uncertainty estimate that split conformal calibration can turn into per-location calibrated bands: the final band is $C_p = [\mu - z q s, \mu + z q s]$, where $\mu$ and $s$ are the ensemble mean and standard deviation of $n_c$ copies of the base operator, and $q$ is computed element-wise as the $\lceil(1-\alpha)(n+1)\rceil/n$ quantile of the calibration scores $|y-\mu|/s$. In Examples I-III (Burgers, Darcy rectangular, Darcy triangular), both CRP-WNO and CRP-VSWNO achieve $\geq 95\%$ coverage at every point of the solution grid, and in Example IV (Helmholtz) they miss at a single grid location in one frequency component. The same per-location performance is not obtained by uncalibrated RP operators, by quantile-trained WNO (Q-WNO), or by conformalized quantile WNO (CQ-WNO), which reach about 95% only on average. Against the three-way-split risk-controlling quantile neural operator (RCQNO), CRP-WNO produces tighter intervals and lower NMSE with fewer calibration samples. The paper additionally claims that mapping $q$ from the training grid to a finer grid with a Gaussian process preserves coverage in zero-shot super-resolution, demonstrated in the first two examples.

Load-bearing premise

The load-bearing premise is that the multiplier $z$ in the band $\mu \pm z q s$ is fixed in advance; the paper never gives a value for $z$, and the conformal guarantee covers $\mu \pm q s$, so an unexamined choice of $z$ could produce the reported coverage.

Editorial extensions

If this is right

CRP-WNO and CRP-VSWNO achieve at least 95% per-location coverage on Burgers (1024 of 1024 locations), Darcy rectangular (7225 of 7225), and Darcy triangular (1102 of 1102), and at all but one of 4900 locations in the Helmholtz example.
Quantile-based baselines, though near 95% on average, undercover at hundreds to thousands of locations, supporting the paper's hypothesis that the initial uncertainty heuristic, not calibration alone, determines per-location performance.
The Gaussian-process extension gives zero-shot super-resolution coverage: bands calibrated at 1024 points remain valid at 2048 points in Example I, and bands calibrated on an 85 by 85 grid remain valid on a 141 by 141 grid in Example II.
CRP-WNO gives tighter calibrated intervals and lower NMSE than RCQNO on the Helmholtz problem while using 150 calibration samples instead of 500.
Only a two-way train/calibration split is needed, unlike RCQNO's three-way split, so the method remains usable when data are limited.

Reading between the lines

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

The value of the width multiplier $z$ is never reported; if $z$ is not fixed in advance, the conformal guarantee in Lemma 1 applies to $\mu \pm q s$, not $\mu \pm z q s$, so the reported coverage could silently depend on an unexamined widening of the bands, and a clean test would report coverage with $z=1$.
Element-wise calibration over a large solution grid raises a multiple-testing concern: with more than seven thousand locations, a few failures would be expected by chance even under a valid method, so the paper's all-locations pass is stronger than the formal guarantee requires.
The Gaussian-process mapping assumes the conformal quantile $q(x)$ varies smoothly over the spatial domain; near sharp features such as the Burgers shock or across velocity discontinuities in the Helmholtz field, this smoothness may fail, which is testable by checking super-resolution coverage near discontinuities.
The choice of the prior network (a smaller WNO inside RP-VSWNO) is an unexamined degree of freedom; ablating the prior's architecture and the weight $\lambda$ would reveal how much of the benefit comes from the prior rather than from ensembling alone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read it straight through. The core move is simple: take the ensemble spread from randomized-prior wavelet/spiking operators, use it as the nonconformity score, and calibrate with split conformal prediction per output-grid location. That is a natural wrapper, and I believe it is the first time it is applied to VSWNO; the GP-based extension of q to unseen super-resolution grids is also new, though heuristic. The empirical story is consistent across four PDE examples: the conformalized RP versions hit roughly 95% per-location coverage, while quantile-based baselines often miss entire regions. That counts for something. The paper also cites the two prior conformal-operator papers (Moya et al., Ma et al.) and does not oversell conditional coverage.

