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Learning by Training: Emergent Physical Memory from Cyclically Tuning Disordered Sphere Packings

Mengjie Zu and Carl P. Goodrich*

  • *Contact author: carl.goodrich@ist.ac.at
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PRX Life 4, 023029 – Published 18 June, 2026

DOI: https://doi.org/10.1103/48k2-cw3b

Abstract

Many living and artificial systems improve their fitness or performance by adapting to changing environments or diverse training data. However, it remains unclear how environmental variation shapes adaptation, what is learned, and when memory of past conditions is retained. Here we show how cyclic environmental change can produce robust memory. Using a model athermal disordered solid trained by inverse design to attain target elastic properties over a prescribed range, we find that the system evolves toward a marginally absorbing manifold (MAM), meaning that training is reversible within the training range but not beyond it, which encodes a memory of that range. We further propose a general mechanism for MAM formation and memory encoding based on discontinuities in the gradient of the trained quantity. These results provide a simple, broadly applicable physical framework for how adaptive systems learn under changing environments and retain memory of past conditions.

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PRX Life Collection on adaptation and learning

PRX Life presents a special collection on ‘Biological systems that learn’, curated by the organizers of the National Institute for Theory and Mathematics in Biology (NITMB) workshop together with early-career researchers. The papers in this collection illustrate how living systems acquire functions, showing it is useful to describe them as tunable matter with many individually adjustable degrees of freedom characterizing microscopic properties. Drawing inspiration from AI and non-biological systems that can acquire desired functions, the featured research highlights the timeliness of the topic and its potential to drive unifying frameworks for emergent collective function in living tunable matter.

Every article in this collection is subject to a rigorous peer review process, upholding the high standards applied to all papers in the journal. The PRX Life editorial team manages peer review and makes all editorial decisions.

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PRX Life 4, 023029– Published 18 June, 2026
  • Received 31 August 2025
  • Accepted 4 May 2026
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