An exact kernel framework for spatio-temporal dynamics

TL;DR

This paper introduces an exact kernel-based framework for analyzing spatio-temporal data governed by dynamic equations, enabling optimal solutions that fit noisy measurements within physical models.

Contribution

It develops a novel representer theorem involving time-dependent kernels for solving dynamic equations with spatio-temporal data, especially in physics-informed contexts.

Findings

Provides a regression and density estimation framework for Fokker-Planck dynamics.

Demonstrates the framework's ability to incorporate initial and boundary conditions.

Offers a method for optimal solution selection among all solutions of a dynamic equation.

Abstract

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.