# Continuous-Time Flows for Efficient Inference and Density Estimation

arXiv: https://arxiv.org/abs/1709.01179 · HTML: https://mlsanity.com/paper/1709.01179/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2017.

## Built on by (most cited first, 21 of 21)

- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Geometrically Equivariant Graph Neural Networks: A Survey](https://mlsanity.com/paper/2202.07230/) (2022)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Implicit Normalizing Flows](https://mlsanity.com/paper/2103.09527/) (2021)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [UniGEM: A Unified Approach to Generation and Property Prediction for Molecules](https://mlsanity.com/paper/2410.10516/) (2024)
- [Deep Generative Models for Offline Policy Learning: Tutorial, Survey, and Perspectives on Future Directions](https://mlsanity.com/paper/2402.13777/) (2024)
- [Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme](https://mlsanity.com/paper/2211.16757/) (2022)
- [Spatial Attention Kinetic Networks with E(n)-Equivariance](https://mlsanity.com/paper/2301.08893/) (2023)
- [Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data](https://mlsanity.com/paper/2210.17230/) (2022)
- [Straight-Line Diffusion Model for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2503.02918/) (2025)
- [High-dimensional density estimation with tensorizing flow](https://mlsanity.com/paper/2212.00759/) (2022)
- [Universality of parametric Coupling Flows over parametric diffeomorphisms](https://mlsanity.com/paper/2202.02906/) (2022)
- [Semi-Equivariant Continuous Normalizing Flows for Target-Aware Molecule Generation](https://mlsanity.com/paper/2211.04754/) (2022)
- [Multimodal Datasets with Controllable Mutual Information](https://mlsanity.com/paper/2510.21686/) (2025)
- [High-dimensional reliability-based design optimization using stochastic emulators](https://mlsanity.com/paper/2604.05759/) (2026)
- [Semi-Equivariant Conditional Normalizing Flows](https://mlsanity.com/paper/2304.06779/) (2023)
- [Binary classification based Monte Carlo simulation](https://mlsanity.com/paper/2307.16035/) (2023)
- [Geometry-Aware Normalizing Wasserstein Flows for Optimal Causal Inference](https://mlsanity.com/paper/2311.18826/) (2023)
- [Deep Generative Sampling in the Dual Divergence Space: A Data-efficient & Interpretative Approach for Generative AI](https://mlsanity.com/paper/2404.07377/) (2024)
- [Flow-Based Density Ratio Estimation for Intractable Distributions with Applications in Genomics](https://mlsanity.com/paper/2602.24201/) (2026)

## Abstract (arXiv)

Two fundamental problems in unsupervised learning are efficient inference for latent-variable models and robust density estimation based on large amounts of unlabeled data. Algorithms for the two tasks, such as normalizing flows and generative adversarial networks (GANs), are often developed independently. In this paper, we propose the concept of em continuous-time flows (CTFs), a family of diffusion-based methods that are able to asymptotically approach a target distribution. Distinct from normalizing flows and GANs, CTFs can be adopted to achieve the above two goals in one framework, with theoretical guarantees. Our framework includes distilling knowledge from a CTF for efficient inference, and learning an explicit energy-based distribution with CTFs for density estimation. Both tasks rely on a new technique for distribution matching within amortized learning. Experiments on various tasks demonstrate promising performance of the proposed CTF framework, compared to related techniques.