Poisson Flow Generative Models
Authors: Yilun Xu
Ziming Liu
Max Tegmark
Tommi S. Jaakkola
Venue: NeurIPS 2022 Accept (2022)
Keywords: poisson equation
generative model
ODE

Abstract: 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 .

Poisson Flow Generative Models
Yilun Xu,, Ziming Liu', Max Tegmark, Tommi Jaakkola
Massachusetts Institute of Technology
(ylxu, zmliu, tegmark] @mit edu; edu; tommi@csall miitt..m.ii............ edu
Abstract
We propose a new "Poisson flow" generative model (PFGM) that maps a uniform distribution on high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the = hyperplane in a space augmented with additional dimension z, generating 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 + 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 reaches zero. Experimentally. PFGM achieves current state-of-the-an performance among the normalizing flow models on CIFAR-IO. 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 -hrtte-atttttttte SDE approaches while offering 10x to 20x acceleration 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 wwwp///eeeeereerrssooooooo.o
image generation tasks. Additionally,
Introduction
Deep generative models are a prominent approach for data generation, and have been used to produce high quality samples in image [1]. text [2] and audio [35], as well improve semi-supervised learning [20], domain generalization [25] and imitation learning [15]. However, current deep generative models also have limitations, such unstable training objectives (GANs [I, 12, 17]) and low sample quality (VAEs [21], normalizing flows [6]). New techniques [12, 24] are introduced to stablize the training of CNN-based or ViT-based GAN models. Although recent advances diffusion [16] and scored-base models [33] achieve comparable sample quality to GAN's without adversarial training, these models have slow stochastic sampling process. [33] proposes backward ODE samplers (normalizing flow) that speed up the sampling process but these methods have not yet performed on par with the SDE counterparts.
We present new "Poisson flow" generative model (PFGM),, exploiting a remarkable physics fact that generalizes to N dimensions. As illustrated in Fig. 1(a), motion in viscous fluid transforms any planar charge distribution into a uniform angular distribution. Specifically, we interpret N- dimensional data points x (images, say) as positive electric charges in the
0 plane of an N + 1-dimensional space (see Fig. (())) filled with a viscous liquid (say honey). ). A positive charge with 0 will be repelled by the other charges and move in the direction of their repulsive force, eventually crossing an imaginary hemisphere of radius r. We show that, remarkably, if the the original charge distribution is let loose just above 0, this law of motion will cause a uniform distribution for their hemisphere crossings in the r - oo limit.
Equal Contribution.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Forward ODE
density high
Backward ODE
y
Forward ODE
2 0
low
Backward ODE
X
(a)
(b)
Figure 1: (a) 3D Poisson field trajectories for a heart-shaped distribution (b) The evolvements of distribution (top) or an (augmented) sample (bottom) by the forward/hackward ODEs pertained to the Poisson field.
Our Poisson flow generative process reverses the forward process: we generate uniform distribution of negative charges on the hemisphere, then track their motion back to the O plane, where they will be distributed as the data distribution. Poisson flow can be viewed as a type of continuous normalizing flows [4, 10, 33] in the sense that it continuously maps between an arbitrary distribution and an casily sampled one: in the previous works an N N-dimensional Gaussian and PFGM a uniform distribution
an N N-dimensional hemisphere. In practice, implement the Poisson flow by solving a pair of forward/backwaar ordinary differential equations (ODEs) induced by the electric field (Fig. 1(b)) given by the N w-dimensional version of Coulomb's law (the gradient of the solution to the Poisson's 's equation with the data as sources). We will interchangeably refer to this gradient as the Poisson field, since electric fields normally refer to the special case N 3.
The proposed generative model PFGM has a stable training objective and empirically outperforms previously state-of-the-an continuous flow methods [30, 33]. As a different iterative method, PFGM offers two advantages compared to score-based methods [32, 33]. First, the ODE process PFGM achieves faster sampling speeds than the SDE samplers in [33]. while retaining comparable perfor- mance. Second, our backward ODE exhibits better generation performance than the reverse-time weaker architecture NSCNv2 [32]. The rationale for robustness is that the time variables in these ODE baselines are strongly correlated with the sample norms during training time, resulting in less error-tolerant inference. In contrast, the tie between the anchored variable and the sample norm in PFGM is much weaker.
ODEs of VE/VP/sub- VP SDEs [33], as well greater stability
Experimentally, we show that PFGM achieves current state-of-the- performance on CIFAR-10 dataset the normalizing flow family, with FID/Inception scores 2.48/9.65 (w/ DDPM++ [33]) and 2.35/9.66 (w/ DDPM++ deep [33]). J). It performs competitively with current state-of-the-art SDF samplers [33] and provides 10x 20x speed up across datasets. Notably, the backward ODE in PFGM is the only ODE-based sampler that can produce decent samples its while other ODE baselines fail without corrections. In addition, PFGM demonstrates the robustness to the step size in the Euler method, with varying number of function evaluations (NFE) ranging from 10 to 100. We further showcase the utility of the invertible field likelihood evaluation and image manipulations, and its scalability to higher resolution images on LSUN bedroom 256 256 dataset.
