Least squares estimation of uncertain partial differential equations

Since parameter estimation of uncertain partial differential equations (UPDEs) is a classical problem with broad applications and relevance across various practical domains, this study introduces a novel approach for estimating unknown parameters, combining least squares estimation with the residuals of UPDEs. At first, analytical formulae of the residuals are established, considering that the residual serves as a link between UPDEs and least squares estimation. Subsequently, the uncertain least squares estimation of UPDEs is performed and is verified using an empirical example. Moreover, uncertain least squares estimation is performed to model the male population density in Chinese cities, with uncertain hypothesis test conducted to evaluate the fitness of the estimated parameters.