# i-flow: High-dimensional Integration and Sampling with Normalizing Flows

arXiv: https://arxiv.org/abs/2001.05486 · HTML: https://mlsanity.com/paper/2001.05486/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2020.

## Built on by (most cited first, 26 of 26)

- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [CaloFlow: Fast and Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2106.05285/) (2021)
- [Generative Networks for Precision Enthusiasts](https://mlsanity.com/paper/2110.13632/) (2021)
- [Understanding Event-Generation Networks via Uncertainties](https://mlsanity.com/paper/2104.04543/) (2021)
- [CaloFlow for CaloChallenge Dataset 1](https://mlsanity.com/paper/2210.14245/) (2022)
- [Convolutional L2LFlows: Generating Accurate Showers in Highly Granular Calorimeters Using Convolutional Normalizing Flows](https://mlsanity.com/paper/2405.20407/) (2024)
- [Latent Space Refinement for Deep Generative Models](https://mlsanity.com/paper/2106.00792/) (2021)
- [Optimising simulations for diphoton production at hadron colliders using amplitude neural networks](https://mlsanity.com/paper/2106.09474/) (2021)
- [Calorimeter shower superresolution](https://mlsanity.com/paper/2308.11700/) (2023)
- [Unifying Simulation and Inference with Normalizing Flows](https://mlsanity.com/paper/2404.18992/) (2024)
- [Scaling laws for amplitude surrogates](https://mlsanity.com/paper/2601.13308/) (2026)
- [Comparison of Affine and Rational Quadratic Spline Coupling and Autoregressive Flows through Robust Statistical Tests](https://mlsanity.com/paper/2302.12024/) (2023)
- [Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling](https://mlsanity.com/paper/2310.09194/) (2023)
- [Rare Event Probability Learning by Normalizing Flows](https://mlsanity.com/paper/2310.19167/) (2023)
- [BUFF: Boosted Decision Tree based Ultra-Fast Flow matching](https://mlsanity.com/paper/2404.18219/) (2024)
- [Extrapolating Jet Radiation with Autoregressive Transformers](https://mlsanity.com/paper/2412.12074/) (2024)
- [Generative Models and Statistical Validation](https://mlsanity.com/paper/2605.30453/) (2026)
- [Comparing Machine Learning and Interpolation Methods for Loop-Level Calculations](https://mlsanity.com/paper/2111.14788/) (2021)
- [BitHEP -- The Limits of Low-Precision ML in HEP](https://mlsanity.com/paper/2504.03387/) (2025)
- [CaloTrilogy: Toward a Breakthrough in One-Step, End-to-End, Physics-Guided Shower Generation for Modern Calorimeters](https://mlsanity.com/paper/2606.04165/) (2026)
- [An AI-powered Technology Stack for Solving Many-Electron Field Theory](https://mlsanity.com/paper/2403.18840/) (2024)
- [Flow Annealed Importance Sampling Bootstrap meets Differentiable Particle Physics](https://mlsanity.com/paper/2411.16234/) (2024)

## Abstract (arXiv)

In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijective mappings between two distributions. i-flow can also be used to sample random points according to complicated distributions in high dimensions. We compare i-flow to other algorithms for high-dimensional numerical integration and show that i-flow outperforms them for high dimensional correlated integrals. The i-flow code is publicly available on gitlab at https://gitlab.com/i-flow/i-flow.