  We propose a randomized physics-informed neural network (PINN) or rPINN
method for uncertainty quantification in inverse partial differential equation
(PDE) problems with noisy data. This method is used to quantify uncertainty in
the inverse PDE PINN solutions. Recently, the Bayesian PINN (BPINN) method was
proposed, where the posterior distribution of the PINN parameters was
formulated using the Bayes' theorem and sampled using approximate inference
methods such as the Hamiltonian Monte Carlo (HMC) and variational inference
(VI) methods. In this work, we demonstrate that HMC fails to converge for
non-linear inverse PDE problems. As an alternative to HMC, we sample the
distribution by solving the stochastic optimization problem obtained by
randomizing the PINN loss function. The effectiveness of the rPINN method is
tested for linear and non-linear Poisson equations, and the diffusion equation
with a high-dimensional space-dependent diffusion coefficient. The rPINN method
provides informative distributions for all considered problems. For the linear
Poisson equation, HMC and rPINN produce similar distributions, but rPINN is on
average 27 times faster than HMC. For the non-linear Poison and diffusion
equations, the HMC method fails to converge because a single HMC chain cannot
sample multiple modes of the posterior distribution of the PINN parameters in a
reasonable amount of time.
