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arxiv: 2605.17107 · v1 · pith:OGOGYP2Znew · submitted 2026-05-16 · 📊 stat.ML · cs.LG· math.OC· math.PR

Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations

Phuoc-Toan Huynh , Richard Archibald , Feng Bao This is my paper

Pith reviewed 2026-05-20 14:57 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OCmath.PR
keywords stochastic partial differential equationsoperator learninguncertainty quantificationdeep operator networksstochastic neural networks
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The pith

Stochastic Operator Networks learn solution operators for SPDEs and quantify uncertainty directly from noisy data.

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

The paper introduces the Stochastic Operator Network to learn solution operators for stochastic partial differential equations without first specifying the noise structure. It builds this by merging the Deep Operator Network architecture with stochastic neural networks, then trains the model by minimizing a Hamiltonian-type loss through the Stochastic Maximum Principle. The result is a method that produces both a predicted mean field and a measure of predictive uncertainty. A reader would care if this holds because many real-world models involve unknown or hard-to-measure uncertainties, and learning them from data could reduce reliance on explicit assumptions about noise.

Core claim

The Stochastic Operator Network is formed by combining the Deep Operator Network structure with Stochastic Neural Networks to model stochasticity and enable probabilistic predictions of SPDE solution operators. Training proceeds by minimizing a Hamiltonian-type loss and optimizing the objective with the Stochastic Maximum Principle. Numerical experiments on benchmark SPDEs with multiple uncertainty sources show the approach captures solution structure and quantifies predictive uncertainty accurately and robustly.

What carries the argument

The Stochastic Operator Network (SON), which merges the DeepONet architecture with Stochastic Neural Networks and trains via minimization of a Hamiltonian-type loss using the Stochastic Maximum Principle.

If this is right

  • The method produces accurate mean solution fields on standard SPDE test problems.
  • It quantifies predictive uncertainty reliably when multiple sources of uncertainty are present.
  • It operates directly on noisy measurements without requiring a separate noise model to be supplied.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same architecture might be tested on inverse problems where the goal is to infer uncertain parameters from observations.
  • Extensions could examine performance on SPDEs with non-Gaussian or spatially correlated noise.
  • Integration with existing physics-informed neural network techniques could further constrain the learned operators.

Load-bearing premise

That combining DeepONet with stochastic neural networks and training through a Hamiltonian loss optimized by the Stochastic Maximum Principle yields well-calibrated uncertainty estimates that generalize across different SPDE problems without an explicit noise model.

What would settle it

Apply the trained SON to a new benchmark SPDE whose true noise statistics are known, then check whether the predicted uncertainty bands contain the actual solution errors at the claimed rate, for instance whether 95 percent intervals contain the true errors in 95 percent of cases.

Figures

Figures reproduced from arXiv: 2605.17107 by Feng Bao, Phuoc-Toan Huynh, Richard Archibald.

