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

• Published in: Journal of computational physics Vol. 462; no. C; p. 111230
• Main Authors: Li, Jing, Tartakovsky, Alexandre M.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.08.2022; Elsevier Science Ltd; Elsevier
• Subjects: Approximation; Artificial neural networks; Computational physics; Conditional Karhunen-Loéve expansions; Cost function; Deep neural networks; Differential equations; Inverse problems; Machine learning; Neural networks; Parameter estimation; Physics; Physics-informed machine learning; Regularization; Time dependence; Physics-informed machine learning; Conditional Karhunen-Loéve expansions; Parameter estimation; Inverse problems; Deep neural networks
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111230

•Unknown parameters are modeled with conditional Karhunen-Loéve (CKL) expansions to enforce known correlation structures.•State variable is approximated with a neural network, which is trained jointly with CKLs subject to differential equation constraints (DEC).•The proposed method is more accurate than the physics-informed neural network (PINN) method, which only enforces DEC. 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.