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

arXiv: https://arxiv.org/abs/2409.14585 · HTML: https://mlsanity.com/paper/2409.14585/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 62 references that are in the map.
- First posted on arXiv 22 Sep 2024.

## Builds on (most cited first, 18 of 18)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Deep learning based numerical approximation algorithms for stochastic partial differential equations](https://mlsanity.com/paper/2012.01194/) (2020)
- [Conditioning diffusion models by explicit forward-backward bridging](https://mlsanity.com/paper/2405.13794/) (2024)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [High-dimensional Filtering using Nested Sequential Monte Carlo](https://mlsanity.com/paper/1612.09162/) (2016)
- [An energy-based deep splitting method for the nonlinear filtering problem](https://mlsanity.com/paper/2203.17153/) (2022)
- [Automatic Backward Filtering Forward Guiding for Markov processes and graphical models](https://mlsanity.com/paper/2010.03509/) (2020)
- [An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations](https://mlsanity.com/paper/2201.03283/) (2022)
- [Deep Learning for the Benes Filter](https://mlsanity.com/paper/2203.05561/) (2022)
- [A High-Dimensional Particle Filter Algorithm](https://mlsanity.com/paper/1901.10543/) (2019)
- [Particle-MALA and Particle-mGRAD: Gradient-based MCMC methods for high-dimensional state-space models](https://mlsanity.com/paper/2401.14868/) (2024)

## Built on by (most cited first, 2 of 2)

- [Nonlinear filtering based on density approximation and deep BSDE prediction](https://mlsanity.com/paper/2508.10630/) (2025)
- [High-dimensional Bayesian filtering through deep density approximation](https://mlsanity.com/paper/2511.07261/) (2025)

## Abstract (arXiv)

A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic H"ormander 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.