Identifying Traces of Learning in Physical Systems*

In artificial neural networks, local parameters (synaptic connections or node weights) are adjusted to navigate downwards in multidimensional learning landscapes, towards minima of a learning cost function. Physical learning networks such as elastic, flow, and resistor networks are additionally subject to nature’s constraints, which translate into the minimization of mechanical energy (mechanical networks) or dissipated power (flow or electrical networks). Functionality in these systems emerges from a dual optimization: a simultaneous minimization in a learning and a physical landscape.

Here we study this interplay within functional linear resistor networks, a paradigmatic model for physical learning systems. By examining various tasks, we reveal how these systems intertwine their learning and physical landscapes, providing a unique perspective, not found in purely artificial systems, to understand their emergent functionality. We specifically show that the high-curvature directions around the learning minima — highlighting the key functional components — are intricately imprinted into the softest modes of the physical landscape. Through a novel and generic method, we retrieve these key components, suggesting new ways to analyze the inner workings of how physical systems learn on their own.