Zuko is a Python package that implements normalizing flows in PyTorch. It relies as much as possible on distributions and transformations already provided by PyTorch. Unfortunately, the Distribution and Transform classes of torch are not sub-classes of torch.nn.Module, which means you cannot send their internal tensors to GPU with .to('cuda') or retrieve their parameters with .parameters(). Worse, the concepts of conditional distribution and transformation, which are essential for probabilistic inference, are impossible to express.

To solve these problems, zuko defines two concepts: the zuko.flows.core.LazyDistribution and zuko.flows.core.LazyTransform, which are any modules whose forward pass returns a Distribution or Transform, respectively. Because the creation of the actual distribution/transformation is delayed, an eventual condition can be easily taken into account. This design enables lazy distributions, including normalizing flows, to act like distributions while retaining features inherent to modules, such as trainable parameters. It also makes the implementations easy to understand and extend.


The zuko package is available on PyPI, which means it is installable via pip.

pip install zuko

Alternatively, if you need the latest features, you can install it from the repository.

pip install git+https://github.com/probabilists/zuko

Getting started#

Normalizing flows are provided in the zuko.flows module. To build one, supply the number of sample and context features as well as the transformations’ hyperparameters. Then, feeding a context \(c\) to the flow returns a conditional distribution \(p(x | c)\) which can be evaluated and sampled from.

import torch
import zuko

# Neural spline flow (NSF) with 3 sample features and 5 context features
flow = zuko.flows.NSF(3, 5, transforms=3, hidden_features=[128] * 3)

# Train to maximize the log-likelihood
optimizer = torch.optim.Adam(flow.parameters(), lr=1e-3)

for x, c in trainset:
    loss = -flow(c).log_prob(x)  # -log p(x | c)
    loss = loss.mean()


# Sample 64 points x ~ p(x | c*)
x = flow(c_star).sample((64,))

Alternatively, flows can be built as custom zuko.flows.core.Flow objects.

from zuko.flows import Flow, MaskedAutoregressiveTransform, Unconditional
from zuko.distributions import DiagNormal
from zuko.transforms import RotationTransform

flow = Flow(
        MaskedAutoregressiveTransform(3, 5, hidden_features=(64, 64)),
        Unconditional(RotationTransform, torch.randn(3, 3)),
        MaskedAutoregressiveTransform(3, 5, hidden_features=(64, 64)),

For more information, check out the tutorials or the API.


NICE: Non-linear Independent Components Estimation (Dinh et al., 2014)
Variational Inference with Normalizing Flows (Rezende et al., 2015)
Masked Autoregressive Flow for Density Estimation (Papamakarios et al., 2017)
Neural Spline Flows (Durkan et al., 2019)
Neural Autoregressive Flows (Huang et al., 2018)