  This paper presents a novel data-driven approach to identify partial
differential equation (PDE) parameters of a dynamical system. Specifically, we
adopt a mathematical "transport" model for the solution of the dynamical system
at specific spatial locations that allows us to accurately estimate the model
parameters, including those associated with structural damage. This is
accomplished by means of a newly-developed mathematical transform, the signed
cumulative distribution transform (SCDT), which is shown to convert the general
nonlinear parameter estimation problem into a simple linear regression. This
approach has the additional practical advantage of requiring no a priori
knowledge of the source of the excitation (or, alternatively, the initial
conditions). By using training data, we devise a coarse regression procedure to
recover different PDE parameters from the PDE solution measured at a single
location. Numerical experiments show that the proposed regression procedure is
capable of detecting and estimating PDE parameters with superior accuracy
compared to a number of recently developed machine learning methods.
Furthermore, a damage identification experiment conducted on a publicly
available dataset provides strong evidence of the proposed method's
effectiveness in structural health monitoring (SHM) applications. The Python
implementation of the proposed system identification technique is integrated as
a part of the software package PyTransKit
(https://github.com/rohdelab/PyTransKit).
