标题： 一种通过守恒定律相互作用的自组织系统的统一方法 显示英文标题

标题： A unifying approach to self-organizing systems interacting via conservation laws

摘要： 我们提出了一种统一的框架，使用基于局部守恒定律的广义投影算子来嵌入和分析动态系统。 通过将物理、生物和工程系统表示为具有关联矩阵和环路矩阵的图，我们推导出一对偶投影算子，用于分解网络通量和势能。 这种形式主义符合非平衡热力学原理，并能够捕捉由通量-驱动关系和局部约束所支配的广泛系统类别。 我们通过PRojective Embedding of Dynamical Systems（PrEDS）方法将这一方法扩展到集体动力学，该方法将低维动力学提升到高维空间，从而实现原始动力学的复制和恢复。 当系统属于PrEDS类别时，其集体行为可以通过投影到平均场空间进行有效近似。 我们在多个领域展示了PrEDS的多功能性，包括电阻和忆阻电路、自适应流网络（如黏菌）、弹性弦网络和粒子群体。 值得注意的是，我们建立了PrEDS与群体动力学之间的直接对应关系，揭示了优化和自组织的新见解。 我们的结果为分析复杂网络系统和设计通过局部相互作用自组织的系统提供了通用的理论基础。 显示英文摘要

摘要： We present a unified framework for embedding and analyzing dynamical systems using generalized projection operators rooted in local conservation laws. By representing physical, biological, and engineered systems as graphs with incidence and cycle matrices, we derive dual projection operators that decompose network fluxes and potentials. This formalism aligns with principles of non-equilibrium 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. When systems fall within the PrEDS class, their collective behavior can be effectively approximated through projection onto a mean-field space. 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. Notably, we establish a direct correspondence between PrEDS and swarm dynamics, revealing new insights into optimization and self-organization. Our results offer a general theoretical foundation for analyzing complex networked systems and for designing systems that self-organize through local interactions.