An Ensemble Score Filter for Tracking High-Dimensional Nonlinear
  Dynamical Systems
Feng Bao Zezhong Zhang Guannan Zhang
We propose an ensemble score filter (EnSF) for solving high-dimensional
nonlinear filtering problems with superior accuracy. A major drawback of
existing filtering methods, e.g., particle filters or ensemble Kalman filters,
is the low accuracy in handling high-dimensional and highly nonlinear problems.
EnSF attacks this challenge by exploiting the score-based diffusion model,
defined in a pseudo-temporal domain, to characterizing the evolution of the
filtering density. EnSF stores the information of the recursively updated
filtering density function in the score function, in stead of storing the
information in a set of finite Monte Carlo samples (used in particle filters
and ensemble Kalman filters). Unlike existing diffusion models that train
neural networks to approximate the score function, we develop a training-free
score estimation that uses mini-batch-based Monte Carlo estimator to directly
approximate the score function at any pseudo-spatial-temporal location, which
provides sufficient accuracy in solving high-dimensional nonlinear problems as
well as saves tremendous amount of time spent on training neural networks.
Another essential aspect of EnSF is its analytical update step, gradually
incorporating data information into the score function, which is crucial in
mitigating the degeneracy issue faced when dealing with very high-dimensional
nonlinear filtering problems. High-dimensional Lorenz systems are used to
demonstrate the performance of our method. EnSF provides surprisingly
impressive performance in reliably tracking extremely high-dimensional Lorenz
systems (up to 1,000,000 dimension) with highly nonlinear observation
processes, which is a well-known challenging problem for existing filtering
methods.