  In this paper, we propose the Adaptive Physics-Informed Neural Networks
(APINNs) for accurate and efficient simulation-free Bayesian parameter
estimation via Markov-Chain Monte Carlo (MCMC). We specifically focus on a
class of parameter estimation problems for which computing the likelihood
function requires solving a PDE. The proposed method consists of: (1)
constructing an offline PINN-UQ model as an approximation to the forward model;
and (2) refining this approximate model on the fly using samples generated from
the MCMC sampler. The proposed APINN method constantly refines this approximate
model on the fly and guarantees that the approximation error is always less
than a user-defined residual error threshold. We numerically demonstrate the
performance of the proposed APINN method in solving a parameter estimation
problem for a system governed by the Poisson equation.
