A Relational Macrostate Theory Guides Artificial Intelligence to Learn Macro and Design Micro

The high dimesionality, non-linearity and emergent properties of complex systems pose a challenge to identifying general laws in the same manner that has been so successful in simpler physical systems. In the seminal work of Anderson on why more is different he pointed to how emergent, macroscale patterns break symmetries of the underlying microscale laws. Yet, less recognized is that these large scale, emergent patterns must also retain some symmetries of the microscale rules. Here we introduce a new, relational macrostate theory (RMT) that defines macrostates in terms of symmetries between two mutually predictive observations, and develop a machine learning architecture, MacroNet, that identifies which symmetries are preserved during the mapping from micro-to-macro. Using this framework, we show how macrostates can be identifed across systems ranging in complexity from the simplicity of the simple harmonic oscillator to the much more complex spatial patterning characteristic of Turing instabilities. Furthermore, we show how our framework can be used for the inverse design of microstates consistent with a given macroscale property - in Turing patterns this allows us to design microstates with a given specification of macroscale spatial patterning, and to identify which parameters most control these patterns. By demonstrating a general theory for how macroscale properties emerge from conservation of symmetries in the mapping from micro-to-macro, we provide a machine learning framework that allows a unified approach to identifying macrostates in systems from the simple to complex, and allows the design of new examples consistent with a given macroscale property.