Non-Markovian Optimal Prediction
Alexandre J. Chorin, Ole H. Hald, Raz Kupferman
TL;DR
This work tackles predicting nonlinear, underresolved dynamical systems by embedding non-Markovian memory of unresolved variables through a Mori–Zwanzig projection framework. It develops a generalized Langevin equation with a Markovian term, a memory integral, and an orthogonal-noise term, and presents a zero-th order orthogonal-dynamics approximation whose memory kernels are estimated via Monte-Carlo averaging over equilibrium data. A first-order optimal prediction yields a reduced set of equations for the resolved variables, and a concrete Hamiltonian example demonstrates the method's reduced dynamics. The key practical insight is that memory kernels can be computed once at equilibrium and reused for new initial data, potentially enabling efficient handling of large-scale problems, including PDEs, by coupling memory effects with equilibrium statistical mechanics concepts. The approach clarifies the role of fluctuation–dissipation-like relations in non-Markovian optimal prediction and outlines avenues for systematic improvements of orthogonal dynamics in future work.
Abstract
Optimal prediction methods compensate for a lack of resolution in the numerical solution of complex problems through the use of prior statistical information. We know from previous work that in the presence of strong underresolution a good approximation needs a non-Markovian "memory", determined by an equation for the "orthogonal", i.e., unresolved, dynamics. We present a simple approximation of the orthogonal dynamics, which involves an ansatz and a Monte-Carlo evaluation of autocorrelations. The analysis provides a new understanding of the fluctuation-dissipation formulas of statistical physics. An example is given.