Spatial Signatures of Learning in Self-Learning Networks*

Recent works have demonstrated the ability of physical systems to be trained for functionality and developed general frameworks for learning in physical systems such as electrical, flow and mechanical networks. A great appeal of physical learning systems is their promise of greater interpretability. In such systems, learning and adaptation occur under physical constraints; this affects both the learning and physical Hessians. Learning encodes the functionality in the eigensystem of the system’s physical Hessian, softening low eigenmodes and aligning them with the desired response [1]. As in deep neural networks, learning raises the high eigenmodes of the learning Hessian [1]; those eigenmodes contain the same information as the low eigenmodes of the physical Hessian. We build on [1,2] and use the low eigenmodes of the physical Hessian to build an understanding of how self-learning electrical circuits learn various machine-learning tasks. 

[1] Menachem Stern, Andrea J. Liu, Vijay Balasubramanian, The Physical Effects of Learning, arXiv:2306.12928

[2] Rocks, Jason W. and Liu, Andrea J. Liu and Katifori, Eleni, Hidden Topological Structure of Flow Network Functionality, Phys. Rev. Lett 126 2 028102