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

Adam Andersson; Filip Rydin; Kasper B{\aa}gmark; Stig Larsson

arxiv: 2409.14585 · v3 · submitted 2024-09-22 · 🧮 math.NA · cs.NA· math.PR· stat.CO· stat.ML

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

Kasper B{\aa}gmark , Adam Andersson , Stig Larsson , Filip Rydin This is my paper

Pith reviewed 2026-05-23 20:58 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PRstat.COstat.ML

keywords nonlinear filteringFokker-Planck equationdeep splittingBayesian filteringconvergence rateFeynman-Kac representationHörmander conditionnumerical approximation

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The pith

A deep splitting scheme approximates the nonlinear filtering density by solving the Fokker-Planck equation and converges under the parabolic Hörmander condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical method for nonlinear filtering that approximates the evolution of the filtering density between observations by solving the Fokker-Planck equation with a deep splitting technique. At each measurement time, it applies an exact Bayesian update using the new observation. The method is designed to work online after an initial training phase and uses sampling to address high-dimensional problems. Theoretical convergence is established when the diffusion satisfies the parabolic Hörmander condition, with supporting numerical evidence in a ten-dimensional example.

Core claim

The central claim is that the proposed prediction-update algorithm, where the prediction step employs a deep splitting scheme based on the Feynman-Kac representation to approximate the Fokker-Planck equation and the update step uses Bayes' formula, converges to the true nonlinear filtering density at a rate determined by the approximation error of the deep splitting method under the parabolic Hörmander condition.

What carries the argument

The deep splitting scheme for approximating solutions to the Fokker-Planck equation via a sampling-based Feynman-Kac approach, which enables the prediction step in the filtering algorithm.

If this is right

The filtering algorithm operates online for new observations after training.
The same convergence rate applies to approximating the Fokker-Planck equation independently.
The sampling approach helps mitigate the curse of dimensionality in high-dimensional settings.
Numerical robustness is demonstrated in a 10-dimensional nonlinear example.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may be extended to other stochastic processes where the Fokker-Planck equation governs the density evolution.
Similar deep splitting techniques could be applied to related problems in stochastic differential equations without the filtering context.
The empirical performance in high dimensions suggests potential for real-world applications in signal processing or data assimilation where analytical solutions are unavailable.

Load-bearing premise

The diffusion process underlying the signal must satisfy the parabolic Hörmander condition for the theoretical convergence rate to hold.

What would settle it

Numerical experiments on a diffusion that violates the parabolic Hörmander condition, checking whether the observed convergence rate matches the theoretical prediction or degrades.

Figures

Figures reproduced from arXiv: 2409.14585 by Adam Andersson, Filip Rydin, Kasper B{\aa}gmark, Stig Larsson.

**Figure 1.** Figure 1: The figure illustrates the L 2 (Ω;L∞(R d ; 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. 100 101 10−1 100 N L 2L ∞-error Drifted Brownian motion 100 101 10−1 100 N Bistable process Instances Average O(N −1 ) O(N −1/2 ) [PITH_… view at source ↗

**Figure 2.** 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 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. For transparency, we also remark that if we let M and L stay const… view at source ↗

read the original abstract

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic H\"{o}rmander condition, and empirically in numerical examples. In a prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, followed by 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. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem. The convergence analysis is complemented by a nonlinear $10$-dimensional numerical example demonstrating the robustness of the method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a convergent deep splitting scheme for the Fokker-Planck prediction step in nonlinear filtering, paired with exact Bayes updates, under the Hörmander condition plus a 10D example.

read the letter

This paper gives a convergent scheme for nonlinear filtering by using deep splitting on the Fokker-Planck prediction step plus exact Bayes update, with a rate under the Hörmander condition. The key is that it operates online after training and uses sampling to mitigate high dimensions. What the paper does well is lay out a classical prediction-update algorithm where the hard part (the FP evolution) is handled by the deep method, and they prove convergence for that under the parabolic Hörmander condition. The corollary for the Fokker-Planck equation by itself is useful on its own. The 10-dimensional nonlinear example shows it can handle something non-trivial. Soft spots are that the convergence is tied to the Hörmander condition, which is a common but restrictive assumption for hypoellipticity. The numerical support is only one example, so it doesn't fully demonstrate robustness. The abstract mentions the scheme but doesn't give implementation specifics like network size or training procedure, which might make it tricky for others to build on immediately. Overall, this is for readers in numerical analysis and computational statistics interested in deep learning for PDEs in filtering. A serious referee should look at it because the theoretical claim is clear and the empirical test is there, even if more examples would strengthen it.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a numerical scheme for the Bayesian filtering problem that approximates the nonlinear filtering density via a deep splitting scheme for the Fokker-Planck prediction step between discrete noisy measurements, followed by an exact Bayes update. It establishes a theoretical convergence rate under the parabolic Hörmander condition on the underlying diffusion, supplies a corollary for the standalone Fokker-Planck equation, employs a sampling-based Feynman-Kac representation to mitigate the curse of dimensionality, and demonstrates empirical performance in a 10-dimensional nonlinear example. The resulting algorithm is online after training.

Significance. If the convergence analysis is rigorous, the work supplies a theoretically grounded, high-dimensional method for nonlinear filtering that combines PDE approximation with exact updates and provides both a filtering result and an independent Fokker-Planck corollary; this is a meaningful contribution to numerical analysis of stochastic filtering problems where dimensionality is a central obstacle.

major comments (1)

The central claim asserts a convergence rate under the parabolic Hörmander condition, yet the abstract and available description do not state the explicit rate (e.g., dependence on time-step size, network width, or sampling error); without this quantitative statement the strength of the result cannot be assessed and the claim remains load-bearing but underspecified.

minor comments (2)

The abstract refers to 'a nonlinear 10-dimensional numerical example' without naming the SDE or observation model; a one-sentence description of the test problem would aid reproducibility and allow readers to judge the relevance of the Hörmander condition.
The distinction between the filtering algorithm and the standalone Fokker-Planck corollary should be emphasized with a dedicated statement or remark early in the introduction to clarify the scope of each result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

Referee: The central claim asserts a convergence rate under the parabolic Hörmander condition, yet the abstract and available description do not state the explicit rate (e.g., dependence on time-step size, network width, or sampling error); without this quantitative statement the strength of the result cannot be assessed and the claim remains load-bearing but underspecified.

Authors: We agree that the abstract would benefit from an explicit quantitative statement of the rate. The body of the manuscript (Theorem 3.4 and the subsequent filtering result) establishes the rate under the parabolic Hörmander condition, with the leading term controlled by the time-step size together with network approximation and Monte-Carlo sampling errors. We will revise the abstract to include this dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes a convergence rate for the deep splitting scheme applied to the Fokker-Planck prediction step (with exact Bayes update) under the standard parabolic Hörmander condition on the diffusion; this is an external hypothesis, not derived from or equivalent to the scheme itself. The corollary for the standalone Fokker-Planck problem is explicitly independent of the filtering application. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims, and the numerical example provides separate empirical support. The argument is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the parabolic Hörmander condition for the diffusion process and on standard assumptions underlying deep neural network approximation of PDEs; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Parabolic Hörmander condition on the underlying stochastic process
Invoked to obtain the theoretical convergence rate of the deep splitting scheme for the Fokker-Planck equation.

pith-pipeline@v0.9.0 · 5695 in / 1100 out tokens · 25654 ms · 2026-05-23T20:58:49.002437+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Reference graph

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