Figure 1
Figure 1. Figure 1: [Reaction-Diffusion] (Left) MSE loss from Phase 1. (Right) MSE loss and Hamiltonian loss (14) from Phase 2. We next investigate the predictive performance of the SON model. We evaluate the performance of SON in terms of both predictive accuracy of the solution profiles and the quality of uncertainty 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: [Reaction-Diffusion] Solution prediction: (Left) Reference solution mean . (Middle) Predicted solution mean. (Right) Prediction mean error. quantification. To this end, we randomly select one input function from the testing dataset and generate 400 predicted samples by using the trained SON. For the same input, we also generate 400 reference solution samples from the corresponding SPDE solver. The left and… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: [Advection-Diffusion Case 1] Solution prediction: (Left) Reference solution mean, (Middle) SON predicted solution mean, (Right) SON prediction mean error [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: [Advection-Diffusion Case 1] Std estimation: (Left) Reference std, (Middle) SON prediction std, (Right) SON std prediction error. To further support this observation, we plot multiple cross-sections of the reference and predicted means and display their corresponding confidence bands in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: [Advection-Diffusion Case 1] Cross-sections at columns 1, 11, 16, 26, 33, and 39 of the reference solution and approximation. each kth entry for k = 1, . . . , r takes the form (σ(A, θn,g))k ≡ (σ(A, θg))k = X Lg l=1 bl,k µ(cl,kA), n = 0, . . . , N − 1, (23) where Lg = 4, and for all k = 1, . . . , r, {bl,k} Lg l=1 and {cl,k} Lg l=1 are sampled uniformly from the intervals [0.035, 0.1] and [0.2, 0.5], respe… view at source ↗
Figure 7
Figure 7. Figure 7: [Advection-Diffusion Case 2] Heatmaps of mean errors: (Left) DeepONet; (Right) SON. We first compare the predictive accuracy between the deterministic DeepONet and the SON. In SON, Phase I and Phase II are trained for 500 and 200 epochs, respectively, for a total of 700 training epochs, while the DeepONet baseline is trained for 500 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: [Advection-Diffusion Case 2] Cross-sections of the predicted solutions with quantitative uncertainty with three different testing inputs: (First row) DeepONet; (Second row) SON. SON achieves higher predictive accuracy than the deterministic DeepONet, and it provides reliable uncertainty quantification [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: [Advection-Diffusion Case 2] Std estimation: (Left) Reference std, (Middle) SON prediction std, (Right) Std prediction Error. To better illustrate SON’s capability in uncertainty quantification, for a representative testing input we plot cross-sections of the reference and predicted means at several selected spatial columns in the solution domain, along with their corresponding confidence bands, in [PITH_… view at source ↗
Figure 10
Figure 10. Figure 10: [Advection-Diffusion Case 2] Cross-sections at columns 1, 11, 16, 26, 33, and 40 of the reference solution and approximation. see that the mean of the reference samples is accurately predicted. We also want to point out that the confidence bands exhibit non-uniform behavior, being narrower in some regions than in others, which makes Case 2 more challenging than Case 1. Although the predicted bands do not … view at source ↗
Figure 11
Figure 11. Figure 11: [Advection-Diffusion Case 2] Errors in the sample mean and std of the B-DeepONet predictions are significantly larger than those of SON under the same input data and training epochs, as shown in [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: [Advection-Diffusion Case 2] Cross-sections at columns 1, 11, 16, 26, 33, and 40 of the B-DeepONet prediction, using the same input as in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: [Heat Equation] DeepONet vs. SON. Cross-sections of the predicted solutions at the boundary (column 1) and time steps tm for m = 1, 10, 20, 30: (First row) DeepONet; (Second row) SON. SON not only provides accurate solution predictions and reliable uncertainty estimates, but also achieves higher predictive accuracy than the deterministic DeepONet. shown in [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: [Heat Equation] SON performance: (Left) Reference solution at the final time; (Middle) SON predicted sample mean at the final time.(Right) SON prediction errors [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: [Heat Equation] SON performance: (Left) Reference std at the final time. (Middle) SON output sample std at the final time.(Right) SON std prediction error. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: [Heat Equation] Cross-sections of the SON predicted solutionand its associated uncertainty at time steps tm for m = 1, 10, 20, 30: (Top row) Column 1; (Middle row) Column 16; (Bottom row) Column 31. 3.4. Burgers’ equation In this numerical example, we consider a more challenging problem: the 2D Burgers’ equation, ut +  u 2 2  x +  u 2 2  y = 0, (x, y, t) ∈ (−1, 1)2 × (0, T), u(x, y, 0) = u0(x, y), x ∈… view at source ↗
Figure 17
Figure 17. Figure 17: [Burger Equation Case 1] Prediction mean at final time: (Left) Reference mean; (Middle) Prediction mean; (Right) Prediction error. We then examine SON’s predictive performance more closely by plotting cross-sections of predicted sample means and their corresponding confidence bands at x = 0, x = 25 and x = 40 for time steps m = 1, 14, 27 and 40 in [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: [Burger Equation Case 1] Prediction std at final time: (Left) Reference std. (Middle) Prediction std. (Third) Std estimation errors [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: [Burger Equation Case 1] Cross-sections of the predicted solution at time steps tm for m = 1, 14, 27, 40. (First row) Column 1. (Second row) Column 25. (Third row) Column 40. tm for m = 1, 14, 27, 40. We visualize these averages as heatmaps in Figures 20 and 21. These results demonstrate the consistent accuracy and reliability of the SON model in capturing the solution’s spatial structure and in quantifyi… view at source ↗
Figure 20
Figure 20. Figure 20: [Burger Equation Case 1] Average mean error over 8 inputs at time step tm for m = 1, 14, 27, 40 [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: [Burger Equation Case 1] Average std error over 8 inputs at time step tm for m = 1, 14, 27, 40. where σm = [σm,1, . . . , σm,H ] ∈ R N is a matrix and ⊙ denotes element-wise multiplication. For all time steps m, the entries of σm are sampled from U(0.15, 0.25) [PITH_FULL_IMAGE:figures/full_fig_p025_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: [Burger Equation Case 2] Prediction performance. (Left) Prediction mean at final time. (Middle) Prediction mean error. (Right) Prediction std error. We first present the performance of SON for a randomly selected representative input. For this input, we generate 400 reference and predicted solution samples and compute the corresponding sample means and stds [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: [Burger Equation Case 2] Cross-sections of the predicted solution in Figure (22) at time steps tm for m = 1, 14, 27, 40. (First row) Column 1. (Second row) Column 25. (Third row) Column 40 [PITH_FULL_IMAGE:figures/full_fig_p026_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: [Burger Equation Case 2] Average mean error over 8 inputs at time step tm for m = 1, 14, 27, 40 [PITH_FULL_IMAGE:figures/full_fig_p026_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: [Burger Equation Case 2] Average std error over 8 inputs at time step tm for m = 1, 14, 27, 40. 4. Conclusion In this work, we introduced the Stochastic Operator Network (SON) framework for learning solution operators associated with various SPDEs and for quantifying intrinsic uncertainty in model predictions. SON is derived from the DeepONet architecture by replacing the branch network with a Stochastic … view at source ↗
read the original abstract

