Combining recent moment and sparse semidefinite programming (SDP) relaxation
techniques, we propose an approach to find smooth approximations for solutions
of problems involving nonlinear differential equations. Given a system of
nonlinear differential equations, we apply a technique based on finite
differences and sparse SDP relaxations for polynomial optimization problems
(POP) to obtain a discrete approximation of its solution. In a second step we
apply maximum entropy estimation (using moments of a Borel measure associated
with the discrete solution) to obtain a smooth closed-form approximation. The
approach is illustrated on a variety of linear and nonlinear ordinary
differential equations (ODE), partial differential equations (PDE) and optimal
control problems (OCP), and preliminary numerical results are reported
