PaperlayerView exact revision on arXiv

Source-linked AI summary

Poisson Flow Generative Models

Yilun Xu, Ziming Liu, Max Tegmark, Tommi Jaakkola

arXiv:2209.11178v4cs.LGcs.CV

TL;DR

Existing generative approaches face training or sample-quality limitations, and stochastic diffusion sampling is slow while earlier backward ODE samplers had not matched SDE quality. PFGM uses reversible Poisson-field flows between data and a uniform hemisphere distribution; on CIFAR-10, it reports state-of-the-art normalizing-flow performance and faster sampling than SDE approaches.

  • Problem

    Existing generative models have limitations including unstable GAN training, low VAE sample quality, slow stochastic diffusion sampling, and backward ODE samplers that had not matched SDE quality.

    Source: §1 · p. 1

  • Method

    PFGM treats data as charges in an augmented space and uses Poisson-field ODEs to map between the data distribution and a uniform hemisphere distribution.

    Source: §1 · pp. 1–2, §3.1 · p. 4

  • Results

    FID/Inception scores of 2.35/9.68 (w/ DDPM++ deep [33]) give PFGM reported state-of-the-art CIFAR-10 normalizing-flow performance; it is competitive with SDE samplers at 10× to 20× speedup.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The invertible ODEs support likelihood evaluation and image interpolation, while the backward ODE enables efficient adaptive sampling.

    Source: §5 · p. 10, §4.4 · p. 10

    The analysis assumes N≥3, which the paper motivates by the typically large dimension of relevant applications.

    Source: §3 · p. 3

Abstract

from arXiv · show

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 while offering $10\times $ to $20 \times$ 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. The code is available at https://github.com/Newbeeer/poisson_flow .

1 Introduction

PFGM models generation as reversible Poisson-field flow between data and a uniform hemisphere distribution, addressing limitations in sample quality and sampling speed. The paper reports strong CIFAR-10 normalizing-flow results alongside faster sampling than SDE approaches.

1 Introduction

  • Earlier models face unstable GAN objectives or low VAE sample quality, while diffusion and score-based models have slow stochastic sampling and prior backward ODE samplers had not matched SDE quality.PFGM is presented against these limitations in generative-model training and sampling.Source: §1 · p. 1
  • In the r →∞ limit, charges released just above the data plane cross a hemisphere with a uniform distribution, extending the described physical flow to N dimensions.Data points are modeled as positive charges in the z = 0 plane, and motion follows their repulsive electric field.Source: §1 · p. 1
  • PFGM reverses this process by moving uniformly distributed negative charges from the hemisphere back to the z = 0 plane, and implements both directions with field-induced ODEs.The model is a continuous normalizing flow between the data distribution and an easily sampled uniform distribution on an N-dimensional hemisphere.Source: §1 · p. 2
  • PFGM has a stable training objective, and its backward ODE is more stable on the weaker NCSNv2 architecture than the compared reverse-time ODE samplers.It is reported as the only ODE-based sampler producing decent samples on NCSNv2 without corrections, and its Euler-method results are robust across 10 to 100 function evaluations.Source: §1 · p. 2
  • FID/Inception scores of 2.48/9.65 (w/ DDPM++ [33]) and 2.35/9.68 (w/ DDPM++ deep [33]) give PFGM state-of-the-art CIFAR-10 normalizing-flow performance.PFGM is also competitive with state-of-the-art SDE samplers and provides 10× to 20× speedup across datasets.Source: §1 · p. 2

2 Background and Related works

The background connects the Poisson equation and its gradient field to particle flows governed by ODEs. It also situates these flows among continuous normalizing-flow and diffusion-based generative approaches.

