# Accelerating Continuous Normalizing Flow with Trajectory Polynomial Regularization

arXiv: https://arxiv.org/abs/2012.04228 · HTML: https://mlsanity.com/paper/2012.04228/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2020.

## Built on by (most cited first, 6 of 6)

- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speed](https://mlsanity.com/paper/2203.10335/) (2022)
- [VoiceShop: A Unified Speech-to-Speech Framework for Identity-Preserving Zero-Shot Voice Editing](https://mlsanity.com/paper/2404.06674/) (2024)
- [Multi-Resolution Continuous Normalizing Flows](https://mlsanity.com/paper/2106.08462/) (2021)
- [Density Descent for Diversity Optimization](https://mlsanity.com/paper/2312.11331/) (2023)
- [Generative Topological Networks](https://mlsanity.com/paper/2406.15152/) (2024)

## Abstract (arXiv)

In this paper, we propose an approach to effectively accelerating the computation of continuous normalizing flow (CNF), which has been proven to be a powerful tool for the tasks such as variational inference and density estimation. The training time cost of CNF can be extremely high because the required number of function evaluations (NFE) for solving corresponding ordinary differential equations (ODE) is very large. We think that the high NFE results from large truncation errors of solving ODEs. To address the problem, we propose to add a regularization. The regularization penalizes the difference between the trajectory of the ODE and its fitted polynomial regression. The trajectory of ODE will approximate a polynomial function, and thus the truncation error will be smaller. Furthermore, we provide two proofs and claim that the additional regularization does not harm training quality. Experimental results show that our proposed method can result in 42.3% to 71.3% reduction of NFE on the task of density estimation, and 19.3% to 32.1% reduction of NFE on variational auto-encoder, while the testing losses are not affected.