Pith Number

pith:BVPBEHEY

pith:2024:BVPBEHEYQX3OFK3OUHRCDUNGAG

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Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

Akil Narayan, John D. Jakeman, John Turnage, Matthew Lowery, Shandian Zhe, Varun Shankar, Zachary Morrow

The Kernel Neural Operator learns maps between function spaces using compositions of kernel integral operators that are universal approximators and require an order of magnitude fewer parameters than existing neural operators.

arxiv:2407.00809 v4 · 2024-06-30 · cs.LG · cs.NA · math.NA

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\pithnumber{BVPBEHEYQX3OFK3OUHRCDUNGAG}

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Claims

C1strongest claim

We present universal approximation theorems showing that both the continuous and fully discretized KNO are universal approximators on operator learning problems. Numerical results demonstrate that on existing benchmarks the training and test accuracy of KNOs is closely comparable to or higher than that of popular neural operators while typically using an order of magnitude fewer trainable parameters.

C2weakest assumption

The decoupling of the choice of kernel from the numerical integration scheme (quadrature) thereby naturally allowing for operator learning with explicitly-chosen trainable kernels on irregular geometries without compromising the universal approximation property or convergence.

C3one line summary

KNOs combine deep compositions of kernel integral operators with neural networks to define expressive kernels, delivering universal approximation for operator learning with geometric flexibility and roughly 10x fewer parameters than prior neural operators.

Formal links

2 machine-checked theorem links

Cited by

2 papers in Pith

Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows

Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators

Receipt and verification

First computed	2026-06-04T01:08:25.537175Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0d5e121c9885f6e2ab6ea1e221d1a601a031ec4a770af001eddbb493a746b3de

Aliases

arxiv: 2407.00809 · arxiv_version: 2407.00809v4 · doi: 10.48550/arxiv.2407.00809 · pith_short_12: BVPBEHEYQX3O · pith_short_16: BVPBEHEYQX3OFK3O · pith_short_8: BVPBEHEY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/BVPBEHEYQX3OFK3OUHRCDUNGAG \
  | 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: 0d5e121c9885f6e2ab6ea1e221d1a601a031ec4a770af001eddbb493a746b3de

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a7f3deb0ff15bbd31327a79fb842a817346bbcead337153c1d88cbb814cd1ce4",
    "cross_cats_sorted": [
      "cs.NA",
      "math.NA"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2024-06-30T19:28:12Z",
    "title_canon_sha256": "c925538d7072917ad3f15ae3df9142178112dd0ad3cf17744ac265be00ba22a4"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2407.00809",
    "kind": "arxiv",
    "version": 4
  }
}