Deep set based operator learning with uncertainty quantification

Hao Wu; Lei Ma; Ling Guo; Tao Zhou

arxiv: 2509.25646 · v2 · submitted 2025-09-30 · 💻 cs.LG · cs.NA· math.NA

Deep set based operator learning with uncertainty quantification

Lei Ma , Ling Guo , Hao Wu , Tao Zhou This is my paper

Pith reviewed 2026-05-18 11:44 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA

keywords operator learninguncertainty quantificationset transformerconditional variational autoencoderscientific machine learningpartial differential equationsNavier-Stokes

0 comments

The pith

UQ-SONet learns solution operators from sparse variable sensors while providing principled uncertainty estimates for PDE predictions.

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

The paper introduces UQ-SONet as a way to learn operators from data without the fixed sensor requirements of standard DeepONets. It uses a set transformer to process inputs at arbitrary locations and a conditional variational autoencoder to model the full distribution over possible solution operators. Minimizing the negative ELBO then yields both accurate point predictions and calibrated uncertainty measures. The method is shown to work on deterministic and stochastic PDEs, including the Navier-Stokes equation, even when observations are incomplete.

Core claim

UQ-SONet integrates a set transformer embedding to handle sparse and variable sensor locations and employs a conditional variational autoencoder to approximate the conditional distribution of the solution operator. By minimizing the negative ELBO, the model supplies uncertainty estimates while preserving predictive accuracy on both deterministic and stochastic PDEs including the Navier-Stokes equation.

What carries the argument

Set transformer embedding paired with a conditional variational autoencoder that approximates the conditional distribution of the solution operator.

If this is right

Operator learning becomes feasible with irregularly placed or incomplete sensor data.
Uncertainty estimates arise directly from the variational approximation rather than post-hoc methods.
The same architecture applies to both deterministic PDEs and PDEs with intrinsic randomness.
Predictive accuracy is retained on benchmark problems such as the Navier-Stokes equation.

Where Pith is reading between the lines

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

The framework could support active sensor placement by using the uncertainty output to decide where to collect new measurements.
Similar set-based variational structures might transfer to other operator-learning tasks such as learning maps between function spaces in inverse problems.
The approach opens a route to hybrid models that combine the learned distribution with physics constraints to tighten uncertainty bounds.

Load-bearing premise

The combination of set transformer embeddings and a conditional variational autoencoder can accurately approximate the conditional distribution of the solution operator when sensor locations are sparse and variable.

What would settle it

A controlled test on the Navier-Stokes equation in which sensor locations are made extremely sparse; if the reported uncertainty intervals fail to contain the true solution variability at the expected rate, the claim is refuted.

read the original abstract

Learning operators from data is central to scientific machine learning. While DeepONets are widely used for their ability to handle complex domains, they require fixed sensor numbers and locations, lack mechanisms for uncertainty quantification, and are thus limited in practical applicability. Recent permutation-invariant extensions, such as the Variable-Input Deep Operator Network, relax these sensor constraints but still rely on sufficiently dense observations and cannot capture uncertainties arising from incomplete measurements or from operators with inherent randomness. To address these challenges, we propose UQ-SONet, a permutation-invariant operator learning framework with built-in uncertainty quantification. Our model integrates a set transformer embedding to handle sparse and variable sensor locations, and employs a conditional variational autoencoder to approximate the conditional distribution of the solution operator. By minimizing the negative ELBO, UQ-SONet provides principled uncertainty estimation while maintaining predictive accuracy. Numerical experiments on deterministic and stochastic PDEs, including the Navier-Stokes equation, demonstrate the robustness and effectiveness of the proposed framework.

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.

