Learning a Vector Field from Snapshots of Unidentified Particles Rather than Particle Trajectories

Abstract

Practitioners frequently aim to infer dynamical system behaviors using snapshots at certain time points. For instance, in single-cell sequencing, to sequence a cell we must destroy it. So we cannot access a full trajectory of the behaviors of a cell, but we can access a snapshot sample. While stochastic differential equations (SDEs) are commonly used to analyze systems with full trajectory access, the availability of only sparse time samples without individual trajectory data makes traditional SDE learning methods inapplicable. Recent works in the deep learning community have explored using Schrödinger bridges for dynamics estimation from such data. However, these methods are primarily tailored for interpolating between two time points and struggle when asked to infer the underlying dynamics that generate all observed data from multiple snapshots. In particular, a naive extension to multiple points performs piecewise perfect interpolation with- out considering the collective information from all snapshots. In contrast, we propose a new method that leverages an iterative projection mechanism inspired by Schrödinger bridges. Our method does not require that the inferred dynamics precisely match every snapshot, offering a substantial advantage in practical applications where perfect data alignment is rare. By incorporating information from the entirety of the dataset, our model provides a more robust and flexible frame- work for dynamics inference. We test our method using well-known simulated parametric models from systems biology and ecology.