PARAMETER ESTIMATION FOR ODES USING A CROSS-ENTROPY APPROACH

Parameter estimation for ODEs is an important topic in numerical analysis. In this paper, we present a novel approach to address this inverse problem that can be applied to differential equations that may include delay terms. Cross-entropy algorithms are general algorithms which can be applied to solve global optimization problems. The main steps of cross-entropy methods are first to generate a set of trial samples from a certain distribution and then to update the distribution based on these generated sample trials. To overcome the prohibitive computation of standard cross-entropy algorithms, we develop a modification combining local search techniques. The modified cross-entropy algorithm can improve the convergence rate and reduce the chances of converging to a local optimum. Two different coding schemes (continuous coding and discrete coding) are introduced (to represent the search space that we are optimizing over). Continuous coding uses a truncated multivariate Gaussian to generate trial samples, while discrete coding reduces the search space to consider only a finite (but relatively dense) subset of the feasible parameter values and uses a Bernoulli distribution to generate the trial samples (which are fixed point approximations to the parameters). Extensive numerical experiments are conducted to illustrate the power and advantages of the proposed methods. Compared to other existing state-of-the-art approaches on some benchmark problems for parameter estimation, our methods have three main advantages: (1) they are robust to noise in the data to be fitted; (2) they are not sensitive to the number of observation points; and (3) the modified versions exhibit faster convergence without sacrificing accuracy.