High-dimensional Bayesian filtering through deep density approximation
Kasper Bågmark, Filip Rydin
TL;DR
This work tackles high-dimensional nonlinear state estimation by recasting nonlinear filtering through the Fokker–Planck equation with Bayes updates and comparing two deep density methods, the Deep Splitting Filter (DSF) and the Deep BSDE Filter (BSDEF), along with their log-density variants. The authors derive log-density formulations to stabilize training and demonstrate a unified simulation-based framework using Euler–Maruyama discretization and importance-sampling normalization, benchmarking these methods against EKF, EnKF, and bootstrap PF across a spectrum of problems up to 100 dimensions. They show that LogBSDEF remains accurate and robust in challenging regimes, including chaotic Lorenz-96 at $d=100$, while particle methods fail or become prohibitively expensive; in linear settings, the deep density methods achieve competitive accuracy, and across nonlinear cases they maintain numerical stability with substantial computational speedups (roughly two to five orders of magnitude) over particle-based filters. The study highlights practical advantages for high-dimensional data assimilation, with clear guidance on when log-density formulations and BSDE-based training offer the most benefit, and discusses limitations related to architecture choice and training stability. Overall, the paper advances scalable, density-based nonlinear filtering by delivering accurate, fast, and robust deep learning-based alternatives to classical methods in high dimensions.
Abstract
In this work, we benchmark two recently developed deep density methods for nonlinear filtering. Starting from the Fokker--Planck equation with Bayes updates, we model the filtering density of a discretely observed SDE. The two filters: the deep splitting filter and the deep BSDE filter, are both based on Feynman--Kac formulas, Euler--Maruyama discretizations and neural networks. The two methods are extended to logarithmic formulations providing sound and robust implementations in increasing state dimension. Comparing to the classical particle filters and ensemble Kalman filters, we benchmark the methods on numerous examples. In the low-dimensional examples the particle filters work well, but when we scale up to a partially observed 100-dimensional Lorenz-96 model the particle-based methods fail and the logarithmic deep density method prevails. In terms of computational efficiency, the deep density methods reduce inference time by roughly two to five orders of magnitude relative to the particle-based filters.