Representation Stability in Domain Space: Cross-Domain Validation for Unsupervised Scientific Discovery

Unsupervised learning is increasingly used for scientific discovery in settings where labeled data are unavailable or ill-defined. Validation is critical but challenging, and internal cluster-quality metrics encode geometric or statistical assumptions that scientific data routinely violate, and external ground truth is typically absent. This paper takes the ensemble-based NEMI workflow (Sonnewald, 2026) and develops it into a formal validation principle for unsupervised scientific discovery, evaluated across four Earth science domains. The principle is representation stability in domain space. If a discovered structure reflects a real property of the underlying system, the partition it implies in geographic, sequence, or property space should persist under perturbations of the learned latent representation. The same ensemble that NEMI uses for clustering yields uncertainty estimates in two coordinate systems: 1) the embedding space where clustering operates, and 2) the domain space in which scientific meaning is found. We argue that the domain-space entropy field is the scientifically meaningful diagnostic, since domain-space output from any single fixed embedding is a deterministic projection of within-embedding uncertainty alone. Across ocean ecology, ocean biogeochemistry, ocean dynamics, sea-ice, and atmospheric dynamics, low-entropy regions identify regimes that support stable scientific interpretation, while the geometry of where uncertainty concentrates generates hypotheses that can be tested. We find a recurring pattern, where low-entropy structure within regions that domain intuition identifies as transitional, suggesting candidate refinements to established regime boundaries. We also analyze sensitivity to cluster alignment strategy across domains and show that the spatial structure of the entropy field is preserved under both Greedy and Hungarian alignment, with quantifiable differences tied to split/merge events. NEMI underlies domain-science publications and manuscripts under peer review across these applications, and is contributing to model development toolkits aimed at improving sea-level predictability.