  We propose a physics-informed machine learning method for uncertainty
quantification in high-dimensional inverse problems. In this method, the states
and parameters of partial differential equations (PDEs) are approximated with
truncated conditional Karhunen-Lo\`eve expansions (CKLEs), which, by
construction, match the measurements of the respective variables. The maximum a
posteriori (MAP) solution of the inverse problem is formulated as a
minimization problem over CKLE coefficients where the loss function is the sum
of the norm of PDE residuals and the $\ell_2$ regularization term. This MAP
formulation is known as the physics-informed CKLE (PICKLE) method. Uncertainty
in the inverse solution is quantified in terms of the posterior distribution of
CKLE coefficients, and we sample the posterior by solving a randomized PICKLE
minimization problem, formulated by adding zero-mean Gaussian perturbations in
the PICKLE loss function. We call the proposed approach the randomized PICKLE
(rPICKLE) method.
  For linear and low-dimensional nonlinear problems (15 CKLE parameters), we
show analytically and through comparison with Hamiltonian Monte Carlo (HMC)
that the rPICKLE posterior converges to the true posterior given by the Bayes
rule. For high-dimensional non-linear problems with 2000 CKLE parameters, we
numerically demonstrate that rPICKLE posteriors are highly informative--they
provide mean estimates with an accuracy comparable to the estimates given by
the MAP solution and the confidence interval that mostly covers the reference
solution. We are not able to obtain the HMC posterior to validate rPICKLE's
convergence to the true posterior due to the HMC's prohibitive computational
cost for the considered high-dimensional problems. Our results demonstrate the
advantages of rPICKLE over HMC for approximately sampling high-dimensional
posterior distributions subject to physics constraints.
