Incorporate physical constraints into latent-space interpolation

Investigate whether incorporating physical constraints, such as volume conservation across breakup events or curvature evolution governed by surface tension, into the SHROM training objective or trajectory fitting can reduce the reconstruction-limited interpolation error for fluid-interface morphologies.

Background

SHROM organizes fluid-interface shapes according to geometric similarity but contains no fluid-mechanical constraints. Consequently, the latent-space regions between observed shapes may include morphologies that are geometrically plausible yet physically inadmissible. The paper identifies two possible extensions: penalizing physically invalid intermediate states during training, or constraining the path used to traverse the latent space between observations.

Neither extension is implemented in the paper. The authors specifically mention conservation of volume between breakup events and curvature evolution consistent with surface tension as candidate physical constraints, and identify constrained path fitting as the cheaper initial experiment.

References

Two routes are open, at different cost. A term in the training objective could penalise physically inadmissible intermediate states: volume that is not conserved between break-up events, or curvature evolution inconsistent with surface tension. This shapes the latent itself, so that the space between observations contains only realisable morphologies, and is the more fundamental of the two. Alternatively the path fitting could be constrained. The schemes used here are purely geometric --- splines and geodesics that know nothing of the dynamics --- and even a scalar conservation law imposed along the trajectory would tailor the traversal between observations without retraining anything. The second is much the cheaper experiment and is where we would begin.

— A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology  (2609.37947 - Hashemi et al., 29 Sep 2026) in Section 5.5, “Limits of a purely geometric representation” (Section \ref{sec:disc:physics})