Pith Number

pith:CEERTUPE

pith:2026:CEERTUPEAEZRE6EGCGP7ZGCP5C

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FlowEqProp: Training Flow Matching Generative Models with Gradient Equilibrium Propagation

Alex Gower

Gradient Equilibrium Propagation trains flow matching generative models by encoding target velocities in equilibrium displacements.

arxiv:2604.08150 v2 · 2026-04-09 · cond-mat.dis-nn

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\pithnumber{CEERTUPEAEZRE6EGCGP7ZGCP5C}

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Claims

C1strongest claim

We introduce Gradient Equilibrium Propagation (GradEP), a mechanism that extends Equilibrium Propagation (EP) to train energy gradients rather than energy minima, enabling EP to be applied to tasks where the learning objective depends on the velocity field of a convergent dynamical system... To our knowledge, this is the first demonstration of EP training a flow-based generative model.

C2weakest assumption

The assumption that a quadratic spring potential allows all units (including visible) to evolve such that the equilibrium displacement encodes the target velocity field of the flow matching objective, while preserving hardware plausibility and enabling stable training dynamics without backpropagation.

C3one line summary

FlowEqProp trains flow matching generative models using gradient equilibrium propagation on a 25k-parameter MLP for digit generation without backpropagation, producing recognizable samples and allowing quality gains from extended inference-time relaxation.

Receipt and verification

First computed	2026-05-26T02:04:10.424090Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

110919d1e40133127886119ffc984fe89199904e7f0b21e3775eed0ace860cd7

Aliases

arxiv: 2604.08150 · arxiv_version: 2604.08150v2 · doi: 10.48550/arxiv.2604.08150 · pith_short_12: CEERTUPEAEZR · pith_short_16: CEERTUPEAEZRE6EG · pith_short_8: CEERTUPE

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CEERTUPEAEZRE6EGCGP7ZGCP5C \
  | 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: 110919d1e40133127886119ffc984fe89199904e7f0b21e3775eed0ace860cd7

Canonical record JSON

{
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    "abstract_canon_sha256": "8f7d51ca7ab2c416776a93bf05e12b59ff0e08fa3e28bb9b7648b03a774db38a",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cond-mat.dis-nn",
    "submitted_at": "2026-04-09T12:10:16Z",
    "title_canon_sha256": "22ce3656c12aac37163f71f3cb85daa76a94f2f0b2ae24c6f6d1a682c8a65234"
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    "kind": "arxiv",
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