# Nonlinear filtering based on density approximation and deep BSDE prediction

arXiv: https://arxiv.org/abs/2508.10630 · HTML: https://mlsanity.com/paper/2508.10630/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 52 references that are in the map.
- First posted on arXiv 14 Aug 2025.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [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)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [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)
- [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)
- [A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2404.08456/) (2024)
- [The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs](https://mlsanity.com/paper/2503.13193/) (2025)
- [Particle-MALA and Particle-mGRAD: Gradient-based MCMC methods for high-dimensional state-space models](https://mlsanity.com/paper/2401.14868/) (2024)
- [A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting](https://mlsanity.com/paper/2409.14585/) (2024)

## Built on by (most cited first, 1 of 1)

- [High-dimensional Bayesian filtering through deep density approximation](https://mlsanity.com/paper/2511.07261/) (2025)

## Abstract (arXiv)

A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using the well-known deep BSDE method and neural networks. The method is trained offline, which means that it can be applied online with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic H"ormander condition. The theoretical convergence rate is confirmed in two numerical examples.