Peridynamics enabled learning partial differential equations, by Prof. Erdogan Madenci

Описание: Title: Peridynamics enabled learning partial differential equations

Speaker: Erdogan Madenci, Professor
Department of Aerospace and Mechanical Engineering, the University of Arizona.

Abstract: This study presents an approach to discover the significant terms in partial differential equations (PDEs) that describe particular phenomena based on the measured data. A sparse linear regression algorithm enabled by the peridynamic differential operator (PDDO) leads to a robust approach for describing a given time series data set as a PDE. The PDEs are approximated by constructing a feature matrix, velocity vector and unknown coefficient vector. Each candidate term (derivatives) appearing in the feature matrix is evaluated numerically by using the PDDO. The numerical results suggest that the PDDO may offer a powerful framework for discovery of PDEs representing data with significant noise. The PD functions can readily be constructed and incorporated into any existing regression algorithm. The solution to the regression model with regularization is achieved through Douglas-Rachford algorithm which is based on proximal operators. This coupling performs well due to their robustness to noisy data and the calculation of accurate derivatives. Its effectiveness is demonstrated by considering several fabricated data associated with challenging nonlinear PDEs such as Burgers, Swift-Hohenberg, Korteweg-de Vries, Kuramoto-Sivashinsky, nonlinear Schrödinger and Cahn-Hilliard equations. The solutions to these equations are constructed with ground truth and recovered coefficients by employing simple discretization and implicit solvers. The field data are divided into training and testing sets to examine the predictive capability of the recovered coefficients for different values of regularization parameters. A transition occurs in the form of a jump in the variation of accuracy metric for a particular value of the regularization parameter. It indicates the change from the correct form of the PDE to its reduced form. Its value can be chosen accordingly in order to ensure the correct form of PDEs during the learning process. This is a joint work with Ali C. Bekar.


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