Bayesian Physics-Informed Deep Learning for Robust Modeling under Uncertainty

Abstract

Physics-Informed Neural Networks (PINNs) have received a lot of interest in the context of forward and inverse problems of partial differential equations (PDEs). But traditional PINNs are deterministic and do not quantify uncertainty due to noisy observations, incomplete boundary conditions, and model discrepancies. This empowers them in real-life scientific and engineering systems. To fill this gap, the current paper suggests a Bayesian Physics-Informed Deep Learning (BPIDL) model that combines both Bayesian inference with physics-constrained neural network as the solution to a sound uncertainty-aware Bayesian modeling. The given solution presents probabilistic weight distributions and uses variational inference to estimate the posterior, allowing the estimation of both epistemic and aleatoric uncertainties at the same time. A single loss is designed as a combination of data fidelity, physics residual constraints and Kullback-Rosenberger divergence-regularization. The framework is assessed on nonlinear PDE tests with sparse and noisy data. Experimentally, it is shown that BPIDL has lower prediction error, and much better uncertainty calibration than standard PINNs and standard deep learning models. Moreover, the suggested approach is more resilient to data distortions and model error. The results of these experiments indicate that BPIDL is an effective and scalable model of uncertainty quantification in physics-informed machine learning tasks.