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

arXiv: https://arxiv.org/abs/2604.17156 · HTML: https://mlsanity.com/paper/2604.17156/

## 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 (most cited first, 21 of 21)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Quality of Uncertainty Quantification for Bayesian Neural Network Inference](https://mlsanity.com/paper/1906.09686/) (2019)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [A statistical theory of cold posteriors in deep neural networks](https://mlsanity.com/paper/2008.05912/) (2020)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning](https://mlsanity.com/paper/2404.08809/) (2024)
- [NekRS, a GPU-Accelerated Spectral Element Navier-Stokes Solver](https://mlsanity.com/paper/2104.05829/) (2021)
- [Repulsive Ensembles for Bayesian Inference in Physics-informed Neural Networks](https://mlsanity.com/paper/2505.17308/) (2025)

## Built on by (most cited first, 4 of 4)

- [Functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks](https://mlsanity.com/paper/2605.07060/) (2026)
- [Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs](https://mlsanity.com/paper/2605.08672/) (2026)
- [Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces](https://mlsanity.com/paper/2608.06428/) (2026)
- [Uncovering and Fixing Collider Bias in Bayesian PINNs](https://mlsanity.com/paper/2610.11737/) (2026)

## 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.