2 Background and Related works

  • The Poisson equation links a source density to a potential; its negative gradient defines the Poisson field, while the divergence relation is identified with Gauss’s law.The field also has an integral representation using an N-dimensional Green’s function.Source: §2 · pp. 2–3
  • The Poisson field points away from sources, and at large distances its lowest-order behavior decays as x/∣∣x∣∣^N, like a point source at the origin.The passage characterizes this power-law decay as long-range relative to exponential decay.Source: §2 · p. 3
  • A Poisson-field gradient flow evolves a particle population through the continuity equation, with the corresponding deterministic trajectories given by the forward ODE dx/dt = E(x).The particle interpretation is an overdamped charged-particle motion under the field.Source: §2 · p. 3
  • Rescaling time by a positive function preserves particle trajectories, and rescaling the stiff field ODE can make it more amenable to sampling.The original dynamics is stiff because of a power-law denominator.Source: §2 · p. 3
  • Prior ODE-based generative work includes maximum-likelihood continuous normalizing flows and reverse-time ODE sampling derived from diffusion processes.The latter approach is described as producing high-quality samples with an improved architecture.Source: §2 · p. 3

3 Poisson Flow Generative Models

PFGM augments data with an additional dimension so Poisson-field trajectories can map between the data distribution and a uniform distribution on an infinite hemisphere. It learns the normalized field and uses a backward ODE anchored by z to generate samples.

3.1 Augment the data with additional dimension

  • Adding a dimension avoids the disk-center collapse of the unaugmented flow, and the backward ODE maps an infinite-hemisphere distribution to the data distribution under mild conditions.Training data are placed on z = 0; particles started uniformly on the upper hemisphere are followed until they reach that plane.Source: §3.1 · p. 4, Figure 2 · p. 4
  • The augmented Poisson field defines forward and backward ODE trajectories between the z = 0 plane and an enclosing hemisphere.The theorem's infinite-radius limit gives the data distribution when particles follow the backward ODE to z = 0.Source: §3.1 · pp. 4–5

3.2 Learning the normalized Poisson Field

  • Training estimates the negative normalized empirical Poisson field, whose normalization preserves the field trajectories while avoiding variation in field magnitude.Training points are generated by perturbing augmented data with noise that grows exponentially with a uniformly sampled power m.Source: §3.2 · p. 5

3.3 Backward ODE anchored by the additional dimension

  • Using z as the backward ODE variable supplies explicit starting and terminal conditions: sampling starts at zmax and reaches the data distribution at z = 0.The ODE supports general-purpose solvers, including RK45 and forward Euler.Source: §3.3 · pp. 5–6
  • The starting prior is obtained by radially projecting a uniform hemisphere distribution onto the z = zmax plane, with radius sampled proportional to ||x||^(N−1) and angle sampled uniformly.Replacing time with a variable that makes z decay exponentially yielded 2× faster sampling with almost no harm to sample quality.Source: §3.3 · p. 6

4 Generative Modeling via the Backward ODE

The section evaluates PFGM for image generation, comparing its quality and speed with flow and SDE/MCMC methods. It also examines weaker-backbone stability, Euler step-size robustness, and forward-ODE utilities for likelihood evaluation and latent-space image manipulation.

4.1 Efficient image generation by PFGM

  • PFGM with DDPM++ deep reaches Inception 9.68 and FID 2.48, leading flow models; it uses 110 NFEs versus SDEs' 1000–2000 at comparable quality.Figure 3 shows uncurated samples on CIFAR-10, CelebA 64 × 64, and LSUN bedroom 256 × 256.Source: Table 1 · p. 7, §4.1 · p. 8, Figure 3 · p. 8

4.2 Failure of VE/VP-ODEs on NCSNv2 architecture

  • On NCSNv2, VE/VP-ODEs have CIFAR-10 FID above 90 and need Langevin correction for decent samples, while PFGM is more resistant to estimation errors.The authors hypothesize that the strong norm–σ correlation in score-based models contributes to their failure; PFGM spans a wider range of training-sample norms, and its z and norm are not strongly correlated.Source: §4.2 · pp. 8–9, Figure 4 · p. 8, Figure 5 · p. 9

4.3 Effects of step size in the forward Euler method

  • As Euler NFEs decrease, PFGM's CIFAR-10 FID degrades gracefully and is more step-size robust than VP-ODE; it beats DDIM at most NFEs, but is marginally worse at 10.Figure 5(c) compares number of steps with FID.Source: Figure 5 · p. 9, §4.3 · p. 9

