Physical learning in dynamical systems*

Recent years have seen the emergence of new classes of self-learning machines - physical systems that learn to perform desired functions when exposed to examples of use in an adaptive fashion, without the use of a processor. The physical learning machines studied thus far are equilibrium systems minimizing scalar functions (e.g. energy in a mechanical network or power in an electrical network), where the interactions between physical degrees of freedom are reciprocal and symmetric. However, most biological learning systems, including neuronal networks and gene regulatory networks, contain non-symmetric interactions that are often directed. While some progress has been made for training fixed points in recurrent neural networks, methods for physically training more complicated dynamical behaviors and attractors like limit cycles have yet to be developed. Such continuous attractors play an important role in neural tasks like orientation perception. In this work we generalize a contrastive learning approach for equilibrium systems, Coupled Learning, to derive local, physically realizable rules for training general dynamical systems described by sets of ordinary differential equations. We discuss how physical learning in dynamical systems is intrinsically constrained by causality, and yet demonstrate how such physical learning rules are practical in training different types of dynamical systems motivated by biology.