NCSNv2 [32],
ODEs of the Poisson
Background and Related works
Poisson equation e R-N and p(x): IR-N be a source function. We assume that the source function has compact support, e C*) and N 2 3. The Poisson equation is
(x)
(1)
8 where pxx) N called the potential function, and 72 = EN Bu? It is usually helpful to define the gradient field E(x) Vx((x) and rewrite the Poisson equation as
is the Laplacian operator.

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V.E= = p, known in physics Gauss's law [11]. The Poisson equation is widely used in physics, giving rise to Newton's 's gravitational theory [9] and the electrostatic theory [11], when p(x) is
interpreted as mass density or electric charge density, respectively. E is the N-dimensional analog of the electric field. The Poisson equation Eq. (1) (with zero boundary condition at infinity) admits a unique simple integral solution 2:
f((x== = f G(x,)))(y)dy, G(x.y) (N 2)SN- (1) ||x y11
1
(2)
where (1) is a geometric constant representing the surface area of the unit (N - 1)-sphe " and y) is the extension of Green's 's function in N-dimensional space (details in Appendix A.3). The negative gradient field of (x),, referred as Poisson field of the source p. is
E(x) =>v+(x)) =- f VzG(xxy)py)dd, VzG(x.y) (x- |x yll
X y
(3)
Qualitatively, the Poisson field Exx) points away from sources, or equivalently -E(x) points towards sources, as illustrated in Fig. 1. It is straightforward to check that when p(x) + 8(x y)). we get (x() G(x. y) and E(x) -Vx y). This implies that G(x.y) y) and -VxG(x.y) can be interpreted as the potential function and the gradient field generated by a unit point source, e.g., point charge, located y. When p(x) takes general forms but has bounded support, simple asymptotics exist for ||x|| >> ||y||. To the lowest order, E(x) VxG(x,
x/||>||** behaves as it were generated by a unit point source at y = O. In physics, the power law decay is considered to be long-range (compared to exponential decay) [1]].
Particle dynamics in a Poisson field The Poisson field immediately defines a flow model, where the probability distribution evolves according to the gradient flow Öp. (x)) = - (pr (p:(x)E(x)) The gradient flow is a special case of the Plkkkr equation [28], where the diffusion coefficient zero. Intuitively we can think of Pt (x) as represented by a population of particles. The corresponding (non-diffusion) case of the Itô process is the forward ODE dx E(x). We can interpret the
trajectories of the ODE as particles moving according to the Poisson field E(z), with initial states drawn from Po- The physical picture of the forward ODE is a charged particle under the influence of electric fields in the overdamped limit (details in Appendix F).
The dynamics is also rescalable in the sense that the particle trajectory remains the same for dx Ex)) for fxx) > fx) C², because the time rescaling dt + f (x(t)) recovers dx +E(x). Note that the dynamics is stiff due to the power law factor in the denominator in Eq. (3). posing computational challenges. Luckily the rescalablility allows us to rescale E(x) properly to get new ODEs (formally defined later in Section 3.3) that are more amenable for sampling.
de
Generative Modeling via ODE Generative modeling can be done by transforming base distribution to data distribution via mappings defined by ODEs. The ODE-based samplers allow for adaptive sampling, exact likelihood evaluation and modeling of continuous-time dynamics [4, 33]. Previous
works broadly fall into two lines. [4, 3] introduce a continuous-time normalizing flow model that can be trained with maximum likelihood by the instantaneous change-of-variables formula [4]. For sampling, they directly integrate the learned invertiable mapping over time. Another work [33] unifies the wored-based model [3 32] and diffusion model [16] into a general diffusion process, and uses the reverse-tue -evese-time ODE of the diffusion process for sampling. They show that the reverse-ting -imere--tmme ODE produces high quality samples with improved architecture.
3 Poisson Flow Generative Models
In this section, we start with the properties of the Poisson flow in the augmented space and show how to draw samples from the data distribution by following the backward ODE of the Poisson flow (Section 3.1). We then discuss how to actually learn normalized Poisson field from data samples through simulations of the forward ODE (Section 3.2) and present an equivalent backward (Section 3.3).
ODE that allows for exponentially decay
FFq. (2) is valid for N 3. When N the Green's function is G(x,y) -log([|x y11//2T. We assume N 2 3 since typically large the relevant applications.
"The N N-sphere with radius is defined as (x e R N-1
||x|| = r)
3

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No augmentation (2D)
Augmentation (3D)
Z
20 d2/S,(1)
Gauss's Law
d
=xx)dA/2
N dim
Figure 2: (a) Poisson field (black arrows) and particle trajectories (blue lines) of a 2D uniform disk (red). Left (no augmentation, 2D): all particles collapse to the disk center. Right (augmentation, 3D): particles hit different points on the disk. (b) Proof idea of Theorem 1. By Gauss's 's Law, the outflow flux duout out equals the inflow flux dd... The factor of two in p(x)dA/2 is due the symmetry of Poisson fields in 2 < 0 and 2 O.
