Pith Number

pith:DXV43W2T

pith:2026:DXV43W2TMT3HZYSFI3XBS7BCJ2

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Neural Networks for Singular Perturbations -- Finite Regularity

Ch. Schwab, C. Xenophontos, F. Rohner

Deep ReLU neural networks with bitstring encoding achieve twice the robust convergence rate of P1 finite elements for singularly perturbed problems with low-regularity data.

arxiv:2605.14459 v1 · 2026-05-14 · math.NA · cs.NA

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\pithnumber{DXV43W2TMT3HZYSFI3XBS7BCJ2}

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Record completeness

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Claims

C1strongest claim

Recent bitstring encoding techniques for deep NNs with ReLU activations afford, still under low data regularity f,b ∈ H^1(I) twice the (robust) convergence rate of P1 Finite-Elements achievable with “eXp” or Shishkin meshes.

C2weakest assumption

The analysis assumes a model linear second-order singularly perturbed elliptic two-point BVP with data f and b in low Sobolev regularity H^1, and relies on specific mesh constructions for FEM and activation choices for NNs.

C3one line summary

Establishes ε-robust algebraic rate bounds for P1 finite elements on special meshes and for ReLU and tanh neural networks in approximating solutions to singularly perturbed boundary value problems under low Sobolev regularity.

References

24 extracted · 24 resolved · 0 Pith anchors

[1] R. Aylwin, F. Henriquez, and C. Schwab. ReLU Neural Network Galerkin BEM. Journ. Sci. Computing, 95(2), 2023 2023

[2] N. S. Bakhvalov. The optimization of methods of solving boundary value problems with a boundary layer.USSR Comput. Math. Math. Phys., 49:139––166, 1969 1969

[3] Brezis.Functional analysis, Sobolev spaces and partial differential equations 2011

[4] T. De Ryck, S. Lanthaler, and S. Mishra. On the approximation of functions by tanh neural networks.Neural Networks, 143:732–750, 2021 2021

[5] W. E and B. Yu. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems.Commun. Math. Stat., 6(1):1–12, 2018 2018

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:06.801461Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1debcddb5364f67ce24546ee197c224ea246f84559f9b9646cf180ead54c9da4

Aliases

arxiv: 2605.14459 · arxiv_version: 2605.14459v1 · doi: 10.48550/arxiv.2605.14459 · pith_short_12: DXV43W2TMT3H · pith_short_16: DXV43W2TMT3HZYSF · pith_short_8: DXV43W2T

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/DXV43W2TMT3HZYSFI3XBS7BCJ2 \
  | 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: 1debcddb5364f67ce24546ee197c224ea246f84559f9b9646cf180ead54c9da4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6c9b5e826bd9fd8c3baa0333bba688968ada94871178d862a6865ccb7cf84d14",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-14T06:55:01Z",
    "title_canon_sha256": "869e39e2b185aed3fd8dc13d282cc710467724fe67e570f428cd4f5cd325692b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14459",
    "kind": "arxiv",
    "version": 1
  }
}