Variational Trajectory Flow Matching with a Non-Linear Interpolant

Simulation-free methods such as Trajectory Flow Matching learn dynamics by regressing the time derivative of a local interpolant placed between consecutive observations. The interpolant defines the regression target, and under sparse, irregular sampling, it underrepresents the dynamics and bounds what can be learned. We show that no fixed interpolant dominates: the best of linear, quadratic, spline and quintic changes with the system and with the sampling density. We propose to learn the interpolant instead. An amortized implicit neural representation (INR) is fitted to the training trajectories and frozen as a teacher supplying the velocity target to a learnable velocity field. We further frame the velocity field within a variational framework, yielding a probabilistic forecaster that, at inference, requires only the velocity field and no SDE solver. We evaluate our method on a range of dynamical systems: it consistently achieves higher accuracy and more stable long-horizon rollouts under sparser observations, at matched network capacity.