4.4 Utilities of ODE: likelihood evaluation and latent representation

  • The invertible forward ODE maps data to a prior on the z = zmax hyperplane, enabling likelihood evaluation and latent-space image manipulation.CelebA latent-space interpolation yields perceptually consistent interpolations.Source: §4.4 · pp. 9–10
  • PFGM achieves better CIFAR-10 likelihoods than discrete normalizing flows without maximum-likelihood training, while sub-VP-ODE has the lowest bits/dim among continuous flows.Sub-VP-ODE's sample quality is worse than VP-ODE and PFGM; the authors leave the likelihood–sample-quality trade-off for future work.Source: Table 2 · p. 10, §4.4 · p. 10

5 Conclusion

PFGM solves the Poisson equation with the data distribution as its source and estimates a normalized field in an augmented space. The conclusion reports strong flow-model performance, faster sampling than stochastic methods, and backward-ODE stability, while identifying near-field normalization as future work.

5 Conclusion

  • PFGM solves the Poisson equation with the data distribution as its source; its forward ODE supports likelihood evaluation and image interpolation, while near-field behavior remains a normalization challenge.The authors identify renormalization as one possible future approach to making the Poisson field well-behaved near the field source.Source: §5 · p. 10

A.1 Formal Proof of Theorem 1

The proof establishes that backward Poisson-field flow maps a uniform distribution on an infinite upper hemisphere to the data distribution, subject to the stated regularity conditions. It supports this result with flux conservation and far-field analysis.

A.1 Formal Proof of Theorem 1

  • For compactly supported p ∈ C0 and N ≥ 3, the augmented Poisson equation has a unique C2 solution outside the data support.The proof constructs a solution using Green’s method; the stated zero boundary condition is at infinity.Source: §A.1 · p. 14
  • The uniqueness argument shows that any two solutions differ at most by a constant, leaving their gradients—and thus the field—identical.It uses the zero boundary condition at infinity and the fact that the squared gradient integrates a positive quantity.Source: §A.1 · p. 15
  • In the r → ∞ limit, the backward ODE maps a uniform upper-hemisphere distribution to the data distribution on z = 0 under Lemma 2’s conditions.The theorem assumes particles follow the Poisson field generated by the source distribution until they reach the data plane.Source: §A.1 · p. 15
  • Far-field flux conservation gives dΩ/dA ∝ p(x), so the forward mapping makes the hemisphere distribution uniform and the reversible backward ODE recovers the data distribution.The tube’s side carries no flux; symmetry splits the source-plane flux between z < 0 and z > 0.Source: Figure 6 · p. 15, §A.1 · p. 16

A.2 Multipole Expansion

  • At large distances, higher multipole corrections decay faster than the monopole term, so the field approaches that of a point source.This follows for a compactly supported data distribution.Source: §A.2 · pp. 16–17
  • The analysis divides space into far, intermediate, and near zones; on CIFAR-10 and CelebA, initial radii are typically above 1000 and lie in the far zone.The far-zone point-source approximation corresponds to κ ≫ 1; the field has moderate curvature at κ ∼ O(1) and high curvature at κ ≪ 1.Source: §A.2 · p. 17

A.3 Extension of Green’s Function in N-dimensional Space

  • For N ≥ 3, the extended Green’s function with the stated source conditions yields a Poisson-equation solution that vanishes at infinity.The passage assumes ρ ∈ C0, finite ∫ρ²(x)dx, and ρ(x) → 0 at infinity.Source: §A.3 · p. 18

A.4 Proof for the Prior Distribution on z = zmax Hyperplane

This section derives the prior on the z = zmax hyperplane by radially projecting a uniform distribution on the upper hemisphere. It also describes sampling that prior by drawing a radius and then an angle.

