In particular at time t equals 1, the marginal probability p_1 is a mixture distribution that closely approximates the data distribution q. Interestingly, we can also define a marginal vector field, by "marginalizing" over the conditional vector fields. We assume p_t(x) is greater than 0 for all t and x. The marginal vector field is an integral of the conditional vector field weighted by the conditional probabilities. Here, u_t(dot given x1) is a conditional vector field that generates p_t(dot given x1). It may not seem apparent, but this way of aggregating the conditional vector fields actually results in the correct vector field for modeling the marginal probability path. Our first key observation is this: The marginal vector field generates the marginal probability path. This provides a surprising connection between the conditional VFs (those that generate conditional probability paths) and the marginal VF (those that generate the marginal probability path). This connection allows us to break down the unknown and intractable marginal VF into simpler conditional VFs, which are much simpler to define as these only depend on a single data sample. We formalize this in a theorem. Subsection 3.2: Conditional Flow Matching  Unfortunately, due to the intractable integrals in the definitions of the marginal probability path and VF, it is still intractable to compute u_t, and consequently, intractable to naively compute an unbiased estimator of the original Flow Matching objective. Instead, we propose a simpler objective, which surprisingly will result in the same optima as the original objective. Specifically, we consider the Conditional Flow Matching (CFM) objective, which is the expected squared norm of the difference between v_t(x) and u_t(x given x1). Here, t is sampled from a uniform distribution between 0 and 1, x1 is sampled from q(x1), and now x is sampled from p_t(x given x1). Unlike the FM objective, the CFM objective allows us to easily sample unbiased estimates as long as we can efficiently sample from p_t(x given x1) and compute u_t(x given x1), both of which can be easily done as they are defined on a per-sample basis. Our second key observation is therefore: The FM and CFM objectives have identical gradients with respect to theta. That is, optimizing the CFM objective is equivalent (in expectation) to optimizing the FM objective. Consequently, this allows us to train a CNF to generate the marginal probability path p_t---which in particular, approximates the unknown data distribution q at t equals 1---without ever needing access to either the marginal probability path or the marginal vector field. We simply need to design suitable conditional probability paths and vector fields. We formalize this property in a theorem. Section 4: Conditional Probability Paths and Vector Fields  The Conditional Flow Matching objective works with any choice of conditional probability path and conditional vector fields. In this section, we discuss the construction of p_t(x given x1) and u_t(x given x1) for a general family of Gaussian conditional probability paths. Namely, we consider conditional probability paths of the form of a normal distribution, where mu is the time-dependent mean of the Gaussian distribution, while sigma describes a time-dependent scalar standard deviation (std). We set the mean at time 0 to be 0 and the standard deviation at time 0 to be 1, so that all conditional probability paths converge to the same standard Gaussian noise distribution at t equals 0. We then set the mean at time 1 to be x1 and the standard deviation at time 1 to be a very small value, so that p_1(x given x1) is a concentrated Gaussian distribution centered at x1. There is an infinite number of vector fields that generate any particular probability path, but the vast majority of these is due to the presence of components that leave the underlying distribution invariant---for instance, rotational components when the distribution is rotation-invariant---leading to unnecessary extra compute. We decide to use the simplest vector field corresponding to a canonical transformation for Gaussian distributions. Specifically, consider the conditional flow which is an affine transformation that maps a standard Gaussian to a normally-distributed random variable with the specified mean and standard deviation. This flow then provides a vector field that generates the conditional probability path. Since this flow is a simple (invertible) affine map we can solve for the vector field u_t in a closed form. A theorem states that for a Gaussian probability path, the unique vector field that defines its corresponding flow map has a specific form related to the time derivatives of its mean and standard deviation. Consequently, this vector field generates the Gaussian path. Subsection 4.1: Special instances of Gaussian conditional probability paths  Our formulation is fully general for arbitrary functions for the mean and standard deviation, and we can set them to any differentiable function satisfying the desired boundary conditions. We first discuss the special cases that recover probability paths corresponding to previously-used diffusion processes.