Randomized physics-informed machine learning for uncertainty quantification in high-dimensional inverse problems

We propose the randomized physics-informed conditional Karhunen-Loève expansion (rPICKLE) method for uncertainty quantification in high-dimensional inverse problems. In rPICKLE, the states and parameters of the governing partial differential equation (PDE) are approximated via truncated conditional Karhunen-Loève expansions (cKLEs). Uncertainty in the inverse solution is quantified via the posterior distribution of cKLE coefficients formulated with independent standard normal priors and a likelihood containing PDE residuals evaluated over the computational domain. The maximum a posteriori (MAP) estimate of the cKLE coefficients is found by minimizing a loss function given (up to a constant) by the negative log posterior. The posterior is sampled by adding zero-mean Gaussian noises into the MAP loss function and minimizing the loss for different noise realizations. For linear and low-dimensional nonlinear problems, we show that the rPICKLE posterior converges to the true Bayesian posterior. For high-dimensional non-linear problems, we obtain rPICKLE posterior approximations with high log-predictive probability. For a low-dimensional problem, the traditional Hamiltonian Monte Carlo (HMC) and Stein Variational Gradient Descent (SVGD) methods yield similar (to rPICKLE) posteriors. However, both HMC and SVGD fail for the high-dimensional problem. These results demonstrate the advantages of rPICKLE for approximately sampling high-dimensional posterior distributions.