mlsanityData to 14 Oct 2026

Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles

arXiv 2604.17156 ↗First posted 18 Apr 2026cs.LG physics.comp-phcited by 4

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Abstract (arXiv)

Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems such as data-driven Reynolds-averaged Navier-Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi-component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE-constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function-space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade-offs between accuracy, computational cost, and uncertainty calibration in physics-informed learning, and offer practical guidance for uncertainty quantification in data-driven turbulence modeling.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 21 of its 49 references that are in the map. First posted on arXiv 18 Apr 2026.

Builds on

21 of 49 references.

References · 21

  1. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  2. 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
  3. 2018Accurate Uncertainties for Deep Learning Using Calibrated Regressioncited by 422
  4. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  5. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  6. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  7. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  8. 2021Repulsive Deep Ensembles are Bayesiancited by 104
  9. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 68
  10. 2019Quality of Uncertainty Quantification for Bayesian Neural Network Inferencecited by 59
  11. 2020Multi-fidelity Bayesian Neural Networks: Algorithms and Applicationscited by 59
  12. 2020A statistical theory of cold posteriors in deep neural networkscited by 54
9 more
  1. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
  2. 2021Learning Functional Priors and Posteriors from Data and Physicscited by 37
  3. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  4. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  5. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  6. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  7. 2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
  8. 2021NekRS, a GPU-Accelerated Spectral Element Navier-Stokes Solvercited by 8
  9. 2025Repulsive Ensembles for Bayesian Inference in Physics-informed Neural Networkscited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networkscited by 0
  2. 2026Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEscited by 0
  3. 2026Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spacescited by 0
  4. 2026Uncovering and Fixing Collider Bias in Bayesian PINNscited by 0