Researchers have developed a theoretical framework explaining how neural networks can naturally discover discrete symbolic structures through gradient-based training.
By lifting neural parameters to a measure space and utilizing Wasserstein gradient flow, the framework demonstrates the emergence of symbolic phenomena under geometric constraints like group invariance.
The framework highlights the decoupling of gradient flow into independent optimization trajectories based on potential functions and a reduction in degrees of freedom, leading to the encoding of algebraic constraints relevant to the task.
The research establishes data scaling laws connecting representational capacity to group invariance, enabling neural networks to transition from high-dimensional exploration to compositional representations aligned with algebraic operations, offering insights for designing neurosymbolic systems.