Poisson Flow Generative Models

Pytorch implementation of the NeurIPS 2022 paper Poisson Flow Generative Models,

by Yilun Xu*, Ziming Liu*, Max Tegmark, Tommi S. Jaakkola

[Slide]

Note: The method has been extended by the subsequent work PFGM++: Unlocking the Potential of Physics-Inspired Generative Models (code) with the following improvements:

😇 Improvements over PFGM / Diffusion Models:

• No longer require the large batch training target in PFGM, thus enable flexible conditional generation and more efficient training!
• More general $D \in \mathbb{R}^+$ dimensional augmented variable. PFGM++ subsumes PFGM and Diffusion Models: PFGM correspond to $D=1$ and Diffusion Models correspond to $D\to \infty$.
• Existence of sweet spot $D^*$ in the middle of $(1,\infty)$!
• Smaller $D$ more robust than Diffusion Models ( $D\to \infty$ )
• Enable the adjustment for model robustness and rigidity!
• Enable direct transfer of well-tuned hyperparameters from any existing Diffusion Models ( $D\to \infty$ )

Hence, we recommend the PFGM++ framework for new projects.

---

We propose a new Poisson flow generative model (PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the $z=0$ hyperplane in a space augmented with an additional dimension $z$, generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the $z=0$ plane transforms into a distribution on the hemisphere of radius $r$ that becomes uniform in the $r \to\infty$ limit. To learn the bijective transformation, we estimate the normalized field {in the augmented space}. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the (unaugmented) data manifold when the $z$ reaches zero.

Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of 9.68 and a FID score of 2.35. It also performs on par with the state-of-the-art SDE approaches (e.g., score-based SDEs or Diffusion models) while offering 10x to 20x acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method, and capable of scale-up to higher resolution datasets.

---

Acknowledgement: Our implementation relies on the repo https://github.com/yang-song/score_sde_pytorch.

Dependencies

We provide two solutions to install a subset of necessary python packages for our code. Please find the best fit for you.

1. The old dependency in repo https://github.com/yang-song/score_sde_pytorch

pip install -r requirements_old.txt

2. Our dependency (Python 3.9.12, CUDA Version 11.6)

pip install -r requirements.txt

Usage

Train and evaluate our models through main.py.

python3 main.py:
  --config: Training configuration.
  --eval_folder: The folder name for storing evaluation results
    (default: 'eval')
  --mode: <train|eval>: Running mode: train or eval
  --workdir: Working directory

For example, to train a new PFGM w/ DDPM++ model on CIFAR-10 dataset, one could execute

python3 main.py --config ./configs/poisson/cifar10_ddpmpp.py --mode train \
--workdir poisson_ddpmpp

• config is the path to the config file. The prescribed config files are provided in configs/. They are formatted according to ml_collections and should be quite self-explanatory.

  Naming conventions of config files: the path of a config file is a combination of the following dimensions:

  • Method: 🌟PFGM: poisson; Score-based models : ve, vp, sub_vp
  • dataset: One of cifar10, celeba64, celebahq_256, ffhq_256, celebahq, ffhq.
  • model: One of ncsnpp, ddpmpp.
  • continuous: train the model with continuously sampled time steps (only for score-based models).

  🌟Important Note 1 : We use a large batch (e.g. current training.batch_size=4096 for CIFAR-10, ~25G GPU memory usage) to calculate the Poisson field for each mini-batch samples (e.g. training.small_batch_size=128 for CIFAR-10). To adjust GPU memory cost, please modify the training.batch_size parameter in the config files.

  🌟Important Note 2 : If rk45 solver exibits unstability for your dataset/neural network, please try to use the forward Euler method or Improved Euler method by modifying the config.sampling.ode_solver parameter to forward_euler or improved_euler.

  Please set some key hyper-parameters for specific dataset by running

  python3 hyper-parameters.py 
  	--data_norm: Average data norm of the dataset 
  	--data_dim: Data dimension

  We also list a few other useful tips in Tips section.

• workdir is the path that stores all artifacts of one experiment, like checkpoints, samples, and evaluation results.

• eval_folder is the name of a subfolder in workdir that stores all artifacts of the evaluation process, like meta checkpoints for pre-emption prevention, image samples, and numpy dumps of quantitative results.

• mode is either "train" or "eval". When set to "train", it starts the training of a new model, or resumes the training of an old model if its meta-checkpoints (for resuming running after pre-emption in a cloud environment) exist in workdir/checkpoints-meta .

• Below are the list of evalutation command-line flags:

  --config.eval.enable_sampling: Generate samples and evaluate sample quality, measured by FID and Inception score.

  --config.eval.enable_bpd : Compute log-likelihoods

  --config.eval.dataset=train/test : Indicate whether to compute the likelihoods on the training or test dataset.

  --config.eval.enable_interpolate : Image Interpolation

  --config.eval.enable_rescale : Temperature scaling

Tips

• 🌟Important : We use a large batch (e.g. current training.batch_size=4096 for CIFAR-10, ~25G GPU memory usage) to calculate the Poisson field for each mini-batch samples (e.g. training.small_batch_size=128 for CIFAR-10). To adjust GPU memory cost, please modify the training.batch_size parameter in the config files.

• 🌟 How to set the hyper-parameters : The prior distribution on the $z=z_{max}$ hyperplane is a long-tail distribution. We recommend clipping the sample norm by the hyper-parameters sampling.upper_norm. Please refer to Appendix B.1.1 and Appendix B.2.1 in the paper (https://arxiv.org/abs/2209.11178) for our recommended setups for hyper-parameters training.M, sampling.z_max and sampling.upper_norm for general datasets.

  We provide a script for easily calculating those hyper-parameters:

  python3 hyper-parameters.py 
  	--data_norm: Average data norm of the dataset 
  	--data_dim: Data dimension

• If rk45 solver exibits unstability for your dataset/neural network, please try to use the forward Euler method or Improved Euler method by modifying the config.sampling.ode_solver parameter to forward_euler or improved_euler.

• TODO

Checkpoints

Please place the pretrained checkpoints under the directory workdir/checkpoints, e.g., cifar10_ddpmpp/checkpoints.

To generate and evaluate the FID/IS of (10k) samples of the PFGM w/ DDPM++ model, you could execute:

python3 main.py --config ./configs/poisson/cifar10_ddpmpp.py --mode eval \ 
--workdir cifar10_ddpmpp --config.eval.enable_sampling --config.eval.num_samples 10000

To only generate and visualize 100 samples of the PFGM w/ DDPM++ model, you could execute:

python3 main.py --config ./configs/poisson/cifar10_ddpmpp.py --mode eval \ 
--workdir cifar10_ddpmpp --config.eval.enable_sampling --config.eval.save_images --config.eval.batch_size 100

The samples will be saved to cifar10_ddpmpp/eval/ode_images_{ckpt}.png.

All checkpoints are provided in this Google drive folder.

Dataset	Checkpoint path	Invertible?	IS	FID	NFE (RK45)
CIFAR-10	poisson/cifar10_ddpmpp/	✔️	9.65	2.48	~104
CIFAR-10	poisson/cifar10_ddpmpp_deep/	✔️	9.68	2.35	~110
LSUN bedroom $256^2$	poisson/bedroom_ddpmpp/	✔️	-	13.66	~122
CelebA $64^2$	poisson/celeba_ddpmpp/	✔️	-	3.68	~110

FID statistics

Please find the statistics for FID scores in the following links:

CIFAR-10, CelebA 64, LSUN bedroom 256