Heteroscedastic uncertainty quantification in Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) provide a machine learning framework to solve differential equations. However, PINNs do not inherently consider measurement noise or model uncertainty. In this paper, we propose the UQ-PINN which is an extension of the PINN with additional outputs to approximate the additive noise. The multi-output architecture enables approximation the mean and standard deviation over data using negative Gaussian log-likelihood loss. The performance of the UQ-PINN is demonstrated on the Poisson equation with additive noise.