Two-Step Bayesian PINNs for Uncertainty Estimation

We use a two-step procedure to train Bayesian neural networks that provide uncertainties over the solutions to differential equation (DE) systems provided by Physics-Informed Neural Networks (PINNs). We take advantage of available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the uncertainties obtained to improve parameter estimation in inverse problems in the fields of cosmology and fermentation.