FM — Flow Matching for Generative Modeling

Continuous Normalizing Flows (CNFs) are a powerful class of generative models that learn invertible transformations between noise and data. However, training CNFs via maximum likelihood requires simulating the ODE during training, which is computationally expensive. We introduce Flow Matching (FM), a simulation-free approach to train CNFs by regressing vector fields that generate probability paths between a source distribution (noise) and a target distribution (data).

We show that FM objectives can be tractably optimized using Conditional Flow Matching (CFM), which has the same gradients as FM but uses conditional probability paths. We demonstrate that FM with Optimal Transport (OT) paths achieves state-of-the-art performance on ImageNet 64×64 and 128×128, with faster training and fewer sampling steps than diffusion models.

Generative modeling has seen remarkable progress with diffusion models achieving state-of-the-art image synthesis. Diffusion models define a fixed probability path from data to noise via a forward SDE/ODE, then learn to reverse it. While effective, diffusion paths are often suboptimal — they require many discretization steps (high NFE) for high-quality samples, and the predefined variance-preserving (VP) paths are not the shortest routes from noise to data.

Continuous Normalizing Flows (CNFs) offer an alternative: they model generation as an ODE $\dot{x}=v_t(x)$ whose flow map transports noise to data. CNFs are flexible and provide exact likelihoods, but training via maximum likelihood requires backpropagating through ODE solvers, making each training step expensive and numerically delicate.

We propose Flow Matching, which directly learns the vector field $v_t$ generating a desired probability path $p_t$, without simulating the ODE during training. Our Conditional Flow Matching (CFM) objective is tractable and provably equivalent (in gradients) to FM. Combined with OT displacement interpolants (straight-line paths), FM achieves better sample quality with fewer NFE than score-based diffusion.

A CNF defines a time-dependent vector field $v_t:\mathbb{R}^d\to\mathbb{R}^d$ and an ODE $\frac{d}{dt}\phi_t(x)=v_t(\phi_t(x))$ with $\phi_0(x)=x$. The flow $\phi_t$ pushes forward a source density $p_0$ (e.g. $\mathcal{N}(0,I)$) to $p_t=[\phi_t]_\# p_0$. Training traditionally maximizes $\log p_1(x_1)$ via the instantaneous change of variables formula, requiring ODE simulation and trace estimation at each step.

Goal: find $v_t$ such that $p_1\approx p_\text{data}$. FM bypasses likelihood by matching $v_t$ to a target field $u_t$ that generates a prescribed path $p_t$ from $p_0$ to $p_\text{data}$.

Given a probability path $p_t$ connecting $p_0$ and $p_1=p_\text{data}$, FM learns $v_\theta$ by minimizing

$$\mathcal{L}_\text{FM}(\theta) = \mathbb{E}_{t\sim\mathcal{U}[0,1],\,x\sim p_t}\big[\,\lVert v_\theta(t,x) - u_t(x)\rVert^2\,\big],$$

where $u_t$ is the marginal vector field that generates $p_t$ via the continuity equation. Problem: $u_t$ is intractable — it depends on the entire data distribution through $p_t(x)$.

Conditional Flow Matching (CFM) uses conditional paths $p_t(x\mid x_1)$ and conditional vector fields $u_t(x\mid x_1)$ for each data point $x_1$:

$$\mathcal{L}_\text{CFM}(\theta) = \mathbb{E}_{t,\,x_1\sim p_1,\,x\sim p_t(\cdot\mid x_1)}\big[\,\lVert v_\theta(t,x) - u_t(x\mid x_1)\rVert^2\,\big].$$

Theorem 2: $\nabla_\theta \mathcal{L}_\text{CFM} = \nabla_\theta \mathcal{L}_\text{FM}$ — CFM 与 FM 梯度相同，但 CFM 的 $u_t(x\mid x_1)$ 是闭式可算的。条件路径由条件流 $\psi_t$ 生成（式6–8）：

$$\psi_t(x\mid x_1): \text{条件流}; \quad p_t(x\mid x_1)=[\psi_t(\cdot\mid x_1)]_\# p_0; \quad u_t(x\mid x_1)=\frac{d}{dt}\psi_t(x\mid x_1)\Big|_{x=\psi_t^{-1}(x\mid x_1)}.$$

We instantiate CFM with Gaussian probability paths. Two choices:

Diffusion path (VP): $p_t(x\mid x_1)=\mathcal{N}(x;\,\alpha_t x_1,\,\sigma_t^2 I)$ matching variance-preserving diffusion schedules — trajectories are curved in data space (Fig. 3).

OT displacement interpolant: for $x_0\sim\mathcal{N}(0,I)$,

$$x_t = \psi_t(x_0\mid x_1) = t\,x_1 + (1-t)\,x_0 \quad\text{(Eq. 21)},$$

with $u_t(x\mid x_1)=x_1-x_0$ — straight-line paths that minimize transport cost between noise and data pairs. We denote FM with diffusion paths as FM-Diff and with OT paths as FM-OT.

We evaluate FM-Diff, FM-OT, and score matching (SM) baselines on CIFAR-10 and ImageNet at 64×64 and 128×128 resolution. All models use the same U-Net architecture. We report NLL (bits/dim), FID, and NFE (number of function evaluations at sampling). Table 1 summarizes main results.

We introduced Flow Matching, a simulation-free framework for training CNFs by regressing vector fields along probability paths. Conditional Flow Matching makes training tractable with provably equivalent gradients. Using OT displacement interpolants, FM-OT achieves strong results on CIFAR-10 and ImageNet with fewer sampling steps and faster training than diffusion-based approaches, opening a path toward efficient continuous-time generative models.