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We present a new physics-informed machine learning approach for the inversionof PDE models with heterogeneous parameters. In our approach, thespace-dependent partially-observed parameters and states are approximated viaKarhunen-Lo\`eve expansions (KLEs). Each of these KLEs is then conditioned ontheir corresponding measurements, resulting in low-dimensional models of theparameters and states that resolve observed data. Finally, the coefficients ofthe KLEs are estimated by minimizing the norm of the residual of the PDE modelevaluated at a finite set of points in the computational domain, ensuring thatthe reconstructed parameters and states are consistent with both theobservations and the PDE model to an arbitrary level of accuracy.  In our approach, KLEs are constructed using the eigendecomposition ofcovariance models of spatial variability. For the model parameters, we employ aparameterized covariance model calibrated on parameter observations; for themodel states, the covariance is estimated from a number of forward simulationsof the PDE model corresponding to realizations of the parameters drawn fromtheir KLE. We apply the proposed approach to identifying heterogeneouslog-diffusion coefficients in diffusion equations from spatially sparsemeasurements of the log-diffusion coefficient and the solution of the diffusionequation. We find that the proposed approach compares favorably againststate-of-the-art point estimates such as maximum a posteriori estimation andphysics-informed neural networks.

Physics-Informed Machine Learning with Conditional Karhunen-Lo\`eve Expansions