Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks
Abstract
We establish inverse inequalities in Gaussian Sobolev norms for shallow ReLUk trial spaces. The estimates identify the sharp regularity threshold and quantify the dependence on parameter separation and bias magnitude. Based on these estimates, we prove convergence of weighted least squares discretizations of linear PDEs on ℝd using shallow networks with prescribed inner parameters and finitely many residual samples in an expanding ball. Under suitable stability and regularity assumptions, we obtain an error bound of order hk+(d+1)/2−rδ±−(S−r) in the Gaussian Sobolev norm of order S, up to logarithmic factors, where r is the PDE order, h is the parameter fill distance, and δ± is the antipodal separation. The convergence results cover deterministic and random sampling and extend to fixed bounded regions by restriction. We construct parameter sets with δ±asymp h, yielding a convergence rate in terms of the network width that matches a lower bound up to logarithmic factors. Numerical experiments support the theoretical results and provide practical guidance on balancing accuracy and stability.
Related
- A comparison of artificial neural networks and partial least squares modelling for the rapid detection of the microbial spoilage of beef fillets based on Fourier transform infrared spectral fingerprints
- On randomized batch-sampling coordinate descent methods for solving linear least-squares problems
- Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous Discretizations
- FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations
- A COVID-19 transmission model with optimal control integrating BERT with LHS and least-squares fitting for real-world consistency
BibTeX
@misc{mao2026convergence,
title = {{Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks}},
author = {Tong Mao and Lei Shi and Hanfei Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14457},
eprint = {2610.14457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14457},
}
RIS
TY - UNPB
TI - Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks
AU - Tong Mao
AU - Lei Shi
AU - Hanfei Zhou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.14457
UR - https://arxiv.org/abs/2610.14457
AB - We establish inverse inequalities in Gaussian Sobolev norms for shallow ReLU^k trial spaces. The estimates identify the sharp regularity threshold and quantify the dependence on parameter separation and bias magnitude. Based on these estimates, we prove convergence of weighted least squares discretizations of linear PDEs on ℝ^d using shallow networks with prescribed inner parameters and finitely many residual samples in an expanding ball. Under suitable stability and regularity assumptions, we obtain an error bound of order h^k+(d+1)/2−rδ_±^−(S−r) in the Gaussian Sobolev norm of order S, up to logarithmic factors, where r is the PDE order, h is the parameter fill distance, and δ_± is the antipodal separation. The convergence results cover deterministic and random sampling and extend to fixed bounded regions by restriction. We construct parameter sets with δ_±asymp h, yielding a convergence rate in terms of the network width that matches a lower bound up to logarithmic factors. Numerical experiments support the theoretical results and provide practical guidance on balancing accuracy and stability.
ER -