Optimal prediction with memory

Optimal prediction methods estimate the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their effect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation, the orthogonal dynamics equation, needed to evaluate a non-Markovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. We present the constructions in detail together with several useful variants, provide simple examples, and point out the relation to the fluctuation–dissipation formulas of statistical physics.