Since we directly work with probability paths, we can simply depart from reasoning about diffusion processes altogether. Therefore, in the second example below, we directly formulate a probability path based on the Wasserstein-2 optimal transport solution as an interesting instance. Example I: Diffusion conditional VFs. Diffusion models start with data points and gradually add noise until it approximates pure noise. These can be formulated as stochastic processes, which have strict requirements in order to obtain closed form representation at arbitrary times t, resulting in Gaussian conditional probability paths with specific choices of mean and standard deviation. For example, the reversed (noise-to-data) Variance Exploding (VE) path and the reversed (noise-to-data) Variance Preserving (VP) diffusion path have specific forms for their mean and standard deviation. Plugging these into our theorem gives the corresponding conditional vector fields. Our construction of the conditional VF does in fact coincide with the vector field previously used in the deterministic probability flow when restricted to these conditional diffusion processes. Nevertheless, combining the diffusion conditional VF with the Flow Matching objective offers an attractive training alternative---which we find to be more stable and robust in our experiments---to existing score matching approaches. Another important observation is that, as these probability paths were previously derived as solutions of diffusion processes, they do not actually reach a true noise distribution in finite time. In practice, p_0(x) is simply approximated by a suitable Gaussian distribution for sampling and likelihood evaluation. Instead, our construction provides full control over the probability path, and we can just directly set the mean and standard deviation, as we will do next. Example II: Optimal Transport conditional VFs. An arguably more natural choice for conditional probability paths is to define the mean and the standard deviation to simply change linearly in time. According to our theorem, this path is generated by a specific vector field which, in contrast to the diffusion conditional VF, is defined for all t between 0 and 1. The conditional flow that corresponds to this vector field is a linear interpolation between the noise and the data point. In this case, the CFM loss takes a simple form. Allowing the mean and standard deviation to change linearly not only leads to simple and intuitive paths, but it is actually also optimal in the following sense. The conditional flow is in fact the Optimal Transport (OT) displacement map between the two Gaussians p_0(x given x1) and p_1(x given x1). The OT interpolant, which is a probability path, is defined as a push-forward of the noise distribution by the OT displacement map. Intuitively, particles under the OT displacement map always move in straight line trajectories and with constant speed. Interestingly, we find that sampling trajectory from diffusion paths can "overshoot" the final sample, resulting in unnecessary backtracking, whilst the OT paths are guaranteed to stay straight. Compared to the diffusion path's conditional score function, the OT path's conditional vector field has constant direction in time and is arguably simpler to fit with a parametric model. An interesting observation is that the OT VF has a constant direction in time, which arguably leads to a simpler regression task. Lastly, we note that although the conditional flow is optimal, this by no means imply that the marginal VF is an optimal transport solution. Nevertheless, we expect the marginal vector field to remain relatively simple. Section 5: Related Work  Continuous Normalizing Flows were introduced as a continuous-time version of Normalizing Flows. Originally, CNFs are trained with the maximum likelihood objective, but this involves expensive ODE simulations for the forward and backward propagation, resulting in high time complexity due to the sequential nature of ODE simulations. Although some works demonstrated the capability of CNF generative models for image synthesis, scaling up to very high dimensional images is inherently difficult. A number of works attempted to regularize the ODE to be easier to solve, for example, using augmentation, adding regularization terms, or stochastically sampling the integration interval. These works merely aim to regularize the ODE but do not change the fundamental training algorithm. In order to speed up CNF training, some works have developed simulation-free CNF training frameworks by explicitly designing the target probability path and the dynamics. For instance, some considered a linear interpolation between the prior and the target density but involves integrals that were difficult to estimate in high dimensions, while others considered general probability paths similar to this work but suffers from biased gradients in the stochastic minibatch regime. In contrast, the Flow Matching framework allows simulation-free training with unbiased gradients and readily scales to very high dimensions. Another approach to simulation-free training relies on the construction of a diffusion process to indirectly define the target probability path. It has been shown that diffusion models are trained using denoising score matching, a conditional objective that provides unbiased gradients with respect to the score matching objective. Conditional Flow Matching draws inspiration from this result, but generalizes to matching vector fields directly. Due to the ease of scalability, diffusion models have received increased attention, producing a variety of improvements such as loss-rescaling, adding classifier guidance along with architectural improvements, and learning the noise schedule.