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Deep set based operator learning with uncertainty quantification

Lei Ma, Ling Guo, Hao Wu, Tao Zhou

TL;DR

UQ-SONet addresses the challenge of learning operators from irregular, sparse observations with built-in uncertainty quantification. By fusing a set-transformer embedding for permutation-invariant sensor data with a conditional variational autoencoder, it models the conditional distribution of the output given incomplete observations, applicable to both deterministic and stochastic PDEs, including Navier–Stokes. The approach achieves accurate operator approximation while providing principled uncertainty estimates, and it outperforms prior permutation-invariant methods like VIDON in uncertainty-aware scenarios. This framework enables robust operator learning in practical settings with noisy data and irregular sampling, with potential for active sensor optimization and long-horizon predictions in time-dependent problems.

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 (UQ), and are thus limited in practical applicability. Recent permutationinvariant extensions, such as the Variable-Input Deep Operator Network (VIDON), 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 UQ. Our model integrates a set transformer embedding to handle sparse and variable sensor locations, and employs a conditional variational autoencoder (cVAE) 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.

Deep set based operator learning with uncertainty quantification

TL;DR

UQ-SONet addresses the challenge of learning operators from irregular, sparse observations with built-in uncertainty quantification. By fusing a set-transformer embedding for permutation-invariant sensor data with a conditional variational autoencoder, it models the conditional distribution of the output given incomplete observations, applicable to both deterministic and stochastic PDEs, including Navier–Stokes. The approach achieves accurate operator approximation while providing principled uncertainty estimates, and it outperforms prior permutation-invariant methods like VIDON in uncertainty-aware scenarios. This framework enables robust operator learning in practical settings with noisy data and irregular sampling, with potential for active sensor optimization and long-horizon predictions in time-dependent problems.

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 (UQ), and are thus limited in practical applicability. Recent permutationinvariant extensions, such as the Variable-Input Deep Operator Network (VIDON), 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 UQ. Our model integrates a set transformer embedding to handle sparse and variable sensor locations, and employs a conditional variational autoencoder (cVAE) 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.

Paper Structure

This paper contains 17 sections, 20 equations, 9 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Methodology
  4. Set transformer embedding
  5. Encoder and decoder
  6. Loss function
  7. Simulation Results
  8. One-dimensional Diffusion Equation.
  9. Two-dimensional Poisson Equation.
  10. One-dimensional Stochastic Differential Equation.
  11. Two-dimensional Stochastic Differential Equation.
  12. Navier-Stokes Equation.
  13. Conclusion
  14. Appendix A
  15. Hyperparameters and neural network architectures used in the numerical examples
  16. ...and 2 more sections

Figures (9)

  • Figure 1: Schematic of the UQ-SONet model. The top figure illustrates the set transformer embedding structure, where $h(\hbox{$\mathcal{O}$})$ is obtained by incorporating the coordinates and values of the input function sensors. The bottom figure depicts the encoder–decoder architecture, which derives the latent variables and reconstructs the target function of the operator.
  • Figure 2: Diffusion Equation \ref{['eq:1d']}.(a): Wasserstein-2 (W2) distance between the reference distribution and the conditional distribution obtained by UQ-SONet, with $m=3$ and $m=5$ (red). (b): Relative $L_2$ errors of the mean and standard deviation between UQ-SONet predictions and reference values with $m=3$ (left) and $m=5$ (right); (c): Mean of the standard deviations of conditional distributions generated by UQ-SONet compared with reference values. Bars indicate results averaged over five independent experiments.
  • Figure 3: Diffusion Equation \ref{['eq:1d']}. Reference solution versus the prediction of UQ-SONet for a representative example with $m=2$ (top left), $m=4$ (top right), $m=7$ (bottom left), $m=10$ (bottom right). Black dashed lines show sampled input values, and red markers indicate sensor locations. Red and blue dashed lines with shaded regions denote the mean and standard deviation of the reference and predicted solutions, respectively.
  • Figure 4: Diffusion Equation \ref{['eq:1d']}. Mean and standard deviation of the conditional distribution obtained by UQ-SONet compared with reference values for $3$ (top) and $7$ (bottom) input sensors under noise conditions: $\epsilon \sim \mathcal{N}(0, 0.5^2)$ (left) and $\epsilon \sim \mathcal{N}(0, 1.0^2)$ (right). Red dots mark the noise-contaminated observations at the corresponding sensor locations. Red and blue curves denote the reference and predicted solutions, respectively, while black dashed lines indicate input samples.
  • Figure 5: 2d Poisson Equation \ref{['eqn:2d-pde']}.Left: reference samples of the two-dimensional operator map. Red markers denote the sensor locations. Top row: reference mean, UQ-SONet predicted mean, and the corresponding absolute error. Bottom row: reference standard deviation, UQ-SONet predicted standard deviation, and the corresponding absolute error.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 3.1