Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings

article Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings published yes Mengjie Zu author 26dd9e7c-e86a-11eb-a854-82ac731c9ae2 Carl Peter Goodrich author EB352CD2-F68A-11E9-89C5-A432E66974250000-0002-1307-5074 CaGo department 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. https://research-explorer.ista.ac.at/download/22248/22251/2026_PRXLife_Zu.pdf application/pdfno American Physical Society2026 eng PRX Life 2835-827910.1103/48k2-cw3b 42 Zu, Mengjie, and Carl Peter Goodrich. “Learning by Training: Emergent Physical Memory from Cyclically Tuning Disordered Sphere Packings.” PRX Life. American Physical Society, 2026. <a href="https://doi.org/10.1103/48k2-cw3b">https://doi.org/10.1103/48k2-cw3b</a>. Zu M, Goodrich CP. Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings. PRX Life. 2026;4(2). doi:<a href="https://doi.org/10.1103/48k2-cw3b">10.1103/48k2-cw3b</a> Zu, Mengjie, and Carl Peter Goodrich. “Learning by Training: Emergent Physical Memory from Cyclically Tuning Disordered Sphere Packings.” PRX Life, vol. 4, no. 2, 023029, American Physical Society, 2026, doi:<a href="https://doi.org/10.1103/48k2-cw3b">10.1103/48k2-cw3b</a>. M. Zu and C. P. Goodrich, “Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings,” PRX Life, vol. 4, no. 2. American Physical Society, 2026. Zu, M., & Goodrich, C. P. (2026). Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings. PRX Life. American Physical Society. <a href="https://doi.org/10.1103/48k2-cw3b">https://doi.org/10.1103/48k2-cw3b</a> Zu M, Goodrich CP. 2026. Learning by Training: Emergent physical memory from cyclically tuning disordered sphere packings. PRX Life. 4(2), 023029. M. Zu, C.P. Goodrich, PRX Life 4 (2026). 222482026-07-05T22:01:37Z2026-07-06T07:28:45Z