# Deep Diffeomorphic Normalizing Flows

arXiv: https://arxiv.org/abs/1810.03256 · HTML: https://mlsanity.com/paper/1810.03256/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Oct 2018.

## Built on by (most cited first, 10 of 10)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs](https://mlsanity.com/paper/2201.05715/) (2022)
- [One Ring to Rule Them All: a simple solution to multi-view 3D-Reconstruction of shapes with unknown BRDF via a small Recurrent ResNet](https://mlsanity.com/paper/2104.05014/) (2021)
- [A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel](https://mlsanity.com/paper/2111.10722/) (2021)
- [Universality of parametric Coupling Flows over parametric diffeomorphisms](https://mlsanity.com/paper/2202.02906/) (2022)
- [Deep Continuous Networks](https://mlsanity.com/paper/2402.01557/) (2024)
- [Empirical evaluation of normalizing flows in Markov Chain Monte Carlo](https://mlsanity.com/paper/2412.17136/) (2024)
- [Multisymplectic Formulation of Deep Learning Using Mean--Field Type Control and Nonlinear Stability of Training Algorithm](https://mlsanity.com/paper/2207.12242/) (2022)
- [Geometry-Aware Normalizing Wasserstein Flows for Optimal Causal Inference](https://mlsanity.com/paper/2311.18826/) (2023)
- [Complex normalizing flows can almost be information K"ahler-Ricci flows](https://mlsanity.com/paper/2604.17954/) (2026)

## Abstract (arXiv)

The Normalizing Flow (NF) models a general probability density by estimating an invertible transformation applied on samples drawn from a known distribution. We introduce a new type of NF, called Deep Diffeomorphic Normalizing Flow (DDNF). A diffeomorphic flow is an invertible function where both the function and its inverse are smooth. We construct the flow using an ordinary differential equation (ODE) governed by a time-varying smooth vector field. We use a neural network to parametrize the smooth vector field and a recursive neural network (RNN) for approximating the solution of the ODE. Each cell in the RNN is a residual network implementing one Euler integration step. The architecture of our flow enables efficient likelihood evaluation, straightforward flow inversion, and results in highly flexible density estimation. An end-to-end trained DDNF achieves competitive results with state-of-the-art methods on a suite of density estimation and variational inference tasks. Finally, our method brings concepts from Riemannian geometry that, we believe, can open a new research direction for neural density estimation.