A Unifying Approach to Self‐Organizing Systems Interacting via Conservation Laws

We present a unified framework for embedding and analyzing dynamical systems using generalized projection operators derived from local conservation laws. By representing physical, biological, and engineered systems as graphs, using incidence and cycle matrices, we construct dual projection operators that decompose network fluxes and potentials. This formalism aligns with principles of nonequilibrium thermodynamics and captures a broad class of systems governed by flux‐forcing relationships and local constraints. We extend this approach to collective dynamics through the PRojective Embedding of Dynamical Systems (PrEDS), which lifts low‐dimensional dynamics into a high‐dimensional space, enabling both replication and recovery of the original dynamics. Further, we show that when systems fall within the PrEDS class, their collective behavior can be effectively approximated by a mean‐field description. We demonstrate the versatility of PrEDS across diverse domains, including resistive and memristive circuits, adaptive flow networks (e.g., slime molds), elastic string networks, and particle swarms. In particular, we establish a formal correspondence between PrEDS and swarm dynamics, revealing new principles of self‐organization and optimization in decentralized systems. Our results offer a general theoretical foundation for analyzing complex networked systems and for designing systems that self‐organize through local interactions.