On the physical interpretation of drifting generative models

Author: Ziming Liu (刘子鸣)

Motivation

I was reading with interest the paper “Generative Modeling via Drifting”, as its use of attraction and repulsion is closely related to our earlier work on PFGM and PFGM++.

The goals of this blog are to:

provide a clear physical interpretation of drifting generative models;
present visualizations across many settings, offering guidance on kernel choice and highlighting potential failure modes.

Physical interpretation

The involvement of neural networks can obscure the intuition, so we first remove them from the story.

The physical picture is straightforward:

There are \(N\) positively charged particles (the data), fixed in space.
There are \(N\) negatively charged particles, randomly initialized.
The negative particles are released and move according to a force field determined by interactions with positive particles (\(V^+\)) and with other negative particles (\(V^-\)). For any pair of particles, the force magnitude is \(f(r)\) (corresponding to their kernel \(K\)). The paper uses \(K(r) = \exp\!\left(-\frac{r}{\tau}\right),\) which corresponds to a Yukawa-type potential. Negative charges are attracted to positive charges and repelled by other negative charges.
Intuitively, the final equilibrium should consist of one-to-one positive–negative pairs, which appear neutral to the outside world, yielding a net force \(V = 0\).

Importantly, the evolution of the negative charges is entirely determined by the force field and does not require a neural network. The role of the network is simply to learn and approximate this evolution, so that it can generalize to unseen configurations at inference time.

In what follows, I focus purely on visualizing the forward dynamics, without training any neural network.

Dependence on \(\tau\)

The paper uses the force \(f(r) = r \exp\!\left(-\frac{r}{\tau}\right).\) We first illustrate how the dynamics depend on the parameter \(\tau\).

\(\tau=0.2\)

\(\tau=1.0\)

\(\tau=5.0\)

Other force fields

We also explore alternative force laws to understand how sensitive the dynamics are to kernel choice.

\(f(r)=1/(r+0.1)\)

\(f(r)=1/(r+1)\)

Explaining balance

The paper reports that the combination \(V^+ - V^-\) performs best, which is exactly what the physical picture suggests. For alternatives such as \(2V^+ - V^-\) or \(V^+ - 2V^-\), the system is no longer neutral. As a result, the negative particles either collapse toward the center of the positive particles or are driven too far away.

We therefore focus on \(f(r) = r \exp(-r),\) and verify this intuition through toy experiments.

\(V^+ - 2V^-\)

\(2V^+ - V^-\)

More examples

Finally, we keep \(f(r) = r \exp(-r)\) fixed and vary the data distribution.

Circle

Square

Overall, drifting dynamics appear surprisingly intricate. While these “failure modes” may not necessarily arise in practical settings, further study is needed to understand how kernels should be chosen and what failure modes may emerge.

Code

Google Colab notebook available here.

I want to thank Ruiqi Ni (@RuiqiNi) for finding a bug in my codes – the reduction should be over axis 0 rather than 1, and mean rather than sum.

Citation

If you find this article useful, please cite it as:

BibTeX:

@article{liu2026diffusion-3,
  title={On the physical interpretation of drifting generative models},
  author={Liu, Ziming},
  year={2026},
  month={February},
  url={https://KindXiaoming.github.io/blog/2026/diffusion-3/}
}