Pith Number

pith:77OMDIUI

pith:2025:77OMDIUIMRATI43SRJAXMN7SJV

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Can Stationary Distributions of Scale-Invariant Neural Networks Be Described by the Thermodynamics of an Ideal Gas?

Dmitry Vetrov, Ekaterina Lobacheva, Ildus Sadrtdinov, Ivan Klimov, Mikhail Burtsev, Mikhail I. Katsnelson

Stationary distributions of SGD for scale-invariant networks correspond to ideal gas thermodynamics.

arxiv:2511.07308 v2 · 2025-11-10 · cs.LG

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Claims

C1strongest claim

Starting with a simplified isotropic noise model, we uncover a close correspondence between SGD dynamics and ideal gas behavior, validated through theory and simulation. Extending to training of neural networks, we show that key predictions of the framework, including the behavior of stationary entropy, align closely with experimental observations.

C2weakest assumption

The derivation begins with a simplified isotropic noise model whose relation to the actual gradient noise in deep networks is not quantified; if this model is a poor approximation, the ideal-gas analogy and its thermodynamic-variable mappings lose their justification.

C3one line summary

A thermodynamic framework maps SGD stationary distributions in scale-invariant networks to ideal-gas behavior, with training hyperparameters acting as thermodynamic variables.

References

19 extracted · 19 resolved · 3 Pith anchors

[1] Layer Normalization 2018 · arXiv:1607.06450

[2] Chaudhari, P., Choromanska, A., Soatto, S., LeCun, Y., Baldassi, C., Borgs, C., Chayes, J., Sagun, L., and Zecchina, R. (2017). Entropy-SGD: Biasing gra- dient descent into wide valleys. InInternation 2017

[3] arXiv preprint arXiv:1711.04623 , year= 2015 · arXiv:1711.04623

[4] Le Ny, A. (2008). Introduction to (generalized) gibbs measures.Ensaios Matemáticos, 15(1-126). Li, Z. and Arora, S. (2020). An exponential learn- ing rate schedule for deep learning. InInternational C 2008

[5] Liu, Z., Liu, Y., Gore, J., and Tegmark, M. (2025). Neural thermodynamic laws for large language model training. Lobacheva, E., Kodryan, M., Chirkova, N., Malinin, A., and Vetrov, D. P. (2021). On the 2025

Receipt and verification

First computed	2026-05-17T23:39:17.168209Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

ffdcc1a28864413473728a417637f24d4147d43dc5dc4d6edb3ce78e26d748c6

Aliases

arxiv: 2511.07308 · arxiv_version: 2511.07308v2 · doi: 10.48550/arxiv.2511.07308 · pith_short_12: 77OMDIUIMRAT · pith_short_16: 77OMDIUIMRATI43S · pith_short_8: 77OMDIUI

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/77OMDIUIMRATI43SRJAXMN7SJV \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: ffdcc1a28864413473728a417637f24d4147d43dc5dc4d6edb3ce78e26d748c6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "fa0483c71cb2b6908fcff72f120e1c968b942d7c0a5f339518bb0b35915dff5f",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2025-11-10T17:10:01Z",
    "title_canon_sha256": "3883034d18afb44f7477dd2d9eb85730aaa04cd022c5c079dabc6f8712ab4f30"
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  "source": {
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}