A.4 Proof for the Prior Distribution on z = zmax Hyperplane

  • Radial projection of the uniform upper-hemisphere distribution gives pprior(x) = 2zmax / (SN(1)r^(N+1)) on the z = zmax hyperplane.The derivation compares the area elements on the hemisphere and hyperplane.Source: §A.4 · p. 19
  • The projection’s area ratio follows by projecting a hyperplane patch orthogonally and comparing areas along the same cone from the origin.The angle between the vertical axis and the projected point determines the orthogonal projection factor.Source: §A.4 · p. 19
  • To sample pprior, draw the norm from its derived radial distribution and then sample the angle uniformly.The passage begins detailing a procedure for generating the radius, but does not provide all steps in the supplied text.Source: §A.4 · p. 20

B.1 Training

The appendices document PFGM’s training, sampling, and evaluation choices. They report that two ODE-sampler adjustments accelerate sampling, while an Euler update improves FID.

B.1 Training

  • Training used a single NVIDIA A100 GPU; PFGM settings include γ = 5 and dataset-specific batch sizes and sampling parameters.The stated larger field-estimation batch sizes are 2048 for CIFAR-10, 256 for CelebA, and 64 for LSUN bedroom.Source: §B.1 · p. 20, §B.1.1 · p. 20

B.1.1 Additional Settings

  • PFGM adapts DDPM++ architectures by using z in the positional embedding and adding a scalar output for the z direction.For LSUN bedroom, the experiments use channel configurations from both NCSN++ and DDPM.Source: §B.1.1 · p. 21

B.2 Sampling

  • The sampling appendix identifies the PFGM and VE/VP implementations and discusses the change-of-variable and ground-truth-z substitution techniques.These are presented as two techniques used in the PFGM ODE sampler.Source: §B.2 · p. 21

B.2.1 Additional settings

  • PFGM’s zmax settings are chosen to satisfy κ ≫ 1, where the data distribution is roughly approximated as a point source.The CIFAR-10 example uses zmax = 40 and has κ greater than 50 with high probability; the authors hypothesize insensitivity to hyperparameters in a reasonable range.Source: §B.2.1 · p. 21, Table 3 · p. 22

B.2.2 Exponential Decay on z Dimension

  • Changing variables to t′ = −log z makes the z-anchored ODE sampler about 2× faster with almost no FID harm.The two ODEs have the same trajectories as dt, dt′ → 0.Source: §B.2.2 · p. 22, Table 4 · p. 22

B.2.3 Substitute the Predicted z Direction with the Ground-truth

  • Using the ground-truth z direction in the near field accelerates sampling, as reported for the comparison with and without substitution.The substitution addresses imperfect neural-network estimates of the z direction when z is small.Source: §B.2.3 · pp. 22–23, Table 5 · p. 23

B.3 Evaluation

  • Evaluation reports FID and Inception scores for sample quality, NFE for inference speed, and bits/dim for likelihood evaluation.FID uses 2048-dimensional Inception-v3 activations; bits/dim is negative log-likelihood divided by data dimension.Source: §B.3 · p. 23

B.4 Effects of Step Size: FID versus NFE

  • Integrating z within each Euler update reduces discretization error, and the authors report that the update scheme significantly improves FID.The update uses z as a function of t′.Source: §B.4 · p. 23, Table 6 · p. 23

C Failure of VE/VP-ODE on NCSNv2 backbone

On the NCSNv2 backbone, PFGM is reported as more robust than VE/VP-ODEs, whose trajectories can diverge from the norm–noise relation. On LSUN bedroom, PFGM is reported to match VE-SDE performance with around 15× acceleration under the DDPM channel configuration.

