Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods

TL;DR

This paper develops methods for optimal finite-dimensional approximations of the Fokker-Planck-Kolmogorov equation, including local and global projection techniques, and explores the use of eigenfunctions and Galerkin methods for improved accuracy.

Contribution

It introduces a unified framework for local and global approximation of the FPK equation using manifold projections and eigenfunction-based families, connecting these to Galerkin methods.

Findings

Local vector field projection yields globally optimal approximation.

Eigenfunction-based families can achieve exact global optimality.

Galerkin methods can coincide with the optimal approximation in certain cases.

Abstract

We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution. For a first local approximation we assign a manifold structure to the family and a metric. We then project the vector field of the partial differential equation (PDE) onto the tangent space of the chosen family, thus obtaining an ordinary differential equation for the family parameter. A second global approximation will be based on projecting directly the exact solution from its infinite dimensional space to the chosen family using the nonlinear metric projection. This will result in matching expectations with respect to the exact and approximating densities for particular functions associated with the chosen family, but this will require knowledge of the exact solution of FPK. A first way around this is a localized version of the metric projection based on the assumed density approximation. While the localization will remove global optimality, we will show that the somewhat arbitrary assumed density approximation is equivalent to the mathematically rigorous vector field projection. More interestingly we study the case where the approximating family is defined based on a number of eigenfunctions of the exact equation. In this case we show that the local vector field projection provides also the globally optimal approximation in metric projection, and for some families this coincides with a Galerkin method.