A Locally Conservative Enriched Linearized Neural Network Approximation to Elliptic PDEs
Abstract
This paper presents a locally conservative Enriched Linearized Neural Network (ENN) method for the Darcy flow model. A linearized shallow ReLUk network, whose hidden-layer parameters are fixed on a quasi-uniform set of the sphere, carries the approximation, and piecewise constant functions on an auxiliary subdivision of the domain are added to ensure local mass conservation. The subdivision may consist of general, possibly curved, elements. Since neural network functions do not satisfy the inverse inequality, the stability and error analysis of the standard discontinuous Galerkin interior penalty formulation do not apply. The missing inverse inequality is remedied by an edge identity combined with a Galerkin least-squares (GLS) term, together with a single neural approximant that is optimal in the L2, H1 and H2 norms simultaneously. We use the nonsymmetric interior penalty (NIPG) form, which is coercive for every positive penalty parameter. With n, the width of the shallow neural network, used in ENN, we prove an error estimate in the EG norm that is of the optimal H1 order, i.e., O(n−(r−1)/d) for solutions in Hr, r≥2, and O(n−(s−1)/d) for solutions in Hs, 3/2<s≤2, for any shape-regular subdivision, when the GLS parameter satisfies min{h,n−1/d}2≲τ≲ n−2/d and the edge terms are scaled with min{he,n−1/d}, i.e., by the resolution of the network rather than of the mesh. A numerical Darcy flux is constructed, which is locally conservative, regardless of the size or the number of the elements. Sample numerical results demonstrate the correctness of the theory. In particular, a fixed 4×4 subdivision gives the same accuracy as a subdivision refined with the network, and the mass loss stays at round-off unlike a non-conservative neural network method.
Related
- Eigenvalues of the linearized Navier-Stokes operator near locally Couette laminar flows
- Parameter Bounds and Inverse Approximation for Linearized ReLUk Networks on the Sphere
- Locally recoverable codes from elliptic surfaces with availability and hierarchical locality
- Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations
- Existence of Dynamical Low-Rank Approximation for SDEs with Locally Lipschitz Coefficients
BibTeX
@misc{kim2026locally,
title = {{A Locally Conservative Enriched Linearized Neural Network Approximation to Elliptic PDEs}},
author = {Seungil Kim and Gwanghyun Jo and Young Ju Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04524},
eprint = {2610.04524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04524},
}
RIS
TY - UNPB
TI - A Locally Conservative Enriched Linearized Neural Network Approximation to Elliptic PDEs
AU - Seungil Kim
AU - Gwanghyun Jo
AU - Young Ju Lee
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04524
UR - https://arxiv.org/abs/2610.04524
AB - This paper presents a locally conservative Enriched Linearized Neural Network (ENN) method for the Darcy flow model. A linearized shallow ReLU^k network, whose hidden-layer parameters are fixed on a quasi-uniform set of the sphere, carries the approximation, and piecewise constant functions on an auxiliary subdivision of the domain are added to ensure local mass conservation. The subdivision may consist of general, possibly curved, elements. Since neural network functions do not satisfy the inverse inequality, the stability and error analysis of the standard discontinuous Galerkin interior penalty formulation do not apply. The missing inverse inequality is remedied by an edge identity combined with a Galerkin least-squares (GLS) term, together with a single neural approximant that is optimal in the L^2, H^1 and H^2 norms simultaneously. We use the nonsymmetric interior penalty (NIPG) form, which is coercive for every positive penalty parameter. With n, the width of the shallow neural network, used in ENN, we prove an error estimate in the EG norm that is of the optimal H^1 order, i.e., O(n^−(r−1)/d) for solutions in H^r, r≥2, and O(n^−(s−1)/d) for solutions in H^s, 3/2<s≤2, for any shape-regular subdivision, when the GLS parameter satisfies min{h,n^−1/d}^2≲τ≲ n^−2/d and the edge terms are scaled with min{h_e,n^−1/d}, i.e., by the resolution of the network rather than of the mesh. A numerical Darcy flux is constructed, which is locally conservative, regardless of the size or the number of the elements. Sample numerical results demonstrate the correctness of the theory. In particular, a fixed 4×4 subdivision gives the same accuracy as a subdivision refined with the network, and the mass loss stays at round-off unlike a non-conservative neural network method.
ER -