Desk Editor's Note private letter to a colleague

UQ-SONet combines set transformers and conditional VAEs to handle variable sensors and add UQ to operator learning, but the abstract gives no results to check if it works.

read the letter

The main takeaway is that UQ-SONet integrates a set transformer embedding with a conditional variational autoencoder to learn solution operators from sparse and variable sensor data while providing uncertainty estimates via negative ELBO minimization. This directly targets the fixed-sensor and no-UQ limitations of DeepONets and their variable-input variants. The work is new in its specific combination aimed at these gaps, particularly for handling uncertainties from incomplete measurements or inherent randomness in operators. The abstract does a solid job describing the motivation and the high-level architecture. It performs well in identifying practical constraints that matter for real PDE applications, like the Navier-Stokes equation, and in proposing a permutation-invariant approach that aligns with standard practices for set-based inputs. The soft spots center on the missing details. Since only the abstract is available, there are no quantitative results, error bounds, or validation of the uncertainty estimates to evaluate. The key assumption that this setup approximates the conditional distribution effectively under sparse conditions remains unexamined here, which makes the claims provisional. This paper would interest readers in scientific machine learning who apply operator networks to physical systems with irregular data collection. It could be useful for those exploring UQ extensions in the field. I recommend sending it for peer review. The approach is consistent and addresses relevant issues, so referees can check the full experiments and any implementation specifics.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes UQ-SONet, a permutation-invariant operator learning framework that integrates a set transformer embedding to accommodate sparse and variable sensor locations with a conditional variational autoencoder to approximate the conditional distribution of the solution operator. Training proceeds by minimizing the negative ELBO to obtain both point predictions and principled uncertainty estimates. The abstract claims that numerical experiments on deterministic and stochastic PDEs, including the Navier-Stokes equation, demonstrate robustness and effectiveness.

Significance. If the integration of set-transformer embeddings with conditional VAEs can be shown to approximate solution-operator distributions accurately under sparse observations while delivering calibrated uncertainty, the work would meaningfully extend existing DeepONet-style methods to more realistic data regimes with incomplete measurements and stochastic operators, which is relevant for scientific machine learning applications.

major comments (1)

[Abstract] Abstract: The central claim that minimizing the negative ELBO yields principled uncertainty quantification while preserving predictive accuracy on stochastic PDEs including Navier-Stokes is stated without any supporting derivation, error analysis, or quantitative results, rendering it impossible to assess whether the architecture actually supports the claim.

minor comments (1)

The abstract would be clearer if it briefly indicated the number or type of sensor configurations tested or the specific stochastic PDE variants considered.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the review and the opportunity to respond. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that minimizing the negative ELBO yields principled uncertainty quantification while preserving predictive accuracy on stochastic PDEs including Navier-Stokes is stated without any supporting derivation, error analysis, or quantitative results, rendering it impossible to assess whether the architecture actually supports the claim.

Authors: We agree that the abstract, by design, is a concise summary and therefore omits derivations, error bounds, and quantitative tables. The full manuscript contains the derivation of the ELBO objective for the conditional VAE that approximates the conditional distribution of the solution operator, together with the associated error analysis and the full set of numerical results (including Navier-Stokes) that quantify both predictive accuracy and uncertainty calibration. To address the concern directly, we will revise the abstract to include a brief clause referencing the theoretical justification and experimental validation provided in the main text. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in abstract

full rationale

The abstract presents a high-level description of UQ-SONet combining a set transformer embedding with a conditional VAE and minimizing the negative ELBO for uncertainty quantification. No equations, derivations, or specific parameter fittings are provided that could reduce to self-definitional inputs or fitted quantities by construction. The approach is framed using standard machine learning components without load-bearing self-citations or uniqueness claims that collapse the central result. This is a self-contained high-level summary relying on established ELBO minimization and permutation-invariant embeddings.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, background axioms, or newly postulated entities are detailed in the provided text.

pith-pipeline@v0.9.0 · 5668 in / 1133 out tokens · 41358 ms · 2026-05-18T11:44:34.138934+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

integrates a set transformer embedding ... conditional variational autoencoder ... minimizing the negative ELBO

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.