Real-world complex systems often miss high-fidelity physical descriptions and are typically subject to partial observability. Learning the dynamics of such systems is a challenging and ubiquitous problem, encountered in diverse critical applications which require interpretability and qualitative guarantees. Our paper addresses this problem in the case of sparsely observed probability distribution flows, governed by ODEs. Specifically, we devise a white box approach -dubbed Symbolic Distribution Flow Learner (SDFL)- leveraging symbolic search with a Wasserstein-based loss function, resulting in a robust model-recovery scheme which naturally lends itself to cope with partial observability. Additionally, we furnish the proposed framework with theoretical guarantees on the number of required snapshots to achieve a certain level of fidelity in the model-discovery. We illustrate the performance of the proposed scheme on the prototypical problem of Kuramoto networks and a standard benchmark of single-cell RNA sequence trajectory data. The numerical experiments demonstrate the competitive performance of SDFL in comparison to the state-of-the-art.
Dakhmouche et al. (Wed,) studied this question.