Pith Number

pith:AQKBSOGG

pith:2026:AQKBSOGGDQ7YYRFKZEPPENCS74

not attested not anchored not stored refs pending

A neural network method for scalar conservation laws with convergence rates for shock-wave solutions

Buyang Li, Hao Li, Jiachuan Cao

Neural networks recover the classical O(h^{1/2}) L1 rate for scalar conservation laws with shocks.

arxiv:2604.27458 v3 · 2026-04-30 · math.NA · cs.NA

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\pithnumber{AQKBSOGGDQ7YYRFKZEPPENCS74}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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Claims

C1strongest claim

when the network size scales comparably to the number of degrees of freedom of a space time mesh of size h, we recover the classical O(h^{1/2}) convergence rate in shock-dominated regimes. The analysis also covers solutions containing rarefaction waves and regular shock interactions; in this extended version, we further treat smooth initial data that develop a shock in finite time, for which an L1 estimate of order O(h^{1/2}|ln h|) is obtained.

C2weakest assumption

The construction of explicit neural network competitors with provably small loss relies on combining shock-adapted continuous piecewise linear approximations with representability results for neural networks, which must hold uniformly for the target entropy solution.

C3one line summary

A neural network method for scalar hyperbolic conservation laws achieves provable O(h^{1/2}) L1 convergence rates for entropy solutions containing shocks by minimizing an entropy-compatible loss.

Receipt and verification

First computed	2026-05-20T01:05:14.724474Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

04141938c61c3f8c44aac91ef23452ff090effadb1e61f2705010c6b498988ff

Aliases

arxiv: 2604.27458 · arxiv_version: 2604.27458v3 · doi: 10.48550/arxiv.2604.27458 · pith_short_12: AQKBSOGGDQ7Y · pith_short_16: AQKBSOGGDQ7YYRFK · pith_short_8: AQKBSOGG

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/AQKBSOGGDQ7YYRFKZEPPENCS74 \
  | 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: 04141938c61c3f8c44aac91ef23452ff090effadb1e61f2705010c6b498988ff

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9164f43ce0bb48994a5cc5c03c7027b931226188029e96b982770bebffbd0fb0",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-04-30T06:00:36Z",
    "title_canon_sha256": "a296a60b6e6f57cde676794e178fd0a0cb77f30a51ed67a5faa163c26aeefaf9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.27458",
    "kind": "arxiv",
    "version": 3
  }
}