A Score-based Nonlinear Filter for Data Assimilation

We introduce a score-based generative sampling method for solving the nonlinear filtering problem with robust accuracy. A major drawback of existing nonlinear filtering methods, e.g., particle filters, is the low stability. To overcome this issue, we adopt the diffusion model framework to solve the nonlinear filtering problem. In stead of storing the information of the filtering density in finite number of Monte Carlo samples, in the score-based filter we store the information of the filtering density in the score model. Then, via the reverse-time diffusion sampler, we can generate unlimited samples to characterize the filtering density. Moreover, with the powerful expressive capabilities of deep neural networks, it has been demonstrated that a well trained score in diffusion model can produce samples from complex target distributions in very high dimensional spaces. Extensive numerical experiments show that our score-based filter could potentially address the curse of dimensionality in very high dimensional problems.