  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.
