Complex error minimization algorithm with adaptive change rate

In this paper, we develop a new algorithm for estimating control parameter values of ordinary differential equation (ODE) models. We call it the complex error minimization algorithm with an adaptive rate of change. It efficiently determines the parameters of ODE models even over extremely short time series. The algorithm is based on the gradient method and successfully solves its major drawback, namely getting stuck in local minima. To detect the getting stuck, we first introduce the calculation of four different errors at once. Our studies show that they achieved the global minimum only when we observe simultaneous minimization of all these errors. When getting stuck in local minima takes place, only one (or several) errors are minimized, while the others remain unchanged or grow. Thus, the algorithm accurately detects getting stuck in local minima. After this, it applies special methods based on simulated annealing to escape from local minima. The algorithm selects the most optimal path for convergence of parameter values to the global minimum by applying random restarts and multi-start methods to avoid local minima. We have tested our method on models with regular and chaotic dynamics and have found that it can work with high accuracy in these cases. Our method is suitable for actual systems with complex dynamics and an extensive set of control parameters but short time series. Thus, the proposed algorithm is an effective method for solving the inverse problem of ODE models.