Geometric Morphology Learning: Empowering Foundation Models from Structure to Spacetime Intelligence
Li Sun, Philip S. Yu
Abstract
Geometry is the very form that connects space, time, and matter, serving as the fundamental fabric of the physical world, from continuous fields to discrete structures. Despite its importance, deep learning is inherently blind to geometry. Riemannian geometry stands as a well-established mathematical foundation for studying geometry, yet it remains largely incompatible with deep learning architectures. To bridge this gap, we establish a new learning framework--Geometric Morphology Learning--that learns geometric or topological structures, shapes or conformations for representation learning, reasoning and action modeling. Moving beyond early attempts limited to specific geometric priors, this theory centers on modeling the intrinsic geometry of objects and learning from such geometric or topological knowledge. Specifically, we construct a neural bundle, a unified scaffolding linking local and global structures, and formulate differentiable operators, including connection and curvature, thereby enabling the study of geometry with advanced geometric concepts such as holonomy and cohomology. Consequently, Geometric Morphology Learning opens new horizons in foundation models, reasoning and planning, graph modeling, physical simulation, PDE systems, and dynamics, marking a fundamental shift toward instilling systematic geometry in artificial intelligence.