{"paper":{"title":"Neural Networks for Singular Perturbations -- Finite Regularity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"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.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Ch. Schwab, C. Xenophontos, F. Rohner","submitted_at":"2026-05-14T06:55:01Z","abstract_excerpt":"We study finite-element and deep feedforward neural network (DNN for short) expressivity rate bounds for solution sets of a model linear, second order singularly perturbed, elliptic two-point boundary value problem, in Sobolev norms on a bounded interval $(-1,1)$, with explicit dependence on the singular perturbation parameter $\\e\\in (0,1]$. Emphasis is on low Sobolev regularity of the data, i.e., source term $f$ and reaction coefficient $b$. A proof of $\\e$-explicit solution regularity based on exponentially weighted energy-norm bounds is developed, and \\emph{$\\e$-robust, algebraic expression"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5223afa6fb9766eb425cfd38bd9a6029e57563e6d16dc5765ab538e42793545d"},"source":{"id":"2605.14459","kind":"arxiv","version":1},"verdict":{"id":"04140860-a803-43bf-97fb-52208286c98e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:02:47.310752Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"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."},"references":{"count":24,"sample":[{"doi":"","year":2023,"title":"R. Aylwin, F. Henriquez, and C. Schwab. ReLU Neural Network Galerkin BEM. Journ. Sci. Computing, 95(2), 2023","work_id":"795278c9-1f81-4a7d-849e-580c91aabe2a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1969,"title":"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","work_id":"abeda47c-f94c-4aa5-9922-c56797362990","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"Brezis.Functional analysis, Sobolev spaces and partial differential equations","work_id":"2149402e-dcad-4dc6-8f51-c3fed5d463fa","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"T. De Ryck, S. Lanthaler, and S. Mishra. On the approximation of functions by tanh neural networks.Neural Networks, 143:732–750, 2021","work_id":"1e95f82d-4db7-4596-9385-9a2f9984b873","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"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","work_id":"c8d8d866-25a9-4021-8f8a-4560d82c016c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"ed5d9a677bf9c79afb69c31021135fe3de587ff69d0900be433a1fbcf3e40150","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b98ea3d47e0331f40df88f183bb6162937db17eefe1477d548d60f1b2c848c53"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}