3.1 Augment the data with additional dimension
We wish to generate samples x R&V from distribution p(x) supported on bounded region. We may set the source p(x) p(x)= ε C and compute the resulting gradient field E(x) from Eq. (3). Since E(x) points towards sources, the backward ODE dx/ dt -xxx will take samples close to the sources. One may naively hope that the backward ODE is generative model that recovers p(x). Unfortunately, the backward ODE has the problem mode collapse. We illustrate this phenomenon with 2D uniform disk. The reverse Poisson field -E(x) on the 2D (æ, /)-pane points towards the center of the disk (Fig. 2(a) left), so all particle trajectories (blue lines) will eventually hit O. If we instead add additional dimension (Fig. 2(a) right), particles can hit different points the disk and faithfully recover the data distribution.
Consequently, instead of solving the Poisson equation <² -p(x) in the original data space, we solve the Poisson equation in an augmented space X = (x, IR N+ with an additional variable IR. We augment the training data in the new space by setting
such that (x, 0). As consequence, the data distribution in the augmented space is p(x) p( x))) where 5 is the Dirac delta function. By Eq. (3), the Poisson field by solving the new Poisson equation v²p(X) -(x) has an analytical form:
VX N+1 E(x) -Vx(
x yl N-I p(Y)dý
(4)
(N
The associated forward/backward ODEs of the Poisson field are dx/dt E(x), E()) Intuitively, theses ODEs uniquely define trajectories of particles between the z 0 hyperplane and an enclosing hemisphere (cf. Fig. (())). the following theorem, we show that the backward infinite hemisphere and the
ODE defines transformation between the uniform distribution data distribution p(x) in the = 0 plane. We present the formal proof to Appendix A, illustrated by Fig. 2(b). The proof is based on the idea that when the radius of hemisphere distribution ş x) can be effectively viewed as a delta distribution at origin. Consequently, the Poisson field points in the radial direction r oo, perpendicular S (r) (Green arrows in Fig. 2(b)).
co, the data
Theorem 1. Suppose particles are sumpled from uniform distribution the upper (z 0) half d% -E(x) until they reach the = O hyperplane, where the Poisson field E(x) is generated by the source p(&). In the r - ca limit, under some mild conditions detailed Appendix A, this process generates particle distribution p(x). i.e.. distribution p(x) in the z = 0 hyperplane.
of the sphere of radius r and evolved by the backward ODE
dt
Proof sketch. Suppose the flux of the backward ODE connects a solid angle di (on S (r)) with an area d.A (on su)p()())) According to Gauss's law, the outflow flux ddo.tt out d (1) on the
A probability distribution p(x) is a special case of "charge density" p(x) because p(x) need to be non- negative and integrates to unity. Here we focus on applications to probability distribution of data, which is the objective to be modeled in generative modeling.

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hemisphere (Green arrows in Fig. 2(b)) equals the inflow flux di p(x)dA/2 (u)p()(X)) (Red
arrows in Fig. 2(b)). d d out gives df//A p(x))Sx(1)22 x p(x). Together, by change-of- variable, we conclude that the final distribution in the 0 hyperplane is (x)).
n
The theorem states that starting from infinite hemisphere, by following the inverse Poisson field -E(x). We defer the formal proof and technical assumptions of the theorem Appendix A. The property allows generative modeling by following the Poisson flow of ( = -p(x).
can recover the data distribution p
3.2 Learning the normalized Poisson Field
Given a set of training data D {x.}__ i.id sampled from the data distribution p(x), we define the empirical version of the Poisson field (Eq. (4)) as follows:
x-
E(x) ((x)
Σ
where the gradient field is calculated on augmented datapoints {&+ - (x,,0)) and c(x) 1/ Ei=1 | is the multiplier for numerical stability. We further normalize the field to re- solve the variations in the magnitude of the norm || E(x) ll,, and fit the neural network to the more amenable negative normalized field vxx) -v (E) E(x) l12. The Poisson field is rescalable (cf. Section 2) and thus trajectories of its forward/backwart ODEs are invariant under normalization. We denote the empirical field calculated on batch data B by Eg and the negative normalized field as vx(*)= = (V Ex(x) ll2.
Similar to the scored-based models, we sample points inside the hemisphere by perturbing the augmented training data. Given a training point x E DD we add noise to its augmented version (x; (x; 0)) to construct the perturbed point 2y,
=
ll 6x (1+ + == ==|c=(11 7)m
where c= ((x; ~N(C,) o N++IN+1), u ((SN=1(11))) (1)) and m ~ [[[,, M]. The upper limit M, hyper-parameters. With fixed and 11 the added noise increases
standard deviation and exponentially with m. The rationale behind the design is that points farther away from the data support play a less important role in generative modeling, sharing similar spirit with the choice of noisy scales in score-based models [32, 33].
