Comparing Two Ways of Inferring a Differential Equation Model Via Grammar-Based Immune Programming

Abstract

An ordinary differential equation (ODE) is a mathematical form to describe physical or biological systems composed by time-derivatives of physical positions or chemical concentrations as a function of its current state. Given observed pairs, a relevant modeling problem is to find the symbolic expression of a differential equation which mathematically describes the concerned phenomenon.
The Grammar-based Immune Programming (GIP) is a method for evolving programs in an arbitrary language by immunological inspiration. A program can be a computer program, a numerical function in symbolic form, or a candidate design, such as an analog circuit. GIP can be used to solve symbolic regression problems in which the objective is to find an analytical expression of a function that better fits a given data set.
At least two ways are available to solve model inference problems in the case of ordinary differential equations by means of symbolic regression techniques. The first one consists in taking numerical derivatives from the given data obtaining a set of approximations. Then a symbolic regression technique can be applied to these approximations. Another way is to numerically integrate the ODE corresponding to the candidate solution and compare the results with the observed data.
Here, by means of numerical experiments, we compare the relative performance of these two ways to infer models using the GIP method.