From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints

We proposed a method for introducing geometric inductive biases into the inference of stochastic systems.
The central idea is to view the deterministic flow field as a scaffold upon which system states fluctuate, and to approximate this scaffold in terms of distortions of a metric induced by the observations.
This effectivelly approximates the low-dimensional invariant density (empirical manifold) without the need to project to a lower dimensional space (whose dimensionality would be hard to estimate due to the presence of fluctuations).
The central premise then is that geodesics computed on the empirical manifold consitute the most probable path between consecutive observations in the Onsager-Machlup sense.

This is the last part of my PhD thesis and the first single author paper I wrote.