In practice, we sample the points by perturbing a mini-batch data B = {x;) in each iteration. We uniformly sample the power m [0. .M] for each datapoint. We select large M (typically around 300) to ensure the perturbed points can reach a large enough hemisphere. We use larger batch B, for the estimation of normalized field since the empirical normalized field is biased, which empirically gives better results. Denoting the set of perturbed points as {y:)!e] we train the neural network fo on these points to estimate the negative normalized field by minimizing the following loss:
((0)= = B
Σ Il fo( - -BB (V:) 11
We summarize the training process in Algorithm 1. In practice, we add small constant y to the
denominator of the normalized field to overcome the numerical issue when li, ||x x =
3.3 Backward ODE anchored by the additional dimension
After estimating the normalized field v, we can sample from the data distribution by the backward ODE dx = v(x) Nevertheless, the boundary condition of the above ODE is unclear: the starting and terminal time I of the ODE are both unknown. To remedy the issue, we propose equivalent backward ODE in which x evolves with the augmented variable 2:
d(x, =) = ( dx dz, dz) (v(x)xv(8)) 11)dz
dt dz
5

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Algorithm Learning the normalized Poisson Field
Input:
Training iteration T. Initial model ʃa. dataset D, constant learning rate
for t = 1...T' do
Sample a large batch 3 from D and subsample batch of datapoints B = [(x})==) from B
Simulate the ODE: (y: perturb x;) ]
Calculate the normalized field by B VB, (3+) = --VNEB. (5.)((I) EB, (Fi) ll2 +7), Vi Calculate the loss: [()) = [8 Σ fe( VB, (5.) 11
Update the model parameter: 0 0 10)
end for return fo
Algorithm 2 perturb(x)
Sample the power m ~ 21 M]
Sample the initial noise (fx, 6z) V (((())))())))))
Uniformly sample the vector from the unit ball u~ u(S((1)))
Construct training point y = x+ || 6x (1 u, === return y = 2y
where v v(x)x. x: v(x); are the corresponding components of x, z in vector vxx). In the new ODE, we replace the time variable with the physically meaningful variable z, permitting explicit starting and terminal conditions: when z O, we arrive at the data distribution and we can freely choose large Zmax the starting point in the backward ODE. The backward ODE is compatible with ODE solvers, e.g., RK45 method [23] and forward Euler method. The popular black-box ODE solvers, such as the one in Scipy library [37]. typically use common starting time for the same batch of samples. Since the distribution on the 2 2max hyperplane is no longer uniform, we derive the prior distribution by radially projecting uniform distribution on the hemisphere with radius r Zmax to the 2max hyperplane:
N-N+1 max
2
Pprior (x)
SN (xmax) (ll
Sx( 11
2 max
maX
where Sw(r) is the surface area of N-sphere with radius The reason behind the radial projection that the Poisson field points the radial direction at r 0o. The new backward ODE also defines bijective transformation between Pprior on the infinite hyperplane (#max 0o) and the data distribution f(x), analogous to Theorem 1. order to sample from Ppeior (x). it is suffice to sample the norm (radius) from the distribution: Pradius (I) x (12) <c x 11 - x x1 +2 Aii and then uniformly sample its angle. We provide detailed derivations and practical sampling procedure in Appendix A.4. We further achieve exponential decay on the dimension by introducing new variable l':
[Backward ODE]
d(x,2) = (x))))((3))2,,2))))
The component in the backward ODE, Le., dz 2dt', can be solved by e" Since 2 reaches zero as t' + -oo, we instead choose a tiny positive number 2min as the terminal condition. The corresponding starting/terminal time of the variable t' are log /max/ log Amin respectively. Empirically this simple change of variable leads to 2x faster sampling with almost no harm to the sample quality. In addition, we substitue the predicted vxx), with more accurate one when is small (Appendix B.2.3). We defer more details of the simulation of backward ODE to Appendix B.2.
4 Generative Modeling via the Backward ODE
In this section, we demonstrate the effectiveness the backward ODE associated with PFGM on image generation tasks. In Section 4.1, , we show that PFGM achieves currently best in class performance the normalizing flow family. In comparison to the existing state-of-the-art SDE or MCMC approaches, PFGM exhibits 10x or 20x acceleration while maintaining competitive or

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Table 1: CIFAR-10 sample quality (FID, Inception) and number of function evaluation (NFE).
Invertible?