The soft spots are real, and one is load-bearing. Eq. (7) and Algorithm 1 define the calibrated band as mu ± z q s, but the conformal quantile q is computed from scores |y-mu|/s. Lemma 1 proves coverage for mu ± q s. Unless z=1, the guarantee is for the modified score |y-mu|/(z s); for z>1 the band is conservative and still covers, but for z<1 it does not. The manuscript never states z, never restricts z≥1, and the code is promised only after acceptance. So the headline coverage claims are not reproducible from the text, and the width comparison against RCQNO in Table 8 is confounded by an unknown multiplicative factor. This is easy to fix: state z=1, or divide calibration scores by z and rerun. But as written it needs to be fixed before the guarantee is taken seriously.

Second, E-IV says the authors trained 20 randomized-prior copies and selected the best 10. That post hoc selection compromises the RP spread as an uncertainty heuristic; it should be justified or removed. Third, all coverage tables come from a single 100-sample test set with no error bars. With 100 samples, per-location coverage has a binomial standard error around 2%, so exact counts of '0 failures' are noisier than they look. Fourth, the GP super-resolution step has no coverage guarantee and should be described as an empirical extension, not as part of the conformal argument.

The Lemma 1 proof has a small order-statistic indexing slip (Eq. 13 should be an inequality or phrased with ranks), but the standard conformal result is invoked correctly in substance. That is minor.

Who is this for? Practitioners doing UQ for neural operators, especially spiking/edge models. Conformal theorists will not learn much. The method is likely to work, and the paper deserves a serious referee, but I would send it back for major revision: specify z, release code, clean up E-IV, and add error bars or repeated splits. My verdict is conditional, leaning positive.

Referee Report

4 major / 8 minor

Summary. The paper proposes a Conformalized Randomized Prior Operator (CRP-O) framework that combines randomized-prior ensembles with split conformal prediction to produce calibrated prediction bands for wavelet and spiking neural operators. The method is applied element-wise on the output grid, with a Gaussian-process extension for zero-shot super-resolution. Four PDE test cases (Burgers, Darcy on rectangular and triangular domains, and Helmholtz) are used to claim that the conformalized bands achieve 95% coverage at every grid location except one location in the fourth example, and that they outperform RP, Q-WNO, CQ-WNO, and RCQNO baselines.

Significance. The paper addresses a timely and practical problem: distribution-free uncertainty quantification for neural operators, including energy-efficient spiking variants. The core idea of using an RP ensemble as a heuristic uncertainty measure and then applying split conformal calibration is sensible, and the standard conformal lemma is invoked in substance. The four PDE experiments cover nontrivial settings, and the comparison with RCQNO is a useful practical addition. However, the central coverage claim currently rests on an unspecified width multiplier z in the prediction band, the super-resolution extension is heuristic rather than guaranteed, and the empirical protocol has several reproducibility gaps. If these issues are resolved, the framework would be a solid contribution to the neural-operator UQ literature.

major comments (4)

[Section 3.2, Eq. (7), Algorithm 1 step 9] The prediction band is defined as Cp = [mu - z q s, mu + z q s], but the conformal guarantee in Lemma 1 applies to the band [mu - q s, mu + q s] induced by the score e = |y - mu|/s. If z is not fixed to 1, the calibrated score is effectively |y - mu|/(z s), and the quantile q computed from the unmodified scores does not provide the stated coverage: for z < 1 the guarantee fails, while for z > 1 the band is conservative. The manuscript never gives the value of z, so the reported per-location coverages and the interval widths in Table 8 are not reproducible from the text, and the comparison with RCQNO is confounded. The authors must either set z = 1 (and state it explicitly) or, if z is chosen a posteriori, prove a conformal guarantee for the modified score and report the chosen z.
[Section 3.3, Figs. 3 and 5] The Gaussian-process extension interpolates the conformal parameter q from the coarse calibration grid to a finer prediction grid and then uses the GP predictive mean as the calibrated q at new locations. The split-conformal guarantee, however, applies to the q computed directly from calibration scores at the original grid locations; the GP-interpolated values are not quantiles of calibration scores for the fine-grid locations. The coverage improvements shown in Figs. 3(a) and 5(a) are therefore empirical observations, not consequences of Lemma 1. The paper should explicitly state that the super-resolution extension is a heuristic (which the text partially does) and should not claim a calibrated guarantee for the fine grid without additional analysis.
[Section 4.4, E-IV and Table 8] The manuscript states that for E-IV the authors train 20 RP copies and 'select the best 10', but it does not specify the selection criterion (e.g., validation loss, calibration score, or test error). If the selection uses the test set, the reported uncertainty calibration is invalid. Additionally, Table 8 compares RCQNO with n = 500 calibration samples against CRP-WNO with n = 150 calibration samples, so the reported interval widths are not directly comparable; the difference in calibration set size could affect the conformal quantile and hence the band width. The comparison should be made at matched calibration sample sizes or the dependence on n should be discussed and quantified.
[Section 4, Tables 3–7] The headline claim is that CRP-WNO and CRP-VSWNO achieve at least 95% observed coverage at all (or all but one) grid locations. Because the test set contains only 100 samples, the binomial uncertainty in an observed 95% coverage is large: e.g., 95 successes out of 100 gives a 95% Clopper-Pearson lower bound of about 88.6%. The paper should report confidence intervals for the coverage proportions, or repeat the evaluation over multiple test splits, before claiming that 'the required coverage is achieved at all locations'.

