Bayesian Solution of Ordinary Differential Equations

Abstract

In the numerical solution of ordinary differential equations, a function y(x) is to be reconstructed from knowledge of the functional form of its derivative: dy/dx = f (x, y), together with an appropriate boundary condition. The derivative f is evaluated at a sequence of suitably chosen points (xk, yk), from which the form of y(•) is estimated. This is an inference problem, which can and perhaps should be treated by Bayesian techniques. As always, the inference appears as a probability distribution prob(y(•)), from which random samples show the probabilistic reliability of the results. Examples are given.