Inception
FID
NFF
PixeICNN [36]
X
4.60
65.9
1024
IGEBM
X
6.02
40.6
60
ViTGAN [24]
X
9.30
6.66
StyleGAN2-ADA [17]
X
9.83
2.92
StyleGAN2-ADA (cond.) [17]
X
10.14
2.42
NCSN [31]
X
8.87
25.32
1001
NCSNv2 [32]
8.40
10.87
DDPM [16]
X
9.46
3.17
1000
NCSN+ VE-SDE [33]
X
9.83
2.38
2000
NCSN+ deep VE-SDE [33]
X
9.89
2.20
2000
Glow [19]
3.92
48.9
DDIM, T=50 [30]
DDIM, T=100 [30]
4.16
100
NCSN+ VE-ODE [33]
9.34
5.29
194
NCSN+ deep VE-ODE [33]
9.17
7.66
194
DDPM+ backbone
VP-SDE [33]
X
9.58
2.55
1000
sub-VP-SDD [33]
X
9.56
2.61
1000
VP-ODE [33]
9.46
2.97
134
sub-VP-ODE [33]
9.30
3.16
146
PFGM (ours)
/
9.65
2.48
104
DDPM+ deep backbone
VP-SDE [33]
X
9.68
2.41
1000
sub-VP-SDE [33]
X
9.57
2.41
1000
VP-ODE [33]
9.47
2.86
134
sub-VP-ODE [33]
/
9.40
3.05
146
PFGM (ours)
9.68
2.35
110
higher generation quality. Meanwhile, unlike existing ODE baselines that heavily rely on corrector to generate decent samples weaker architectures, PFGM exhibits greater stability against error (Sec- tion 4.2). Finally, we show that PFGM is robust to the step size the Fuler method (Section 4.3), and its associated ODE allows for likelihood evaluation and image manipulation by editing the latent space (Section 4.4).
4.1 Efficient image generation by PFGM
Setup For image generation tasks, consider the CIFAR- 10 [22], CelebA 64 64 [38] and LSUN bedroom 256 x 256 [39]. Following [32], we first center-crop the CelebA images and then resize them 64 x 64. We choose M 291 (CIFAR-10 and CelehA) (LSUN bedroom).
a 0.01 and т = 0.03 for the perturbation Algorithm 2, and 2min
1e 3, 2max
40 (CIFAR- /60 (CelebA //2) (LSUN bedroom) for the backward ODE, We further clip the norms of initial samples into (0, 3000) for CIFAR-IO, (0, 6000) for CelebA 64 and (0, 30000) for LSUN bedroom. We adopt the DDPM+ and DDPM+ deep architectures [33] our backbones. We add the scalar 2 (resp. predicted direction on 2) as input (resp. output) to accommodate the additional dimension. We take the same of paper-parameters such as batch size, learning rate and training iterations from [33]. We provide more training details in Appendix B. and discuss how to set these hyper-parameters for general datasets B. and B.2.1.
Baselines We compare PFGM to modern autoregressive model [36], GAN [17, 24], normalizing flow [19] and EBM [8]. We also compare with variants cosse-based models such as DDIM [30] and current oftte-oftthe--ee SDE/ODE methods [33]. We denote the methods that use forward-time SDES in [33] such as Variance Exploding (VE) SDE/Variance Preserving (VP) SDE/ variinie Preserving (sub-VIL cuub and the corresponding backward SDE/ODE, A-B, where A (VE, VP, sub-VP) and ε (SDE, ODE). We follow model selection protocol in [33], which selects the checkpoint with the smallest FID score over the course of training every 50k iterations.

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Figure 3: Uncurated samples on datasets of increasing resolution. From left to right: CIFAR-10
32 x 32, CelebA 64 x 64 and LSUN bedroom 256 x 256.
Numerical Solvers The backward ODE (Eq. (6)) is compatible with any general purpose ODE solver. In our experiments, the default solver of ODEs the black box solver the Seipy library [37] with the RK45 [7] method (RK45), unless otherwise specified. For VE/VP/subV--SDEs. we use the predictor-corrector (PC) sampler introduced in [33]. For VP-SDE, we apply the
predictor-only sampler, because its performance is on par with the PC sampler while requiring half computation.
Results For quantitative evaluation CIFAR-10, report the Inception [29] (higher is better) and FID [13] scores (lower is better) in Table We also include our preliminary experimental results on weaker architecture NCSNv2 [32] in Appendix D.2. We measure the inference speed by the average NFE (number of function evaluation). We also explicitly indicate which methods belong to the invertible flow family.
Our main findings are: (1) PFGM achieves the best Inception scores and FID scores among the normalizing flow models. Specifically, PFGM obtains Inception score of 9.68 and FID score of 2.48 using the DDPM++ deep architecture. To our best knowledge, these are the highest FID and Inception scores by flow models CIFAR-10. (2) PFGM achieves 10x 20x faster inference speed than the SDE methods using the similar architectures, while retaining comparable sample quality. As shown in Table PFGM requires NFEs of 10 whereas the SDE methods typically use 1000 2000 inference steps. PFGM outperforms all the baselines on DDPM++ in all metrics. In addition, PFGM generally samples faster than other ODE baselines with the same RK45 solver. (3) The backward ODE in PFGM is compatible with architectures with varying capacities. PFGM consistently outperforms other ODE baselines on DDPM++ (Table 1) or NCSNv2 (Appendix D.2) backbones. (4) PFGM shows scalability to higher resolution datasets. In Appendix D.I. we show that PFGM are capable scale-up up to LSUN bedroom 256 256. In particular, PFGM has comparable performance with VE-SDE with 15x fewer NFE.
In Fig. 3, visualize the uncurated samples from PFGM CIFAR-10, CelebA 64 x 64 and LSUN bedroom 256 x 256. We provides more samples in Appendix E.
