Extension to heterogeneous multi-type particle systems from unlabeled data

Develop an extension of the trajectory-free self-test framework to heterogeneous interacting particle systems with multiple particle types using unlabeled snapshots by jointly identifying particle types and learning the associated coupled weak-form PDEs governing the empirical distributions.

Background

The proposed method assumes a homogeneous particle population so that unlabeled snapshots can be summarized by a single empirical distribution obeying a weak-form stochastic PDE. In many applications, particles belong to different types with distinct interactions and external forces.

Handling multi-type systems from unlabeled data requires disentangling types (a latent-class identification problem) and formulating/estimating coupled weak-form PDEs, which substantially complicates identifiability and inference.

References

Extension to heterogeneous systems with multiple particle types from unlabeled data would require type identification and coupled weak-form PDEs, which is a challenging open problem beyond the scope of the current framework.

— Learning interacting particle systems from unlabeled data  (2604.02581 - Wei et al., 2 Apr 2026) in Section 6 (Conclusion), Limitations paragraph