C Failure of VE/VP-ODE on NCSNv2 backbone

  • The figure marks noisier VE-ODE samples in red and noisier corrected samples in green, with matching spatial positions generated from identical initial latents.The remaining samples in the uncorrected panel are identified as cleaner.Source: C Failure of VE/VP-ODE on NCSNv2… · pp. 23–24, Figure 8 · p. 24
  • PFGM’s weaker correlation between norm and z is offered as the reason for its greater robustness on the lighter NCSNv2 backbone.At large z, its norm distribution has a wider span for the high-density region.Source: C Failure of VE/VP-ODE on NCSNv2… · p. 24

D.1 LSUN Bedroom 256 × 256

  • On LSUN bedroom, PFGM with DDPM channels has comparable performance to VE-SDE while achieving around 15× acceleration.The authors also report better FID with a similar DDPM configuration and that the VE-ODE baseline produces only noisy samples.Source: Table 7 · p. 24, §D.1 · p. 25

D.2 Results on NCSNv2 Architecture

The NCSNv2 experiments evaluate PFGM on CIFAR-10 and CelebA using a smaller-capacity architecture, with CIFAR-10 results showing competitive quality at lower sampling cost.

D.2 Results on NCSNv2 Architecture

  • Experiments use NCSNv2, a smaller-capacity predecessor to NCSN++ and DDPM++, on CIFAR-10 and CelebA 64 × 64.The extra dimension z is supplied as a constant input channel, and an added output channel is pooled to estimate the z direction.Source: §D.2 · p. 25
  • The revised signal-to-noise configurations in Table 8 empirically give better results on NCSNv2.Table 8 covers dataset-method pairs.Source: §D.2 · p. 25, Table 8 · p. 25

D.2.1 CIFAR-10

  • On CIFAR-10, PFGM’s RK45 sampler matches Langevin dynamics competitively with 10× less NFE and outperforms the other ODE samplers.The backward ODE is the only compared ODE sampler reported to generate high-quality samples without a Langevin corrector; it also beats baselines with a corrector.Source: §D.2.1 · p. 25

D.2.2 CelebA

On CelebA, PFGM reports the best FID among the tested baselines and about 20× faster inference. The section also describes wall-clock sampling, latent-space uses, extended examples, and a physical interpretation of the ODE.

D.2.2 CelebA

  • On CelebA 64 × 64, PFGM achieves the best FID among the tested baselines while accelerating inference around 20×.These preliminary experiments use NCSNv2; Table 10 reports FID and NFE.Source: §D.2.2 · p. 25, Table 10 · p. 27
  • Despite lower FID, PFGM’s NCSNv2 CelebA samples show more obvious artifacts than Langevin samples; DDPM++ samples are diverse with fewer artifacts.The authors hypothesize that sample diversity affects FID more than the artifacts do.Source: §D.2.2 · pp. 25–26, Figure 9 · p. 26

D.3 Wall-clock Sampling Time

  • PFGM achieves the smallest total wall-clock sampling time, while similar architectures and solvers have nearly equal time per ODE step.The reported wall-clock comparison generates a batch of 1,000 CIFAR-10 images on one NVIDIA A100 GPU.Source: §D.3 · pp. 26–27, Table 11 · p. 27

D.4 Image Interpolations

  • PFGM’s invertible ODE enables spherical interpolation between image latent representations.The illustrated CelebA interpolation uses the first- and last-column images as endpoints.Source: §D.4 · p. 27, Figure 10 · p. 28

D.5 Temperature Scaling

E Extended Examples

F Physical Interpretation of the ODEs in PFGM

  • The ODE sets particle motion along the Poisson field; in the overdamped fluid interpretation, particle velocity follows the electric field.The discussion frames the motion as a first-order equation in the overdamped limit, such as a particle moving in honey.Source: F Physical Interpretation of the ODEs… · pp. 27, 29

G Limitations and Future Directions

The paper identifies bias in the minibatch normalized field as an unresolved training limitation and discusses potential social risks and needs for further research.

G Limitations and Future Directions

  • The minibatch normalized Poisson field is biased; larger training batches alleviate but do not fundamentally solve the problem.The authors suggest renormalization or modeling data points as quantum particles as possible directions.Source: G Limitations and Future Directions · p. 29

H Potential Social Impact

  • Fast generation may support deepfake-enabled social scams, while generative models remain susceptible to backdoor attacks; the authors call for robustness research and socio-technical collaboration.Source: H Potential Social Impact · p. 29

Related papers

Cited by

Loading 2209.11178v4…