4.2 Failure of VE/VP-ODEs on NCSNv2 architecture
In our preliminary experiments on NCSNv2 architectures. we empirically observe that the VE/VP-ODES have FID scores greater than 90 on CIFAR-10. In particular. VE/VP-ODES can only generate decent samples when apply- ing the Langevin dynamics corrector, and even then, their performances are still inferior to PFGM (Table 9, Table 10). The poor perfor- mance on NCSNv2 stands in striking contrast to their high sample quality on NCSN++/DDPM++ in [33]. It indicates that the VE/VP-ODEs are
PEGM
Score-based VE-ODE
Pill
ott)
16
Figure 4: Sample norm distributions with varying time variables (o for VE-ODE and for PFGM)

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VE-ODE (Buler)
PFGM (Euler)
CIFAR-10
Genner COrIgI
1239
FROM (tsker
Noxeer PARtOIN
Noxerr sartsine Cormacer
150
norm
100
200
FID
100
1
12
200
00
5.3
15.0 12.5
(tt)
z(t')
NFE
(a) Ntrm-o(t) in VE-ODE
(b) Norm-z((') in PFGM
(c) FID vs. NFE on CIFAR-I0
Figure 5: (a) (trm-a(t) relation during the backward sampling of VE-ODE (Euler). (b) Norm-z(t') relation during the backward sampling of PFGM (Euler). The shaded areas mean the standard deviation of norms. (c) Number of steps versus FID score.
more susceptible to estimation errors than PFGM. We hypothesize that the strong norm-o correla- tion seen during the training of score-hased models causes the problem.
For score-based models, the la norms of perturbed training samples and the standard deviations (t) of Gaussian noises have strong correlation, e.g., norm (tt for large o (t) in VE [33]. In contrast, as shown in Fig. 4, PFGM allocates high mass across a wide spectrum of the training sample norms. During sampling, VE/VP-ODEs could break down when the trajectories of backward ODEs deviate from the norm-o (1) relation to which most training samples pertain. The weaker NCSNv2 backbone incurs larger errors and thus leads to their failure. The PFGM is more resistant to estimate errors because of the greater range of training sample norms.
To further verify the hypothesis above, split batch of VE-ODE samples into cleaner and noisier samples according to visual quality (Fig. 8(a)). In Fig. 5(a), we investigate the relation for cleaner and noisier samples during the forward Euler simulation of VE-ODE when (t) 15. We the trajectory of cleaner samples stays close to the norm-o(t) relation (the red dash line), whereas that of the noisier samples diverges from the relation. The Langevin dynamics corrector changes the trajectory of noisier samples to align with the relation. Fig. 5(b) further shows that the anchored variable z(t') and the norms in the backward ODE of PFGM to the robustness against the imprecise estimation on NCSNv2. We defer more details to Appendix C.
that
strongly correlated, giving rise
4.3 Effects of step size in the forward Euler method
In order to accelerate the inference speed of ODEs, we can increase the step size (decrease the NFEs) in numerical solvers such as the forward Euler method. It also enables the trade-off between sample quality and computational efficiency in real-world deployment. We study the effects of increasing step size on PFGM, VP-ODE and DDIM [30] using the forward Euler method, with a varying NFE ranging from 10 to 100.
In Fig. 5(c), we report the sample quality measured by FID scores on CIFAR-10. As expected, all the methods have higher FID scores when decreasing the NFE. We observe that the sample quality of PFGM degrades gracefully as we decrease the NFE. Our method shows significantly better robustness to step sizes than the VP-ODE, especially when only taking a few Buler steps. In addition, PFGM obtains better FID scores than DDIM on most NFES except for 10 where PFGM is marginally worse. This suggests that the PFGM is promising method for accommodating instantancous resource availability, as high-quality samples can be generated in limited steps.
4.4 Utilities of ODE: likelihood evaluation and latent representation
Similar to the family of discrete normalizing flows [6, 19, 14] and continuous probability flow [33], the forward ODE in PFGM defines an invertible mapping between the data space and latent space with a known prior. Formally, we define the invertible forward M mapping by integrating the corresponding forward ODE di x, = vV))) z, =)dl'' of Eq. (6):
f log San
x(log 2max) MMxxllog Zmin)) ))= x(log Emin)
v(x(")))
Swin

---

where log 2min lmg 2max are the starting/terminal time in the forward ODE. The forward mapping transfers the data distribution the prior distribution Pprier on the 3max hyperplane (cf. Section 3.3):
Ppeior x(log ?max) = MMp( (log *)in))). The invertibility enables likelihood evaluation and creates a meaningful latent space on the 2 Zmax hyperplane. In addition, we can adapt the computational constraints by adjusting the step size or the precision in numerical ODE solvers.
