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

2026-04-18

AI Summary

This work addresses the insufficient quantification of epistemic uncertainty in physics-informed neural networks (PINNs) for turbulence modeling by proposing a probabilistic PINNs framework for data-driven inverse problems in Reynolds-averaged Navier–Stokes (RANS) equations. The approach integrates Bayesian inference, Monte Carlo dropout, and a function-space repulsive deep ensemble, augmented with a temperature-scaled multi-component likelihood and physics-constrained loss terms. Notably, diversity among ensemble members is enhanced through a function-space repulsion mechanism, significantly improving the calibration of uncertainty estimates in turbulent flow reconstruction under sparse data conditions. In benchmark cases such as flow past a cylinder, Bayesian PINNs yield the most reliable uncertainty quantification, while the repulsive ensemble achieves high-fidelity predictions of primary flow variables at substantially reduced computational cost.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

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.

Problem

• Uncertainty Quantification
• Physics-informed Neural Networks
• Turbulent Flows
• Epistemic Uncertainty
• Inverse Problems

Innovation

• Bayesian PINNs
• Uncertainty Quantification
• Repulsive Ensembles
• Likelihood Tempering
• Turbulent Flow Modeling