# Temporal Normalizing Flows

arXiv: https://arxiv.org/abs/1912.09092 · HTML: https://mlsanity.com/paper/1912.09092/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2019.

## Built on by (most cited first, 8 of 8)

- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)
- [FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula](https://mlsanity.com/paper/2503.11427/) (2025)
- [Generative modeling of time-dependent densities via optimal transport and projection pursuit](https://mlsanity.com/paper/2304.09663/) (2023)
- [Fully differentiable model discovery](https://mlsanity.com/paper/2106.04886/) (2021)
- [Flow to Rare Events: An Application of Normalizing Flow in Temporal Importance Sampling for Automated Vehicle Validation](https://mlsanity.com/paper/2407.07320/) (2024)

## Abstract (arXiv)

Analyzing and interpreting time-dependent stochastic data requires accurate and robust density estimation. In this paper we extend the concept of normalizing flows to so-called temporal Normalizing Flows (tNFs) to estimate time dependent distributions, leveraging the full spatio-temporal information present in the dataset. Our approach is unsupervised, does not require an a-priori characteristic scale and can accurately estimate multi-scale distributions of vastly different length scales. We illustrate tNFs on sparse datasets of Brownian and chemotactic walkers, showing that the inclusion of temporal information enhances density estimation. Finally, we speculate how tNFs can be applied to fit and discover the continuous PDE underlying a stochastic process.