However, these works only consider a restricted setting of Gaussian conditional paths defined by simple diffusion processes with a single parameter---in particular, it does not include our conditional OT path. In another line of works, some proposed finite time diffusion constructions via diffusion bridges theory resolving the approximation error incurred by infinite time denoising constructions. While existing works make use of a connection between diffusion processes and continuous normalizing flows with the same probability path, our work allows us to generalize beyond the class of probability paths modeled by simple diffusion. With our work, it is possible to completely sidestep the diffusion process construction and reason directly with probability paths, while still retaining efficient training and log-likelihood evaluations. Lastly, concurrently to our work, other researchers arrived at similar conditional objectives for simulation-free training of CNFs, while some derived an implicit objective when the vector field is assumed to be a gradient field. Section 6: Experiments  We explore the empirical benefits of using Flow Matching on the image datasets of CIFAR-10 and ImageNet at resolutions 32, 64, and 128. We also ablate the choice of diffusion path in Flow Matching, particularly between the standard variance preserving diffusion path and the optimal transport path. We discuss how sample generation is improved by directly parameterizing the generating vector field and using the Flow Matching objective. Lastly we show Flow Matching can also be used in the conditional generation setting. Unless otherwise specified, we evaluate likelihood and samples from the model using a standard ODE solver at absolute and relative tolerances of 1e-5. Subsection 6.1: Density Modeling and Sample Quality on ImageNet  We start by comparing the same model architecture, i.e., the U-Net architecture from Dhariwal and colleagues with minimal changes, trained on CIFAR-10, and ImageNet 32/64 with different popular diffusion-based losses: DDPM, Score Matching (SM), and Score Flow (SF). Our results alongside these baselines report negative log-likelihood (NLL) in units of bits per dimension (BPD), sample quality as measured by the Frechet Inception Distance (FID), and averaged number of function evaluations (NFE) required for the adaptive solver to reach its a prespecified numerical tolerance, averaged over 50k samples. All models are trained using the same architecture, hyperparameter values and number of training iterations, where baselines are allowed more iterations for better convergence. Note that these are unconditional models. On both CIFAR-10 and ImageNet, Flow Matching with Optimal Transport (FM-OT) consistently obtains best results across all our quantitative measures compared to competing methods. We are noticing a higher than usual FID performance in CIFAR-10 compared to previous works that can possibly be explained by the fact that our used architecture was not optimized for CIFAR-10. Secondly, a model trained using Flow Matching with the OT path on ImageNet at resolution 128x128 shows state-of-the-art FID with the exception of IC-GAN which uses conditioning with a self-supervised ResNet50 model, and therefore is left out of this comparison. Faster training. While existing works train diffusion models with a very high number of iterations, we find that Flow Matching generally converges much faster. For ImageNet 64x64, FM-OT is able to lower the FID faster and to a greater extent than the alternatives. For ImageNet-128, previous work trained for 4.36 million iterations with a batch size of 256, while FM (with a 25% larger model) used 500,000 iterations with a batch size of 1.5k, i.e., 33% less image throughput. Furthermore, the cost of sampling from a model can drastically change during training for score matching, whereas the sampling cost stays constant when training with Flow Matching. Subsection 6.2: Sampling Efficiency  For sampling, we first draw a random noise sample and then compute the final sample by solving the ODE with the trained vector field on the interval t from 0 to 1 using an ODE solver. While diffusion models can also be sampled through an SDE formulation, this can be highly inefficient and many methods that propose fast samplers directly make use of the ODE perspective.