Q1: is it possible to train the Flow Matching model to directly predict x1​ (the target data) rather than the velocity v_target?

Yes.

1. The Standard Flow Matching: “pred_vel” Objective
   In the standard Flow Matching framework with a straight-line path $zt=(1-t)z0+tx1$​:

2. The “pred_data” Objective (Predicting x1​ directly)
   Let’s consider an alternative where your neural network, let’s call it $g\theta (zt,t)$, is trained to directly predict the final data point x1​.

Mathematical Equivalence (for Straight Paths)

For the specific straight-line path $zt=(1-t)z0+tx1$, the velocity $v_{target}$​ and the target data x1​ are directly related:

We know:
$$zt=(1-t)z0+tx1$$

From this, we can rearrange to solve for x1​:

$$\begin{gathered} tx1=zt-(1-t)z0 \\ x1=(zt-(1-t)z0)/t(t!=0) \\ x1=(zt-z0+tz0)/t \\ x1=(zt-z0)/t+z0 \end{gathered}$$

And we also know $v_{target}=x1-z0$. So, we can see the relationship: $x1=z0+v_{target}$

This means:

• If your model fθ​(zt​,t) predicts vtarget​, then it implicitly predicts x1​ as fθ​(zt​,t)+z0​.
• If your model gθ​(zt​,t) predicts x1​, then it implicitly defines a velocity as gθ​(zt​,t)−z0​.

Conclusion: Because of this direct algebraic relationship for the straight path, minimizing one loss function can be equivalent to minimizing the other, provided the models are parameterized appropriately.

Connection to Diffusion Models

“pred_data” is very perceptive because it highlights the strong conceptual overlap with Denoising Diffusion Probabilistic Models (DDPMs).

• In DDPMs, the model is typically trained to predict the noise added to a sample (ϵ).
• However, it can be mathematically shown that predicting the noise ϵ is equivalent to predicting the denoised image x0​ (the clean data source). So, many diffusion models effectively learn a function that maps a noisy xt​ to its denoised counterpart x0​. This is essentially a “pred_data” objective in a noisy context.

Q2: Can We Just Predict x1​ in One Shot When Sampling?

No.

The reason is the complexity of the mapping and the design of the model’s capacity.

1. Complexity of z0​→x1​: The transformation from a simple noise distribution to a complex data distribution is highly non-linear and often involves high dimensionality. A single neural network trying to learn this full mapping directly from t=0 might struggle.
2. Learning Local Dynamics: The ODE-based approach (whether Flow Matching or Diffusion) breaks down this enormous task into learning many simpler local transformations. The neural network gθ​(zt​,t) (or fθ​(zt​,t)) is trained to understand how to move from any intermediate point zt​ at time t towards the data manifold. It’s learning the “micro-steps” of the transformation, not the “macro-step” in one go.
3. Accuracy and Robustness: By taking many small steps, the ODE solver can accumulate these local transformations precisely. The model doesn’t need to be perfect at predicting x1​ from t=0 directly; it just needs to be good at predicting the velocity (or implied x1​) from any zt​ along the path. This makes the learning problem much more tractable.

Conclusion: So, even if your network outputs x1​ during training, it’s still participating in an ODE-solving process during inference. The ODE is the mechanism that allows us to compose many simple, learned local changes into a complex global transformation.