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A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting

Kasper Bågmark, Adam Andersson, Stig Larsson, Filip Rydin

TL;DR

This work tackles the challenge of online nonlinear Bayesian filtering in high dimensions by propagating the filtering density through the Fokker–Planck equation between discrete observations and updating with Bayes' rule. The authors introduce an online deep splitting filter that leverages a Feynman–Kac representation and a neural-network-based, energy-enhanced minimization to approximate the prediction density, enabling efficient online inference after offline training. They prove a strong convergence rate of order $O(\tau)$ under a parabolic Hörmander condition, with Malliavin calculus providing key control of stochastic terms, and they also obtain convergence results for the Fokker–Planck equation in isolation. Numerical experiments in one dimension demonstrate the method’s convergence behavior and online applicability, highlighting its potential for higher-dimensional filtering tasks while illustrating the impact of model regularity on the observed rates.

Abstract

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in two numerical examples. For the prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, combined with an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. The convergence proof relies on stochastic integration by parts from the Malliavin calculus. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem.

A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting

TL;DR

This work tackles the challenge of online nonlinear Bayesian filtering in high dimensions by propagating the filtering density through the Fokker–Planck equation between discrete observations and updating with Bayes' rule. The authors introduce an online deep splitting filter that leverages a Feynman–Kac representation and a neural-network-based, energy-enhanced minimization to approximate the prediction density, enabling efficient online inference after offline training. They prove a strong convergence rate of order under a parabolic Hörmander condition, with Malliavin calculus providing key control of stochastic terms, and they also obtain convergence results for the Fokker–Planck equation in isolation. Numerical experiments in one dimension demonstrate the method’s convergence behavior and online applicability, highlighting its potential for higher-dimensional filtering tasks while illustrating the impact of model regularity on the observed rates.

Abstract

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in two numerical examples. For the prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, combined with an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. The convergence proof relies on stochastic integration by parts from the Malliavin calculus. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem.
Paper Structure (17 sections, 18 theorems, 133 equations, 2 figures)

This paper contains 17 sections, 18 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The online deep splitting filter and main results
  4. Notation
  5. Setting
  6. Solution to the filtering problem and deep splitting approximations
  7. Optimization based formulation of the deep splitting scheme
  8. Convergence of deep splitting approximation of the Fokker--Planck equation
  9. Convergence proofs
  10. Preliminaries for the analysis
  11. Local convergence
  12. Global convergence
  13. Numerical experiments
  14. Energy-based approximation
  15. Evaluation
  16. ...and 2 more sections

Key Result

Proposition 2.1

There exists a unique $p_{k}\in L^\infty(\mathbb{Y};C([t_k,t_{k+1}]\times\mathbb{R}^d,\mathbb{R}))$, $k=0,\dots,K$, satisfying eq: global Fokker--Planck with update. Moreover, $p_{k}(y)\in C_{\mathrm b}^{1,\infty}([t_k,t_{k+1}]\times \mathbb{R}^d; \mathbb{R}))$ for all $k=0,\dots,K$, $y\in\mathbb{Y}

Figures (2)

  • Figure 1: The figure illustrates the $L^2(\Omega;L^\infty(\mathbb{R}^d;\mathbb{R}))$-error over time for five different discretizations averaged over 10 instances. To the left we see the error for the drifted Brownian motion example and to the right we see the error for the example with the bistable process.
  • Figure 2: The figure presents the convergence for the numerical scheme for 10 individual instances of the scheme in red, their average in blue, and in black we see reference lines of order 1 and $\frac{1}{2}$ respectively. To the left we have the errors corresponding to the drifted Brownian motion example and to the right the example with the bistable process.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2: Feynman--Kac representation formula
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.1
  • Proposition 2.4
  • Lemma 2.2
  • ...and 18 more