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

Kasper Bågmark, Adam Andersson, Stig Larsson

TL;DR

A deep splitting method is combined with an energy-based model for the approximation of functions by a deep neural network to result in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received.

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.

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

TL;DR

A deep splitting method is combined with an energy-based model for the approximation of functions by a deep neural network to result in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received.

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.
Paper Structure (23 sections, 56 equations, 7 figures)

This paper contains 23 sections, 56 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The filtering problem
  5. Deep splitting method
  6. A splitting scheme for the SPDE
  7. Derivation of a local optimization problem
  8. Extension to non-fixed observation sequence
  9. Accounting for a limited number of samples
  10. The energy-based approximation scheme
  11. Comparison to related approaches
  12. The original approach based on the Zakai equation
  13. An approach based on the Fokker--Planck equation
  14. Numerical examples
  15. Reference solution
  16. ...and 8 more sections

Figures (7)

  • Figure 1:
  • Figure 2: The figure presents numerical results for the three examples with respect to time. Left to right: Linear, mean reverting and bistable example. Top to bottom: Underlying densities of $X_T$, example trajectories, MAE, FME and KLD. Our method (EBDS) is illustrated in blue, the reference solution in red and a baseline in orange.
  • Figure 3: In (a) we see the time evolution of the density given by our model in blue, from time $t=0.01$ at the top to $t=1.00$ at the bottom. (b)--(d) are snapshots of the true filtering density given by the KF in red, the density from our model (EBDS) in blue, the PF-1000 in orange as well as the true state $X_t$ in green.
  • Figure 4: In (a) we see the time evolution of the density given by our model in blue, from time $t=0.01$ at the top to $t=1.00$ at the bottom. (b)--(d) are snapshots of the true filtering density given the PF in red, the density from our model (EBDS) in blue and the EKF in orange as well as the true state $X_t$ in green.
  • Figure 5: In (a) we see the time evolution of the density given by our model in blue, from time $t=0.01$ at the top to $t=0.50$ at the bottom. (b)--(d) are snapshots of the true filtering density given by the PF in red, the density from our model (EBDS) in blue and the EKF in orange as well as the true state $X_t$ in green.
  • ...and 2 more figures