论文标题
时空动力学的确切内核框架
An exact kernel framework for spatio-temporal dynamics
论文作者
论文摘要
引入了基于内核的时空数据分析框架，该框架适用于底层系统动力学受动态方程管辖的情况。关键成分是涉及时间依赖性内核的代表定理。这种内核通常发生在偏微分方程的溶液的扩展中。代表定理用于在动态方程的所有解决方案中找到通过给定时空样本最小化误差的解决方案。这是由于经常给予差分方程的事实而引起的，例如物理定律），从业者寻求与她嘈杂的测量相兼容的最佳解决方案。我们的指导示例是Fokker-Planck方程，它描述了随机扩散过程中密度的演变。在具有初始和边界条件的Fokker-Planck动力学下，引入了用于时空建模的回归和密度估计框架。
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.
