High-dimensional Bayesian filtering through deep density approximation

TL;DR

This paper benchmarks two deep density approximation methods for high-dimensional nonlinear filtering, demonstrating their superior efficiency and robustness over classical particle filters, especially in high-dimensional scenarios.

Contribution

It introduces and compares deep splitting and deep backward SDE filters based on Feynman--Kac formulas, extending them to logarithmic formulations for high-dimensional filtering.

Findings

Deep density methods significantly outperform particle filters in high dimensions.

Logarithmic formulations provide robust, positivity-preserving density estimates.

Deep filters reduce inference time by 2-5 orders of magnitude compared to classical methods.

Abstract

In this work, we systematically benchmark two recently developed deep density methods for nonlinear filtering. We model the filtering density of a discretely observed stochastic differential equation through the associated Fokker--Planck equation, coupled with Bayesian updates at discrete observation times. The two filters: the deep splitting filter and the deep backward stochastic differential equation filter, are both based on Feynman--Kac formulas, Euler--Maruyama discretizations and neural networks. The two methods are extended to logarithmic formulations providing sound, robust, and positivity-preserving density approximations in increasing state dimension. Comparing to the classical bootstrap particle filter and an ensemble Kalman filter, 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 backward stochastic differential equation filter 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.