Physics-Informed Neural Networks Embedded Bayesian Framework for Longitudinal Tunnel Performance Analysis

Physics-Informed Neural Networks (PINNs) have gained significant attention in geotechnical engineering in recent years. Their high efficiency in solving complex partial differential equations (PDEs) and estimating unknown parameters with limited observational data make PINNs particularly suitable and practical for geotechnical applications. However, when dealing with uncertain inputs, standard PINNs trained using nominal values of input parameters may become inaccurate. Moreover, PINNs for solving standard inverse problems lack built-in uncertainty quantification capabilities. These limitations make it challenging to account for uncertainties in many applications, leading to potentially inaccurate results, especially in scenarios with large measurement noise or strong parameter variability. In this paper, we present a physics-informed neural networks embedded Bayesian framework to address these limitations. We demonstrate its application to a practical problem in tunnel engineering: estimating the posterior distributions of the uncertain inputs and characterizing the longitudinal tunnel performance while considering the parameter variability, thereby quantifying the uncertainty arising from input parameter variations, noisy measurements and the physical modelling process. To this end, a parameterized physics-informed neural network is first trained by embedding the governing equations of the soil-tunnel interaction. This surrogate model is trained using an ensemble of tunnel performances generated from a set of realizations of the uncertain inputs. Then Bayesian updating is conducted to estimate the posterior distributions based on the surrogate model and on-site measurements. The effectiveness of the proposed framework is demonstrated through comparison with results from Bayesian updating using a standard numerical model. The results highlight the advantages of the framework in capturing uncertainties and improving predictive performance, while decreasing computational cost, showcasing its potential for practical applications in geotechnical engineering.

Keywords: Tunnel longitudinal performance, Parameterized Physics-informed neural networks, Bayesian updating, Uncertain quantification.