Macroscopic fluctuation theory has shown that a wide class of non-equilibrium stochastic dynamical systems obey a large deviation principle, but except for a few one-dimensional (1D) examples these large deviation principles are in general not known in closed form. We consider the problem of constructing successive approximations to an (unknown) large deviation functional and show that the non-equilibrium probability distribution then takes a Gibbs-Boltzmann form with a set of auxiliary (non-physical) energy functions. The expectation values of these auxiliary energy functions and their conjugate quantities satisfy a closed system of equations which can imply a considerable reduction of dimensionality of the dynamics. We show that the accuracy of the approximations can be tested self-consistently without solving the full non-equilibrium dynamical equations. We test the general procedure on the simple model problem of a relaxing 1D Ising chain.