Flow reconstruction with uncertainty quantification from noisy measurements based on Bayesian physics-informed neural networks

Flow reconstruction based on limited measurement data, which can be considered as a state estimation problem, constitutes a fundamental task within the realm of fluid mechanics. In recent years, the physics-informed neural networks (PINNs) have been proposed to achieve flow field reconstruction by integrating the measurements with governing equations during network training. However, the performance is compromised by the presence of high-level data noise, and the uncertainty of the reconstructed flow fields remains unattainable. In this paper, we first perform a systematic study to investigate the impact of data noise on the reconstruction result of PINNs. Subsequently, we present strategies of early stopping and loss regularization, which can suppress the overfitting issue to some extent. Ensemble learning is also employed to quantify the uncertainty of the results from vanilla PINNs. In addition, we propose to use a Bayesian framework of PINNs (BPINNs) for flow field reconstruction, which incorporates the Bayesian neural network with PINNs. It is demonstrated that BPINNs are capable of reconstructing the velocity and pressure fields from sparse and noisy velocity measurements, while providing comprehensive uncertainty quantification of the flow fields simultaneously. Compared to the vanilla PINNs, BPINNs are more accurate and robust when there is a high level of data noise. We conduct experiments on two-dimensional cavity flow and the flow past a cylinder to validate the effectiveness of the proposed methods throughout the paper.