Lie Meets Network Dynamics: Exact Macroscopic Reductions (Finite Systems)

What the math gives to ML

The paper provides a constructive mechanism for exact dimensional reduction of finite mean-field networks: if the nodal vector field belongs to a finite-dimensional Lie algebra of vector fields, its solutions obey a nonlinear superposition principle. An n-node, d-dimensional network can then be represented by m fundamental solutions plus time-independent node constants, with macroscopic dimension md rather than nd and invariant-leaf dimension d(n-m). A direct neural-network transfer is to build recurrent or neural-ODE layers from Lie-algebraic nodal vector fields, evolve only the fundamental macroscopic trajectories, and reconstruct all node states through the superposition map. This creates an exactly compressible population layer whose reconstruction error and long-horizon stability can be tested against the unreduced system.