minor comments (8)

[Section 3.2] The acronym SCP is expanded as 'Stochastic Cross-Validation Procedure', but the correct term is 'Split Conformal Prediction'; this should be fixed.
[Eqs. (9) and (11)] The quantile definition is written as Quantile({e}, ceil((1-alpha)(n+1))/n) in Eq. (9) but as q = e_{ceil((1-alpha)(n+1))} in Eq. (11). These are different formulas; the notation should be aligned and the order-statistic indexing clarified.
[Lemma 1 proof] The proof writes the sorted scores as e1 <= ... <= en and then states P(et <= ei) = i/(n+1), reusing the symbol ei for both the raw and sorted scores; this is a notational flaw in an otherwise standard argument.
[Eq. (3)] There is a typo in 'thetth time step'; it should read 'the t-th time step'.
[Algorithm 1, step 2] The symbol ⊘ is used for element-wise division but is not defined; please define it in the text or in the algorithm.
[Section 4.4, Eq. (31)] The domain is given as x, z in [0,690] with delta x = delta z = 70, but the output is described as a 70 x 70 grid; the relation between these numbers should be clarified.
[Section 4, Table 1 and text] The nomenclature in Table 1 uses CQ-WNO, but the text at the start of Section 4 refers to 'CRQ-WNO'; this inconsistency should be fixed.
[Appendix A] The statement that sample code 'is to be released after acceptance' is incompatible with the reproducibility needs of the central coverage claim, especially given the unspecified z; please release code or provide complete implementation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported coverage is an external held-out test benchmark produced by standard split-conformal calibration, not a restatement of the calibration fit.

full rationale

The paper's derivation chain is not circular. The conformal multiplier q is computed from a separate calibration set via Eq. (9) and Algorithm 1, and all reported coverages are measured on a disjoint test set, so the headline claim is an empirical check rather than a tautology. The RP ensemble mean and spread in Eq. (5) are ordinary ensemble statistics trained on the training set; they are not fitted to a coverage target, so the calibrated band is not forced by construction. The Gaussian-process interpolation in Section 3.3 is a heuristic extension for super-resolution and does not feed test coverage back into the method. The self-citations to VSWNO [27], RP-WNO [38], and the variable spiking neuron [43] are prior architectural and algorithmic works with independent content; they are used as building blocks, not as an authority that predetermines the calibration outcome. The main limitation is non-circular: Eq. (7) and Algorithm 1 step 9 introduce a width multiplier z that is never given a numeric value, while Lemma 1's proof only establishes the guarantee for z = 1; Appendix A also states that sample code will be released only after acceptance. These are reproducibility and correctness risks, not reductions of the claimed prediction to its own inputs, so they do not raise the circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are implementation constants and one post hoc selection rule; the unspecified z is the most consequential. The axioms are the standard exchangeability condition for conformal prediction plus the heuristic assumptions behind element-wise calibration and the GP super-resolution extension.

free parameters (4)

z = not specified
Multiplier in the calibrated band Cp = µ ± z q s; never stated, so coverage and interval width depend on an unknown constant.
lambda (prior loss weight) = not specified
Weight of the non-trainable prior loss in the augmented loss of Eq. (4); no value or tuning range is given.
alpha_w, beta_w (spiking loss weights) = not specified
Weights in the spiking loss function Ls (Appendix C); chosen as hyperparameters and not reported.
best_10_of_20 copies in E-IV = best 10 of 20 trained
In E-IV the authors train 20 RP copies and select the best 10, a post hoc choice that affects the uncertainty estimate.

