Learning the inhomogenous term of a linear ODE

Abstract

A new method to explain the action of a non-interpretable predictive model correcting the homogeneous solution of a linear ordinary differential equation (ODE) to fit given observations is presented. The prediction correcting a particular solution is ’explained’ by an estimate of the ODE's inhomogeneous term. The method uses the ODE to establish an alternate description of the predicted solution and thus belongs to the surrogate modeling approaches within the explainable AI (XAI) method families. Useful applications include processes which allow fair modeling by a homogeneous linear differential equation but have potential to be improved by including an inhomogeneous term. A study of perturbed heat transfer described by Newton's cooling law illustrates the advantages of explanations given in the form of terms of a differential equation.