Determining the Ordinary Differential Equation From Noisy Data

Abstract

A challenging inverse problem is to identify the smooth function and the differential equation it represents from uncertain data. This paper extends the procedure previously developed for smooth data. The approach involves two steps. In the first step the data is smoothed using a recursive Bezier filter. For smooth data a single application of the filter is sufficient. The final set of data points provides a smooth estimate of the solution. More importantly, it will also identify smooth derivatives of the function away from the edges of the domain. In the second step the values of the function and its derivatives are used to establish a specific form of the differential equation from a particular class of the same. Since the function and its derivatives are known, the only unknowns are parameters describing the structure of the differential equations. These parameters are of two kinds: the exponents of the derivatives and the coefficients of the terms in the differential equations. These parameters can be determined by defining an optimization problem based on the residuals in a reduced domain. To avoid the trivial solution a discrete global search is used to identify these parameters. An example involving a third order constant coefficient linear differential equation is presented. A basic simulated annealing algorithm is used for the global search. Once the differential form is established, the unknown initial and boundary conditions can be obtained by backward and forward numerical integration from the reduced region.