Likelihood evaluation We evaluate the data likelihood by the instantaneous change-of-variabe formula [4, 33]. In Table 2, we report the bits/dim on the uniformly de- quantized CIFAR-10 10 test set and compare with exist- ing baselines that use the same setup. We observe that PFGM achieves better likelihoods than discrete normaliz- ing flow models, even without maximum likelihood train- ing. Among the continuous flow models, sub-VPa VP-ODE shows the lowest bits/dim, although its sample quality is worse than VP-ODE and PFGM (Table 1). The explo- ration of the seeming trade-off between likelihood and sample quality is left for future works.
Table 2: Bits/dim on CIFAR-10
bitsddim ;
RealNVP [6]
3.49
[19]
3.35
Residual Flow [3]
3.28
Flow++ [14]
3.29
DDPM (Z) [16]
3.70"
DDPM- + backbone
VP-ODE [33]
3.20
VE-VP-OOR D.b-VP--DE [33]
3,02
PFGM (ours)
3.19
Latent representation Since the samples are uniquely identifiable by their latents via the invertible mapping M, PFGM further supports image manipulation using its latent representation on the 2 Amax hyperplane. We include the results of image interpolation and the temperature scaling [6, 19, 33] to Appendix D..4 and Appendix D.5. For interpolation, it shows that we can travel along the latent space to obtain perceptually consistent interpolations between CelebA images.
Conclusion
We present a deep generative model by solving the Poisson equation whose source term is the data distribution. We estimate the normalized gradient field of the solution in an augmented space with additional dimension. For sampling, we devise a backward ODE that exponential decays on the physically meaningful additional dimension. Empirically, our approach has currently best performance over the stochastic methods. Our backward ODE shows greater stability against errors than popular ODE-based methods, and enables efficient adaptive sampling. We further demonstrate the utilities of the forward ODE on likelihood evaluation and image interpolation. Future directions include improving the normalization of Poisson fields. More principled approaches be used to get around the divergent near- field behavior. For example, we may exploit renormalization, a useful tool in physics, to make the Poisson field well-behaved in near fields.
other normalizing flow baselines, and achieving 10x to 20x acceleration
Acknowledgements
We are grateful to Shangyuan Tong, Timur Garipov and Yang Song for helpful discussion. We would like to thank Octavian Ganea and Wengong Jin for reviewing an early draft of this paper. YX and TJ acknowledge support from MIT-DSTA Singapore collaboration, from NSF Expeditions grant (award 1918839) "Understanding the World Through Code", and from MIT-IBM Grand Challenge project. ZL and MT would like to thank the Center for Brains, Minds, and Machines (CBMM) for hospitality, ZL and MT are supported by The Casey and Family Foundation, the Foundational Questions Institute, the Rothberg Family Fund for Cognitive Science and IAIFI through NSF grant 7H1-901
10

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Appendix G.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix H.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Ap- pendix A.
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed reproduce the main ex- perimental results (either the supplemental material or as a URL)? [Yes] See the abstract.
(b) Did you specify all the training details (c.g.. data splits, hyperparameters, how they were chosen)? [Yes] See Appendix B.
(c) Did you report error bars (e.g.. with respect to the random seed after running experi- ments multiple times)? [N/A]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] All the experiments single NVIDIA 1100 GPU.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [N/A] The assets are public/open-source datasets and codes.
(c) Did you include any assets either in the supplemental material as URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you'l 're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
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14


Reviews:
[{"review_id": "c2GIWcy00bh", "reviewer": "Reviewer_RM6S", "summary": "This paper propose a generative model based on Poisson equation. The authors augment each data point with a scalar variable z and model the true data point as points centered as $z = 0$  hyperplane. During training, the authors define a empirical Poisson field based on the current batch of data and run the forward ODE to calculate the normalized field $V_{B_L}$. They train the model $f_\\theta$ to regress on $V_{B_L}$ via L2 loss. During the backward process, the author start from a large $z = z_{max}$ and graduately walk back to $z=0$ to generate samples. Using this model, the authors try to solve the problem in the original VE/VP ODES that the $l2$ norm of perturbed training data and the standard variance of Gaussain perturbation kernel may have strong correlation. They use various experiments to demonstrate the performance and effectiveness of the propose model.", "questions": "1. The sample quality: The authors claim that their model achieves comparable and even better performance as [1, 2] based on the scores reported in Table 1 and 2. However, the samples shown in Figure 3 seems to be worse than that shown in [1]. For example, comparing Figure 3 with Figure 14 in [1], it seems that the face samples in Figure3b contain more obvious artifacts. Do you have any explanation for that?\nAlso, the current paper only shows 64 x 64 samples while [1,2] can do 256 x 256. \nThus, it might be hard to judge whether the proposed method does improve the baseline.\n\n2. The sampling time: Table1 and Table2 compare the sampling time through NFE. But since PFGM uses different ODE equation, would it be any time difference for runing each ODE step? Or would you please also comparing the total sampling time (maybe based on wall-clock time)?\n\n3. The sensitivity of the hyperparameters: The proposed model introduce several hyperparametes like MAX_STEP, $z_{min}$, $z_{max}$, etc. How sensitive is the model if we change these parameters. Especially, Theorem 1 assumes $r \\rightarrow \\infty$. Then do we need to  carefully choose $z_{max}$ to meet this requirement?