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

arXiv cs.LG / 4/21/2026

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Key Points

The paper addresses a key limitation of standard Physics-Informed Neural Networks (PINNs) by introducing probabilistic PINN variants that can quantify epistemic uncertainty for ill-posed PDE inverse problems in turbulence modeling.
It proposes and evaluates three uncertainty-focused approaches: Bayesian PINNs using Hamiltonian Monte Carlo with tempered multi-component likelihood, Monte Carlo dropout, and repulsive deep ensembles designed to enforce diversity in function space.
The authors highlight that ensemble diversity and likelihood tempering substantially improve uncertainty calibration, especially for PDE-constrained inverse problems.
Experiments on multiple test cases—including the Van der Pol oscillator and turbulent flow past a circular cylinder at Re=3,900 (DNS) and Re=10,000 (experimental PIV)—show Bayesian PINNs yield the most consistent uncertainty estimates across inferred variables.
Repulsive deep ensembles are presented as a more computationally efficient alternative that can achieve competitive accuracy for the main flow variables, enabling practical trade-off guidance between accuracy, cost, and calibration quality.

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.