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  • Open Access

Microscopic Imprints of Learned Solutions in Tunable Networks

Marcelo Guzman1,*, Felipe Martins1,*, Menachem Stern2, and Andrea J. Liu1

  • 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
  • 2AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands

  • *These authors contributed equally to this work.
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Phys. Rev. X 15, 031056 – Published 27 August, 2025

DOI: https://doi.org/10.1103/f2hb-c9s1

Abstract

In physical networks trained using supervised learning, physical parameters are adjusted to produce desired responses to inputs. An example is an electrical contrastive local learning network of nodes connected by edges that adjust their conductances during training. When an edge conductance changes, it upsets the current balance of every node. In response, physics adjusts the node voltages to minimize the dissipated power. Learning in these systems is therefore a coupled double-optimization process, in which the network descends both a cost landscape in the high-dimensional space of edge conductances and a physical landscape—the power dissipation—in the high-dimensional space of node voltages. Because of this coupling, the physical landscape of a trained network contains information about the learned task. Here, we derive a structure-function relation for trained tunable networks and demonstrate that all the physical information relevant to the trained input-output relation can be captured by a tuning susceptibility, an experimentally measurable quantity. We supplement our theoretical results with simulations to show that the tuning susceptibility is correlated with functional importance and that we can extract physical insight into how the system performs the task from the conductances of highly susceptible edges. Our analysis is general and can be applied directly to mechanical networks, such as networks trained for protein-inspired function such as allostery.

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Popular Summary

Traditional digital neural networks are infamously difficult to interpret, making it hard to understand how microscopic parameter changes affect task performance. Recently, a new type of network—electrical contrastive local learning networks—has emerged, where learning occurs directly via physical resistors that individually adjust their conductances. These networks are bound by physical laws, such as minimizing electrical power. Here, we show that this physical constraint yields powerful insight into how the networks perform tasks.

We demonstrate that the resistors most critical to a task can be identified by their physical properties alone. These key resistors often have higher resistance values than their neighbors, forming barriers that shape the flow of electrical current through the network. This current flow determines the network’s output in response to given inputs, meaning that the physical configuration of the resistors is directly linked to its function. Unlike digital models, these physical networks allow us to see how specific components contribute to the task.

Our findings go beyond electrical systems. The same principles apply to mechanical networks, which naturally minimize elastic energy instead of electrical power. This suggests a universal framework for understanding how complex functionality can emerge in physical platforms—from engineered materials to biological systems.

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Outline

Information

Phys. Rev. X 15, 031056– Published 27 August, 2025
  • Received 29 January 2025
  • Accepted 15 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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