Flow models VI: equilibrium matching

One of the particularities with flow matching methods is that they're inherently out-of-equilibrium. When solving the ODE $\dot{X}_t = v_t(X_t)$, we're not really minimizing a natural function; indeed, we chose to stop at a finite time horizon $t=1$, but in general we could very well continue after this time, leading to non-sensical results. Energy-Based Models, on the contrary, were known to be quite natural: EBMs are just a fancy name for classical likelihood learning. They consist, given a dataset $x_1, ..., x_n$ from $p$, in directly learning $p$ by parametrizing its log-likelihood: the model we're learning is $p_\theta(x) = e^{-U_\theta(x)}/Z_\theta$, where $Z_\theta = \int e^{-U_\theta}$ and $\theta$ are the parameters of the neural network. Sampling from an EBM is just sampling from a Gibbs density using any Monte-Carlo or Langevin method, until equilibrium is reached.

How this training is done in practice is a vast topic and I won't cover it; however, right now in the second half of 2025, it seems to me that while EBMs represent a kind of holy grail from a theoretical point of view, they've always been suffering from very hard training difficulties. In spite of many techniques for scaling them, no EBM can rival diffusion-based models in the tasks of image or video generation.

In a recent paper, the EqFM paper by R. Wang and Y. Du introduced an interesting variation of flow matching, interpolating between FMs and EBMs. Suppose that we have an interpolation

where $X_1$ is drawn from the unknown distribution of interest, $\varepsilon$ is a Gaussian and $t$ is between 0 and 1.

The Flow Matching objective consists in learning $\mathbb{E}[x-\varepsilon \mid X_t]$ by minimizing

It had been noted in many works that the ultimately, $v_t(x_t)$ should approximate $x-\varepsilon$ given $x_t$: but $x-\varepsilon$ does not depend on $t$. One could try to get rid of the "noise conditioning" induced by the dependency on $t$, and to learn a time-independent field $v(x)$ to approximate $\varepsilon - x$ without the need to know $t$ (a "blind denoiser" as sometimes read on the litterature), by minimizing the objective

That didn't work better. In the intriguing paper Is Noise Conditioning Necessary for Denoising Generative Models?, the authors showed that getting rid of noise conditioning didn't result in really bad results, but but didn't result in any gains too.

EqFM

Equilibrium Flow Matching tries to learn an energy function which vanishes at the true data points, i.e.

where $\sigma_t$ is a function which should be zero when $t=1$. Suppose that such a function exists and ...

Suppose that $\sigma_t = 1-t$. Indeed, in this case $X_t = X+\sigma_t(\varepsilon-X_1)$, and

This is zero when $t=1$.

problems : what's the energy ? does it converge to local zero-gradient points or to minima ?

References

EqFM