We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under uncertainty, their practical use typically requires specifying the magnitude and structure of model uncertainties that are often unknown and difficult to infer from noisy measurements. To address this challenge, we develop a stochastic operator-learning framework that learns directly from noisy data and outputs both a mean solution field and a quantification of uncertainty. The proposed method, namely the Stochastic Operator Network (SON), is constructed by combining the structure of the Deep Operator Network (DeepONet) with Stochastic Neural Networks (SNNs) to model stochasticity and enable probabilistic prediction. The training procedure is carried out by minimizing a Hamiltonian-type loss and optimizing the resulting objective using the Stochastic Maximum Principle. Numerical experiments on benchmark SPDEs under multiple uncertainty sources demonstrate the accuracy and robustness of the proposed method in capturing solution structure and quantifying predictive uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Stochastic Operator Network (SON) for uncertainty quantification in SPDEs. It combines the DeepONet architecture with Stochastic Neural Networks to model stochasticity, and trains the model by minimizing a Hamiltonian-type loss optimized via the Stochastic Maximum Principle. This allows learning both mean solution operators and predictive uncertainty directly from noisy data without an explicit noise model. Numerical experiments on benchmark problems such as the stochastic heat equation and Burgers equation under multiple uncertainty sources are used to demonstrate accuracy in capturing solution structure and robustness in uncertainty quantification.

