Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems

We present the PI-CKL-NN method for parameter estimation in differential equation (DE) models given sparse measurements of the parameters and states. In the proposed approach, the space- or time-dependent parameters are approximated by Karhunen-Loéve (KL) expansions that are conditioned on the parameters' measurements, and the states are approximated by deep neural networks (DNNs). The unknown weights in the KL expansions and DNNs are found by minimizing the cost function that enforces the measurements of the states and the DE constraint. Regularization is achieved by adding the l₂ norm of the conditional KL coefficients into the loss function. Our approach assumes that the parameter fields are correlated in space or time and enforces the statistical knowledge (the mean and the covariance function) in addition to the DE constraints and measurements as opposed to the physics-informed neural network (PINN) and other similar physics-informed machine learning methods where only DE constraints and data are used for parameter estimation. We use the PI-CKL-NN method for parameter estimation in an ordinary differential equation with an unknown time-dependent parameter and the one- and two-dimensional partial differential diffusion equations with unknown space-dependent diffusion coefficients. We also demonstrate that PI-CKL-NN is more accurate than the PINN method, especially when the observations of the parameters are very sparse.