Learning a vector field from snapshots of unidentified particles rather than particle trajectories

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, preventing us from accessing individual trajectories but only snapshot samples. 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. This is because of their inherent design for piecewise perfect interpolation without considering the collective information from all snapshots. In contrast, we propose a new method that leverages an iterative projection mechanism inspired by Schrödinger bridges. This method does not necessitate the inferred dynamics to precisely intersect with every snapshot, offering a significant 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 framework for dynamics inference. We test our method using well-known simulated parametric models from ecology and systems biology.