Karhunen-Loève deep learning method for surrogate modeling and approximate Bayesian parameter estimation

In the surrogate model, the Karhunen–Loève (KL) expansions are used for the dimensionality reduction of the number of unknown parameters and variables, and a deep neural network is employed to relate the reduced space of parameters to that of the state variables.

The KL-DNN surrogate model is used to formulate a maximum-aposteriori-like least-squares problem, which is randomized to draw samples of the posterior distribution of the parameters. We test the proposed framework for a hypothetical unconfined aquifer via comparison with the forward MODFLOW and inverse PEST++ iterative ensemble smoother (IES) solutions as well as the state-of-theart Fourier neural operator (FNO) and deep operator networks (DeepONets) operator learning surrogate models.