assumptions (5)

domain assumption Calibration and test samples are exchangeable (i.i.d.)
Required for the split conformal guarantee in Lemma 1; stated in Section 3.2.
domain assumption VSN dynamics and surrogate gradient training are valid
The spiking operator inherits VSWNO from [27]; UQ results depend on the trained network being a meaningful operator.
ad hoc to paper Element-wise SCP across output grid points provides per-location coverage
The paper applies SCP independently to each grid point (Algorithm 1), which gives marginal coverage per location, not a joint guarantee.
ad hoc to paper GP interpolation of q preserves coverage on the fine grid
Super-resolution extension has no formal proof; it assumes the coarse-grid q field is smooth and that GP predictive mean at new points yields valid calibration (Section 3.3).
domain assumption Spiking activity is a valid surrogate for energy consumption
Taken from [27,28]; used to claim energy efficiency in Appendix C.

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Cite this review

Pith. "Pith review of Distribution free uncertainty quantification in neuroscience-inspired deep operators." pith.science (2026). https://pith.science/paper/PPZLD46C

@misc{pith2026241209369,
  author       = {Pith},
  title        = {Pith review of: Distribution free uncertainty quantification in neuroscience-inspired deep operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PPZLD46C}},
  note         = {Machine review of arXiv:2412.09369}
}

read the original abstract

Energy-efficient deep learning algorithms are essential for a sustainable future and feasible edge computing setups. Spiking neural networks (SNNs), inspired from neuroscience, are a positive step in the direction of achieving the required energy efficiency. However, in a bid to lower the energy requirements, accuracy is marginally sacrificed. Hence, predictions of such deep learning algorithms require an uncertainty measure that can inform users regarding the bounds of a certain output. In this paper, we introduce the Conformalized Randomized Prior Operator (CRP-O) framework that leverages Randomized Prior (RP) networks and Split Conformal Prediction (SCP) to quantify uncertainty in both conventional and spiking neural operators. To further enable zero-shot super-resolution in UQ, we propose an extension incorporating Gaussian Process Regression. This enhanced super-resolution-enabled CRP-O framework is integrated with the recently developed Variable Spiking Wavelet Neural Operator (VSWNO). To test the performance of the obtained calibrated uncertainty bounds, we discuss four different examples covering both one-dimensional and two-dimensional partial differential equations. Results demonstrate that the uncertainty bounds produced by the conformalized RP-VSWNO significantly enhance UQ estimates compared to vanilla RP-VSWNO, Quantile WNO (Q-WNO), and Conformalized Quantile WNO (CQ-WNO). These findings underscore the potential of the proposed approach for practical applications.

Figures

Figures reproduced from arXiv: 2412.09369 by the authors.

**Figure 1.** Schematic for the overall framework. marginal likelihood). Details on training GP is shown in Algorithm 2). Once trained, the predictions at a new location x ∗ can be obtained by computing the predictive distribution, p(q ∗ |x ∗ , xˆ, q), p(q ∗ |x ∗ , xˆ, q) ∼ N (q ∗ |µ ∗ , K∗ ), (20) where µ ∗ = K(xˆ, x ∗ ) ⊤K(xˆ, xˆ ′ ) −1 q (21) and K∗ = K(x ∗ , x ∗ ) − K(xˆ, x ∗ ) ⊤K(xˆ, xˆ ′ ) −1K(xˆ, x ∗ ). (22) Here, K(x ∗ , … view at source ↗

**Figure 2.** Mean predictions for test inputs obtained using CRP-WNO and CRP-VSWNO compared against the ground [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Super-resolution results for E-I. despite the original calibration being done on a coarser grid [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Mean predictions for test inputs obtained using CRP-WNO and CRP-VSWNO compared against the ground [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Super-resolution results for E-II. However, the equation is now defined on a triangular domain Ω with a rectangular notch. The boundary condition u(x, y)|∂Ω is assumed to be random, and sampled from a Gausian random field, u(x) = GP(0, K(x, x ′ )), K(x, x ′ ) = exp −…

**Figure 6.** Figure 6: Mean predictions for test inputs obtained using CRP-WNO and CRP-VSWNO compared against the ground [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Mean predictions corresponding to one test input obtained using CRP-VSWNO compared against the ground [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.

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