Establish latent-manifold curvature, injectivity, and closure properties

Establish whether the nine-dimensional normalized latent-shape representation has sufficiently small curvature, is injective along relevant coagulation trajectories, and provides exact closure for the coagulation dynamics.

Background

The AeroMELD-Coag model represents aerosol populations using total particle number and a nine-dimensional normalized latent-shape coordinate. The paper proves regularity properties for the latent trajectory and shows that the complete state cannot recur while coagulation is active, but the projected latent curve may still bend or intersect itself.

These unresolved geometric and closure properties determine whether distinct particle populations can share the same latent coordinate and yet require different coagulation tendencies. Resolving them is therefore important for establishing whether the learned latent state is mathematically sufficient for reliable reduced dynamics.

References

Small curvature, injectivity, and exact closure remain unresolved.

— Learning Aerosol Coagulation Dynamics in Latent Space with a Scale-Covariant Neural ODE  (2609.31271 - Tang et al., 25 Sep 2026) in Section 2.2, subsection “Latent coagulation paths”

The relationship between these diagnostics and aerosol mass beyond the selected cases remains unresolved.

— Learning Aerosol Coagulation Dynamics in Latent Space with a Scale-Covariant Neural ODE  (2609.31271 - Tang et al., 25 Sep 2026) in Appendix D, subsection “Decoded total-mass retention”