Significance. If the central claim holds, the framework would offer a practical advance in operator learning for SPDEs by enabling data-driven UQ when noise structure is unknown, which is common in physical modeling. The integration of DeepONet with SNNs and SMP-based optimization is a novel technical contribution to probabilistic operator networks. The work is grounded in learning from data rather than assuming a specific noise law, which aligns with real-world needs.

major comments (2)
  1. [Numerical Experiments] Numerical Experiments section: results are limited to pointwise errors and visual uncertainty bands on standard benchmarks; no proper scoring rules, coverage probabilities, calibration plots, or out-of-distribution tests (e.g., deliberately changing from additive to multiplicative noise) are reported. This leaves the claim of well-calibrated predictive uncertainty without formal verification.
  2. [Method] Method and Experiments: the implicit assumption that SNN stochasticity optimized under the SMP-derived Hamiltonian loss recovers accurate posterior predictive distributions for unknown noise structures is load-bearing for the central claim, yet no sensitivity analysis or formal checks against violated modeling assumptions are provided.
minor comments (2)
  1. [Numerical Experiments] Add explicit details on data splits, training/validation/test ratios, and any error bars or statistical significance measures for the reported errors.
  2. [Related Work] Clarify notation for the stochastic components in the SON architecture relative to prior stochastic DeepONet variants.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments identify opportunities to strengthen the empirical validation of uncertainty calibration and to provide additional robustness checks. We address each major comment below and describe the revisions we will implement.

read point-by-point responses

  1. Referee: [Numerical Experiments] Numerical Experiments section: results are limited to pointwise errors and visual uncertainty bands on standard benchmarks; no proper scoring rules, coverage probabilities, calibration plots, or out-of-distribution tests (e.g., deliberately changing from additive to multiplicative noise) are reported. This leaves the claim of well-calibrated predictive uncertainty without formal verification.

    Authors: We agree that the current numerical results rely primarily on pointwise errors and visual inspection of uncertainty bands. To provide a more rigorous assessment of calibration, we will augment the Numerical Experiments section with the Continuous Ranked Probability Score (CRPS), empirical coverage probabilities at multiple nominal levels, and reliability (calibration) diagrams. We will also add an out-of-distribution experiment that deliberately changes the noise structure (additive to multiplicative) on the stochastic heat and Burgers equations. These additions will be included in the revised manuscript. revision: yes

  2. Referee: [Method] Method and Experiments: the implicit assumption that SNN stochasticity optimized under the SMP-derived Hamiltonian loss recovers accurate posterior predictive distributions for unknown noise structures is load-bearing for the central claim, yet no sensitivity analysis or formal checks against violated modeling assumptions are provided.

    Authors: The Stochastic Maximum Principle supplies a principled optimality condition for the Hamiltonian loss that enables learning without an explicit parametric noise model; this is the theoretical motivation for the framework. Nevertheless, we recognize that empirical sensitivity checks are important for practical credibility. In the revised manuscript we will add a dedicated sensitivity study that perturbs the noise structure and distribution during training and reports the resulting changes in predictive mean and uncertainty metrics. These results will be presented alongside the existing benchmark experiments. revision: yes

Circularity Check

0 steps flagged

No circularity: method is a data-driven combination of architectures with external benchmark validation

full rationale

The paper presents a Stochastic Operator Network formed by combining DeepONet with Stochastic Neural Networks, trained via a Hamiltonian-type loss derived from the Stochastic Maximum Principle. This is a constructive modeling choice applied to noisy data, followed by numerical experiments on standard SPDE benchmarks. No step reduces a claimed prediction to a fitted input by construction, invokes a self-citation as the sole justification for uniqueness, or renames an empirical pattern as a first-principles result. The derivation chain remains self-contained because the architecture and loss are explicitly assembled from known components and then evaluated against independent benchmark problems rather than being tautological with the training data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Because only the abstract is available, the ledger is necessarily incomplete and inferred from the high-level description; the central claim rests on the unstated assumption that the Stochastic Maximum Principle optimization converges to a useful posterior over operators and that the stochastic neural network parameterization is sufficiently expressive for the target SPDEs.

axioms (1)
  • domain assumption The Hamiltonian-type loss combined with the Stochastic Maximum Principle yields a well-defined optimization problem whose solution corresponds to accurate mean and uncertainty estimates for the SPDE solution operator.
    Invoked in the description of the training procedure.
invented entities (1)
  • Stochastic Operator Network (SON) no independent evidence
    purpose: To model stochasticity in operator learning for SPDEs and enable probabilistic predictions.
    New named architecture introduced in the abstract.

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