Pith Number

pith:WW65ATP3

pith:2026:WW65ATP3VJFPHKAC2CKKY5XM4M

not attested not anchored not stored refs pending

Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

Diego Marcondes

The H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.

arxiv:2605.03542 v2 · 2026-05-05 · math.NA · cs.LG · cs.NA

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\usepackage{pith}
\pithnumber{WW65ATP3VJFPHKAC2CKKY5XM4M}

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4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that the H^{-1} norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain.

C2weakest assumption

That there exists a random test function distribution depending only on the domain such that averaging squared evaluations exactly recovers the H^{-1} norm and the weak topology independently of the differential operator.

C3one line summary

H^{-1} norm equivalence to expected squared evaluations on domain-dependent random test functions enables SV-PINNs that recover accurate solutions to challenging second-order elliptic PDEs faster than standard PINNs.

Receipt and verification

First computed	2026-06-26T01:15:19.025453Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b5bdd04dfbaa4af3a802d094ac76ece31accb5968cc32374ee04f2f9e66a6682

Aliases

arxiv: 2605.03542 · arxiv_version: 2605.03542v2 · doi: 10.48550/arxiv.2605.03542 · pith_short_12: WW65ATP3VJFP · pith_short_16: WW65ATP3VJFPHKAC · pith_short_8: WW65ATP3

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WW65ATP3VJFPHKAC2CKKY5XM4M \
  | 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: b5bdd04dfbaa4af3a802d094ac76ece31accb5968cc32374ee04f2f9e66a6682

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a13a9ac148bcc5c55b537927ea518c4a14aab4885806e510b8f6ab7e92e92e42",
    "cross_cats_sorted": [
      "cs.LG",
      "cs.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-05T09:14:38Z",
    "title_canon_sha256": "71dde23fc17813f8d5dc6151af7f29125fb7046d34f01b973f2b592d8101b3b3"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.03542",
    "kind": "arxiv",
    "version": 2
  }
}