---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '22248'
abstract:
- lang: eng
  text: 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.
acknowledgement: "We thank Nathan Keim, Aayush Desai, Nicholas Barton,\r\nand Gašper
  Tkacik for important and stimulating discussions. ˇ\r\nThe work was funded by the
  Institute of Science and Technology Austria."
article_number: '023029'
article_processing_charge: Yes
article_type: original
author:
- first_name: Mengjie
  full_name: Zu, Mengjie
  id: 26dd9e7c-e86a-11eb-a854-82ac731c9ae2
  last_name: Zu
- first_name: Carl Peter
  full_name: Goodrich, Carl Peter
  id: EB352CD2-F68A-11E9-89C5-A432E6697425
  last_name: Goodrich
  orcid: 0000-0002-1307-5074
citation:
  ama: 'Zu M, Goodrich CP. Learning by Training: Emergent physical memory from cyclically
    tuning disordered sphere packings. <i>PRX Life</i>. 2026;4(2). doi:<a href="https://doi.org/10.1103/48k2-cw3b">10.1103/48k2-cw3b</a>'
  apa: 'Zu, M., &#38; Goodrich, C. P. (2026). Learning by Training: Emergent physical
    memory from cyclically tuning disordered sphere packings. <i>PRX Life</i>. American
    Physical Society. <a href="https://doi.org/10.1103/48k2-cw3b">https://doi.org/10.1103/48k2-cw3b</a>'
  chicago: 'Zu, Mengjie, and Carl Peter Goodrich. “Learning by Training: Emergent
    Physical Memory from Cyclically Tuning Disordered Sphere Packings.” <i>PRX Life</i>.
    American Physical Society, 2026. <a href="https://doi.org/10.1103/48k2-cw3b">https://doi.org/10.1103/48k2-cw3b</a>.'
  ieee: 'M. Zu and C. P. Goodrich, “Learning by Training: Emergent physical memory
    from cyclically tuning disordered sphere packings,” <i>PRX Life</i>, vol. 4, no.
    2. American Physical Society, 2026.'
  ista: 'Zu M, Goodrich CP. 2026. Learning by Training: Emergent physical memory from
    cyclically tuning disordered sphere packings. PRX Life. 4(2), 023029.'
  mla: 'Zu, Mengjie, and Carl Peter Goodrich. “Learning by Training: Emergent Physical
    Memory from Cyclically Tuning Disordered Sphere Packings.” <i>PRX Life</i>, vol.
    4, no. 2, 023029, American Physical Society, 2026, doi:<a href="https://doi.org/10.1103/48k2-cw3b">10.1103/48k2-cw3b</a>.'
  short: M. Zu, C.P. Goodrich, PRX Life 4 (2026).
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: "The data that support the findings of this article are
  not\r\npublicly available. The data are available from the authors\r\nupon reasonable
  request."
date_created: 2026-07-05T22:01:37Z
date_published: 2026-06-18T00:00:00Z
date_updated: 2026-07-06T07:28:45Z
day: '18'
ddc:
- '570'
department:
- _id: CaGo
doi: 10.1103/48k2-cw3b
file:
- access_level: open_access
  checksum: e2d13c30bf9c036951fd2ba3455cf72a
  content_type: application/pdf
  creator: dernst
  date_created: 2026-07-06T07:24:43Z
  date_updated: 2026-07-06T07:24:43Z
  file_id: '22251'
  file_name: 2026_PRXLife_Zu.pdf
  file_size: 2758728
  relation: main_file
  success: 1
file_date_updated: 2026-07-06T07:24:43Z
has_accepted_license: '1'
intvolume: '         4'
issue: '2'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: PRX Life
publication_identifier:
  eissn:
  - 2835-8279
publication_status: published
publisher: American Physical Society
quality_controlled: '1'
researchdata_availability: upon request
scopus_import: '1'
status: public
supplementarymaterial: no
title: 'Learning by Training: Emergent physical memory from cyclically tuning disordered
  sphere packings'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2026'
...