\n\n[1] Improved techniques for training score-based generative models.\n\n[2] Score-Based Generative Modeling through Stochastic Differential Equations \n", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "For the strength of the paper:\n\nI think this model proposed a new form of the SDE/ODE based generative model. By using Poisson's equation, the proposed PFGM model may have weaker $norm-z(t')$ correlation and propose good samples with less computation. The paper is well written and the ideas are easy to follow.  \n\nFor the weakness:\n1. The first weakness of the proposed method is the large computation needed during training. As shown and discussed in Limitations and Future Directions in the appendix, using more complex Poisson field, the PFGM is more than 2 times slower than the score-based models. This may limit the model from going to high-resolution and high-quality data. \n\n2. Also as pointed out by the authors, besides the training time, the model defines an empirical Poisson field based only on a subset of the data, which means at each step, the  model $f_\\theta$ is trying to fit a different underlying distribution and the estimation is biased. Although the authors try to mitigate the problem by using larger batch size, as the image space is highly multimodal, just enlarging batch size is not the final solution. This might also prevent the model from scaling up.\n3. I also have some questions, please check the next part.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "MIpJKQ95NwL", "reviewer": "Reviewer_cmyU", "summary": "The manuscript proposes using the Poisson equation in higher dimension to use as a generative model. Several numerical case studies are reported to illustrate the benefit of the proposed model in terms of computational speed and accuracy.", "questions": "- What is the mathematically rigorous justification of making the time derivative of $\\boldsymbol{x}$  proportional to $\\boldsymbol{E}(\\boldsymbol{x})$? Can you do away with the heuristic argument in lines (84)-(85) and give mathematical arguments? Otherwise, one can think of infinitely many generative models (why just hemisphere?) and infinitely many vector fields for the same geometry of the Poisson equation.\n\n", "limitations": "The authors have stated some limitations. However, the main limitation I see, that is, the mathematically rigorous motivation/justification of the proposed method, is not discussed. I would have expected the authors to acknowledge this and put it in the Section I.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- The paper proposes an interesting physics-motivated heuristic for generative models. Despite the initial idea being interesting, the reviewer gets the feeling that the authors did not spend efforts to make the ideas rigorous instead focused more on showcasing the idea on some examples through more fixes and heuristics along the way.\n\n- Typos and carelessness galore. Some examples: $G(\\boldsymbol{x},\\boldsymbol{y})$ in equation (2) is valid only if the dimension $N\\geq 3$, otherwise (for $N=2$) the formula is not valid and we need a logarithmic fundamental solution. The manuscript did not state this assumption on $N$. Line 191: \"radical projection\" --> \"radial projection\". In line 86: the positive function $f(\\boldsymbol{x})$ is introduced without any other restrictions. We cannot guarantee Lipschitzness of the vector field $f(\\boldsymbol{x})\\boldsymbol{E}(\\boldsymbol{x})$, and hence the existence-uniqueness of the forward and backward flows without any assumptions whatsoever. Notice that here regularity of $\\boldsymbol{E}$ inherited from $\\varphi$ which is in turn inherited from the regualrity of $\\rho$, is not enough for the forward and backward ODE solutions' existence-uniqueness. In line 108: $p(\\boldsymbol{x})$ is introduced as a probability distribution while in line 109 it is equated with $\\rho(\\boldsymbol{x})$ which was defined to be a density in line 68.\n\n- The proofs, like the writing throughput the paper, are all heuristic lacking mathematical rigor and ignores all technical issues similar to the above. For instance, the crucial issue of whether the Poisson equation is guaranteed to have unique $C^2$ solution is not treated. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "AhszDnhbxPd", "reviewer": "Reviewer_F5nf", "summary": "The authors of this paper propose a physics based generative model named Possion flow generative model (PFGM). In this model, the source function corresponds to the data distribution; and a dual ODEs serve as mapping between the data distribution and the positive hemisphere with radius r. The generative process starts from a point on the hemisphere, follows the negative electric field, and finally generate a sample. This model was tested on CIFAR-10 and CelebA and compared with state of the art image generation models including score-based methods and obtained comparable or better performance. Furthermore, it is shown that PFGM's generation process is much faster than existing methods.", "questions": "1. It is highly recommended that PFGM should be investigated on a third dataset.\n2. It would be interesting to see a direct comparison with ViT based image generation models. ", "limitations": "Potential social impacts are discussed in the supplementary file.  ", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. Inspired by Poisson equation in physics, the idea of this deep generative model is interesting. To avoid generated data points collapse at one point, augmented dimension is applied.  \n2. The generation process in PFGM is more efficient than existing models for image generation. \n\nWeaknesses:\n1. Quantitative analysis unveils that PFGM seems to work better on larger datasets. This should be furthered investigated on a third dataset. \n2. The authors should discuss and compare (if possible) with ViT based generative model, e.g., https://arxiv.org/abs/2107.04589. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept