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arxiv: 2509.01296 · v2 · submitted 2025-09-01 · ⚛️ physics.comp-ph · cond-mat.dis-nn· cond-mat.soft

Learning by training: emergent return-point memory from cyclically tuning disordered sphere packings

Mengjie Zu , Carl P. Goodrich This is my paper

Pith reviewed 2026-05-18 20:00 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cond-mat.dis-nncond-mat.soft
keywords return-point memorymarginally absorbing manifoldsphere packingsinverse designcyclic trainingdisordered materialselastic propertiesadaptive systems
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The pith

Cyclically tuned sphere packings evolve toward a marginally absorbing manifold that remembers the training range via return-point memory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines athermal disordered sphere packings that undergo cyclic inverse design to reach target elastic properties over a selected range. These systems move toward a marginally absorbing manifold that stores memory of the training interval, producing behavior akin to return-point memory in cyclically driven materials. The authors trace this outcome to discontinuities in the gradients of the trained elastic quantities and present this as a general mechanism. A reader would care because the result supplies a concrete physical model for how adaptation under varying conditions can generate lasting memory without separate storage mechanisms.

Core claim

Athermal disordered sphere packings subjected to cyclic inverse design evolve toward a marginally absorbing manifold. This manifold encodes memory of the training range and produces return-point memory that matches observations in other cyclically driven systems. The mechanism rests on gradient discontinuities in the trained elastic quantities, which the authors propose as a general route to such manifolds and their associated memory.

What carries the argument

The marginally absorbing manifold (MAM), a structure in configuration space that absorbs cyclic training trajectories and encodes the range of elastic targets through return-point memory.

If this is right

  • Trained packings retain information about the full range of past target properties through their configuration on the manifold.
  • Memory formation occurs automatically from the training process without requiring explicit encoding steps.
  • The same gradient-discontinuity mechanism can generate analogous memory in other adaptive physical systems.
  • Design of materials that adapt under repeated loading can exploit this manifold structure to store history.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mechanism may apply to living systems that adapt to cyclic environmental stresses, such as cells or tissues under repeated mechanical loading.
  • Testing the model with different particle interaction potentials would check whether the gradient-discontinuity route remains dominant.
  • The framework could connect to machine-learning settings where models are trained on data drawn from varying distributions.

Load-bearing premise

Gradient discontinuities in the trained elastic quantities are both necessary and sufficient to produce the marginally absorbing manifold and its return-point memory, and this mechanism applies beyond the specific sphere-packing model.

What would settle it

A concrete counterexample would be a system of cyclically tuned packings that develops return-point memory while showing no gradient discontinuities in the trained elastic quantities.

Figures

Figures reproduced from arXiv: 2509.01296 by Carl P. Goodrich, Mengjie Zu.

Figure 1
Figure 1. Figure 1: FIG. 1. Training and cyclic training. (a) A schematic of our gradient-based optimization routine to tune the Poisson’s ratio [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Memory in four representative examples. Columns (a)–(c) show readout results following cyclic training targeting the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The evolution of the parameters during cyclic [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Gradient Discontinuity Learning. (a) An example training function [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Confirming predictions made by Gradient [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Cyclic training on Poisson’s ratio between [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Robustness of MAMs. Column (a) shows a [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Memory in a representative example trained on [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Cyclic training on two elements of elastic modulus [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
read the original 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 such environmental variation influences adaptation, what is learned in the process, and whether memory of past conditions is retained. In this work, we investigate these questions using athermal disordered systems that are subject to cyclic inverse design, enabling them to attain target elastic properties spanning a chosen range. We demonstrate that such systems evolve toward a marginally absorbing manifold (MAM), which encodes memory of the training range that closely resembles return-point memory observed in cyclically driven systems. We further propose a general mechanism for the formation of MAMs and the corresponding memory that is based on gradient discontinuities in the trained quantities. Our model provides a simple and broadly applicable physical framework for understanding how adaptive systems learn under environmental change and how they retain memory of past experiences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies athermal disordered sphere packings subjected to cyclic inverse design that targets elastic properties over a chosen range. It reports that the packings evolve toward a marginally absorbing manifold (MAM) whose geometry encodes memory of the training range in a manner that closely resembles return-point memory. The authors propose that gradient discontinuities in the trained elastic quantities provide a general mechanism for both MAM formation and the emergence of this memory.

Significance. If the central claims hold, the work supplies a concrete, minimal physical model for how adaptive systems acquire and retain memory of past conditions under environmental variation. The link between cyclic training, gradient discontinuities, and return-point memory is conceptually interesting and could inform broader studies of learning in physical and biological systems. The sphere-packing inverse-design protocol is a reasonable choice for an athermal, disordered setting, and the introduction of the MAM as an emergent structure is a useful organizing idea.

major comments (2)
  1. [§4 (proposed general mechanism)] §4 (proposed general mechanism): The assertion that gradient discontinuities in the trained quantities are both necessary and sufficient for MAM formation and return-point memory is not tested by any control in which the discontinuities are removed or smoothed (for example by replacing the elastic response with a differentiable surrogate) while the cyclic inverse-design protocol is held fixed. Without such a control, it remains possible that the MAM and its memory properties arise from the geometry of configuration space or the form of the inverse-design objective rather than from the non-differentiable points.
  2. [Results on evolution to the MAM] Results on evolution to the MAM: The claim that the systems evolve toward the MAM and encode memory of the training range is presented without quantitative support such as a distance-to-manifold metric tracked over training cycles, convergence statistics across independent realizations, or error bars. This absence makes it difficult to assess how robust or complete the evolution is.
minor comments (2)
  1. [Methods / early Results] The definition of the MAM would be clearer if accompanied by an explicit mathematical characterization (e.g., a condition on the Hessian or on the set of admissible strains) rather than a purely descriptive statement.
  2. [Figure captions] Figure captions should explicitly state the number of independent realizations used for averaging and the precise definition of any shaded regions or error bars.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. We are encouraged by the positive assessment of the significance and the usefulness of the MAM concept. We address each of the major comments below.

read point-by-point responses

  1. Referee: §4 (proposed general mechanism): The assertion that gradient discontinuities in the trained quantities are both necessary and sufficient for MAM formation and return-point memory is not tested by any control in which the discontinuities are removed or smoothed (for example by replacing the elastic response with a differentiable surrogate) while the cyclic inverse-design protocol is held fixed. Without such a control, it remains possible that the MAM and its memory properties arise from the geometry of configuration space or the form of the inverse-design objective rather than from the non-differentiable points.

    Authors: We agree that an explicit control experiment would provide stronger support for the proposed mechanism. However, in the context of athermal sphere packings, the gradient discontinuities stem directly from the discrete nature of contact formation and breaking, which is fundamental to the system's response. Implementing a fully differentiable surrogate while maintaining the inverse-design protocol and athermal conditions is challenging and would likely require significant modifications to the physical model. In the revised manuscript, we will add a dedicated paragraph in §4 discussing this point, including why such a control is difficult to implement without changing the essence of the system, and we will present additional supporting analysis from our existing data that links the memory properties specifically to the observed discontinuities. We believe this will clarify the scope of our claims. revision: partial

  2. Referee: Results on evolution to the MAM: The claim that the systems evolve toward the MAM and encode memory of the training range is presented without quantitative support such as a distance-to-manifold metric tracked over training cycles, convergence statistics across independent realizations, or error bars. This absence makes it difficult to assess how robust or complete the evolution is.

    Authors: We appreciate this feedback. While the manuscript includes several figures illustrating the evolution and memory encoding, we acknowledge that quantitative metrics would improve the presentation. In the revised manuscript, we will include new quantitative analyses: specifically, we will track and plot the distance to the MAM over the course of training cycles for multiple realizations, include error bars representing standard deviations across independent packings, and provide statistics on convergence rates. These additions will allow readers to better evaluate the robustness of the reported behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: MAM emerges from training dynamics without self-referential definition or fitted prediction

full rationale

The paper presents the marginally absorbing manifold (MAM) as an emergent outcome of cyclic inverse design applied to athermal disordered sphere packings, with memory of the training range arising from the adaptation process rather than being presupposed in the definition. The proposed mechanism based on gradient discontinuities is offered as a general explanation derived from observed model behavior, not as a tautological fit or self-citation that reduces the central result to its inputs. No equations or derivations in the abstract or described claims show a 'prediction' that is statistically forced by construction from fitted parameters, nor does the argument rely on load-bearing self-citations or imported uniqueness theorems. The derivation remains self-contained, with the resemblance to return-point memory serving as an external analogy rather than an internal circular loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of gradient discontinuities created by the inverse-design training loop and on the assumption that athermal disordered packings can be driven to a manifold whose boundary encodes the training range.

axioms (1)
  • domain assumption Athermal disordered sphere packings can be subjected to cyclic inverse design that tunes their elastic properties across a chosen range
    This is the experimental setup stated in the abstract.
invented entities (1)
  • marginally absorbing manifold (MAM) no independent evidence
    purpose: Encodes memory of the training range and produces return-point memory
    New state space region introduced to explain the observed memory; no independent falsifiable prediction outside the model is given in the abstract.

pith-pipeline@v0.9.0 · 5683 in / 1293 out tokens · 48564 ms · 2026-05-18T20:00:22.348087+00:00 · methodology

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