mlsanityData to 7 Oct 2026

Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks

arXiv 2108.13054 ↗First posted 30 Aug 2021math.NA cs.LG cs.NAcited by 16

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Abstract (arXiv)

In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of samples are sufficiently taken. According to our established error bound, we also find that our physics-informed WGANs have higher requirement for the capacity of discriminators than that of generators. Numerical results on synthetic examples of partial differential equations are reported to validate our theoretical results and demonstrate how uncertainty quantification can be obtained for solutions of partial differential equations and the distributions of initial/boundary data. However, the quality or the accuracy of the uncertainty quantification theory in all the points in the interior is still the theoretical vacancy, and required for further research.

Key facts. Cited by 16 papers in the map (data through 7 Oct 2026). Builds on 37 of its 68 references that are in the map. First posted on arXiv 30 Aug 2021.

Builds on

37 of 68 references.

References · 37

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 2; 2024: 2; 2025: 2; 2026: 3.

16 papers.

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  1. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 169
  2. 2023A Survey on Uncertainty Quantification Methods for Deep Learningcited by 78
  3. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  4. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  5. 2022A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problemscited by 20
  6. 2022Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfirescited by 10
  7. 2022A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacousticcited by 7
  8. 2022Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problemscited by 7
  9. 2024A local squared Wasserstein-2 method for efficient reconstruction of models with uncertaintycited by 2
  10. 2025SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertaintycited by 2
  11. 2023PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantificationcited by 1
  12. 2022Solving Coupled Differential Equation Groups Using PINO-CDEcited by 0
4 more
  1. 2025A generalized Wasserstein-2 distance approach for efficient reconstruction of random field models using stochastic neural networkscited by 0
  2. 2026A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equationscited by 0
  3. 2026Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEscited by 0
  4. 2026Structure-preserving uncertainty quantification for GENERIC dynamicscited by 0