An exact kernel framework for spatio-temporal dynamics
This work provides a method for practitioners to find the best solution compatible with noisy measurements when a differential equation is known a priori, particularly useful for spatio-temporal modeling under Fokker-Planck dynamics.
This paper introduces a kernel-based framework for analyzing spatio-temporal data where system dynamics are governed by a dynamic equation. It applies a representer theorem involving time-dependent kernels to find the solution of a given dynamic equation that best fits noisy spatio-temporal samples, specifically demonstrating this for the Fokker-Planck equation to perform regression and density estimation.
A kernel-based framework for spatio-temporal data analysis is introduced that applies in situations when the underlying system dynamics are governed by a dynamic equation. The key ingredient is a representer theorem that involves time-dependent kernels. Such kernels occur commonly in the expansion of solutions of partial differential equations. The representer theorem is applied to find among all solutions of a dynamic equation the one that minimizes the error with given spatio-temporal samples. This is motivated by the fact that very often a differential equation is given a priori (e.g.~by the laws of physics) and a practitioner seeks the best solution that is compatible with her noisy measurements. Our guiding example is the Fokker-Planck equation, which describes the evolution of density in stochastic diffusion processes. A regression and density estimation framework is introduced for spatio-temporal modeling under Fokker-Planck dynamics with initial and boundary conditions.