Pith Number

pith:Z7RBPEBC

pith:2026:Z7RBPEBC52JAD2JYMOJVTYRVRV

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A multigrid and neural network approach to reduce the computational cost of phi-FEM

Killian Vuillemot, Michel Duprez, Rapha\"el Bulle, Vanessa Lleras

A multigrid approach combined with neural networks reduces the computational cost of phi-FEM while preserving accuracy.

arxiv:2605.13718 v1 · 2026-05-13 · math.NA · cs.NA

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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Claims

C1strongest claim

A combination of a multigrid approach and the phi-FEM immersed boundary finite element method reduces its computational cost while preserving its accuracy; further reduction is achieved by combining the previous technique with neural network methods, illustrated with numerical test cases in 2D and 3D.

C2weakest assumption

That the neural-network components can be trained or applied in a way that does not introduce new errors or require problem-specific retraining that offsets the reported speed gains.

C3one line summary

Multigrid plus neural-network acceleration reduces the cost of phi-FEM while preserving accuracy in 2D and 3D test cases.

References

18 extracted · 18 resolved · 0 Pith anchors

[1] I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. S. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, and G. N. Wells. DOLFINx: the next generation FEniCS problem solving environment 2023

[2] H. Barucq, M. Duprez, F. Faucher, E. Franck, F. Lecourtier, V. Lleras, V. Michel-Dansac, and N. Vic- torion. Enriching continuous Lagrange finite element approximation spaces using neural networks. ES 2026

[3] J. H. Bramble and B. E. Hubbard. On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation.Numer. Math., 4:313–327, 1962 1962

[4] E. Burman and P. Hansbo. Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method.Computer Methods in Applied Mechanics and Engineering, 199(41):2680–268 2010

[5] E. Burman and P. Hansbo. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method.Applied Numerical Mathematics, 62(4):328–341, 2012 2012

Receipt and verification

First computed	2026-05-18T02:44:16.682840Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cfe2179022ee9201e938639359e2358d5c82dd3900ac993dcfd5731c6db3c2e0

Aliases

arxiv: 2605.13718 · arxiv_version: 2605.13718v1 · doi: 10.48550/arxiv.2605.13718 · pith_short_12: Z7RBPEBC52JA · pith_short_16: Z7RBPEBC52JAD2JY · pith_short_8: Z7RBPEBC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/Z7RBPEBC52JAD2JYMOJVTYRVRV \
  | 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: cfe2179022ee9201e938639359e2358d5c82dd3900ac993dcfd5731c6db3c2e0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "050f7df6a02f86c4c6f7f25d4e4322a1ed973b50171be021acd19071d5fbd775",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-13T16:03:14Z",
    "title_canon_sha256": "440f4851a62fa7d6b2209c28bb31567759bd6f37c27f8ef849a7def3a6a87a9f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13718",
    "kind": "arxiv",
    "version": 1
  }
}