# Physics-informed neural particle flow for the Bayesian update step

arXiv: https://arxiv.org/abs/2602.23089 · HTML: https://mlsanity.com/paper/2602.23089/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 74 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Particle Flow Bayes' Rule](https://mlsanity.com/paper/1902.00640/) (2019)
- [A New Parameterized Family of Stochastic Particle Flow Filters](https://mlsanity.com/paper/2103.09676/) (2021)
- [Stiffness Mitigation in Stochastic Particle Flow Filters](https://mlsanity.com/paper/2107.04672/) (2021)
- [Analytic solution of the exact Daum-Huang flow equation for particle filters](https://mlsanity.com/paper/2209.00089/) (2022)
- [How to implement the Bayes' formula in the age of ML?](https://mlsanity.com/paper/2411.09653/) (2024)

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

- [DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables](https://mlsanity.com/paper/2606.05247/) (2026)
- [Population-Aware Physics-Informed Neural Particle Flow for Robust Spacecraft Bayesian Navigation](https://mlsanity.com/paper/2606.10959/) (2026)

## Abstract (arXiv)

The Bayesian update step poses significant computational challenges in high-dimensional nonlinear estimation. While log-homotopy particle flow filters offer an alternative to stochastic sampling, existing formulations usually yield stiff differential equations. Conversely, existing deep learning approximations typically treat the update as a black-box task or rely on asymptotic relaxation, neglecting the exact geometric structure of the finite-horizon probability transport. In this work, we propose a physics-informed neural particle flow, which is an amortized inference framework. To construct the flow, we couple the log-homotopy trajectory of the prior to posterior density function with the continuity equation describing the density evolution. This derivation yields a governing partial differential equation (PDE), referred to as the master PDE. By embedding this PDE as a physical constraint into the loss function, we train a neural network to approximate the transport velocity field. This approach enables purely unsupervised training, eliminating the need for ground-truth posterior samples. We demonstrate that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity. Experimental validation on multimodal benchmarks and a challenging nonlinear scenario confirms better mode coverage and robustness compared to state-of-the-art baselines.