Multimarginal Flow Matching with Adversarially Learnt Interpolants

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at a few discrete time steps, the problem of modelling the dynamics -- and thus inferring the underlying trajectories -- can be solved by multimarginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes certain limitations of existing multimarginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by achieving strong results on trajectory prediction in single-cell RNA sequencing data.