Multi-output physics-informed neural networks for forward and inverse PDE problems with uncertainties

Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network (MO-PINN) which can provide solutions with uncertainty distributions for both forward and inverse PDE problems with noisy data. In this framework, the uncertainty arising from the noisy data is first translated into multiple measurements regarding the prior noise distribution using the bootstrap method, and then the outputs of neural networks are designed to satisfy the measurements as well as the underlying physical laws.The posterior estimation of target parameters can be obtained at the end of training, which can be further used for uncertainty quantification and decision making. In this paper, MO-PINNs are demonstrated with a series of numerical experiments including both linear and nonlinear, forward and inverse problems. The results show that MO-PINN is able to provide accurate predictions with noisy data.In addition, we also demonstrate that the prediction and posterior distributions from MO-PINNs are consistent with the solutions from traditional a finite element method (FEM) solver and Monte Carlo methods given the same data and prior knowledge. Finally, we show that additional statistical knowledge can be incorporated into the training to improve the prediction if available.