A principled distance-aware uncertainty quantification approach for enhancing the reliability of physics-informed neural network

• Published in: Reliability engineering & system safety Vol. 245; p. 109963
• Main Authors: Li, Jinwu, Long, Xiangyun, Deng, Xinyang, Jiang, Wen, Zhou, Kai, Jiang, Chao, Zhang, Xiaoge
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2024
• Subjects: Gaussian process; Model uncertainty; PINN; Spectral normalization; Uncertainty quantification; Uncertainty quantification; Gaussian process; Spectral normalization; PINN; Model uncertainty
• ISSN: 0951-8320; 1879-0836
• DOI: 10.1016/j.ress.2024.109963

•Develop a principled approach to estimate model uncertainty of physics-informed neural network (PINN).•Define performance measures tailored to PINN when evaluating the quality of estimated uncertainty.•The proposed method outperforms Monte Carlo dropout in detecting PINN's prediction failure. Physics-Informed Neural Network (PINN) is a special type of deep learning model that encodes physical laws in the form of partial differential equations as a regularization term in the loss function of neural network. In this paper, we develop a principled uncertainty quantification approach to characterize the model uncertainty of PINN, and the estimated uncertainty is then exploited as an instructive indicator to identify collocation points where PINN produces a large prediction error. To this end, this paper seamlessly integrates spectral-normalized neural Gaussian process (SNGP) into PINN for principled and accurate uncertainty quantification. In the first step, we apply spectral normalization on the weight matrices of hidden layers in the PINN to make the data transformation from input space to the latent space distance-preserving. Next, the dense output layer of PINN is replaced with a Gaussian process to make the quantified uncertainty distance-sensitive. Afterwards, to examine the performance of different UQ approaches, we define several performance metrics tailored to PINN for assessing distance awareness in the measured uncertainty and the uncertainty-informed error detection capability. Finally, we employ three representative physical problems to verify the effectiveness of the proposed method in uncertainty quantification of PINN and compare the developed approach with Monte Carlo (MC) dropout using the developed performance metrics. Computational results suggest that the proposed approach exhibits a superior performance in improving the prediction accuracy of PINN and the estimated uncertainty serves as an informative indicator to detect PINN's prediction failures.