Emergence of Dynamic Similarity in Neural Representations

Physical systems can look very different in dimensional coordinates while still obeying the same governing dynamics. When neural networks learn such systems, do their internal representations organize around the dimensionless relationships identified by classical physics? We study neural networks trained on publicly available physics datasets from seven dynamical systems spanning transport, mechanics, reaction–diffusion, and waves. We find that training consistently reorganizes hidden representations around the dimensionless physical relationships that govern the observed dynamics. This organization can strengthen in two ways: by emphasizing changes that alter the underlying physics or by becoming less sensitive to equivalent dimensional descriptions. Yet independently trained networks need not share the same neural coordinates, suggesting that representational convergence can occur at the level of physical relationships rather than a common latent basis. These results motivate the broader hypothesis that independently learned representations may converge on the relationships that structure the underlying physical world.