An energy-based deep splitting method for the nonlinear filtering problem

TL;DR

This paper introduces a deep splitting method combined with energy-based neural networks to efficiently solve the nonlinear filtering problem, providing a fast, re-train-free filtering approach that outperforms traditional methods in various scenarios.

Contribution

It presents a novel deep learning approach that solves the Zakai equation for nonlinear filtering using energy-based models, enabling fast, re-train-free filtering.

Findings

The method is computationally fast and does not require re-training with new observations.

It performs well on both linear and nonlinear filtering problems in multiple dimensions.

Benchmark tests show promising results compared to Kalman and bootstrap particle filters.

Abstract

The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution of (stochastic) partial differential equations. This is combined with an energy-based model for the approximation of functions by a deep neural network. This results in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received. The method is tested on four examples, two linear in one and twenty dimensions and two nonlinear in one dimension. The method shows promising performance when benchmarked against the Kalman filter and the bootstrap particle filter.