Approximation of solutions of parameter-dependent problems by residual neural networks
Abstract
We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.
Versions
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BibTeX
@misc{carpio2026approximation,
title = {{Approximation of solutions of parameter-dependent problems by residual neural networks}},
author = {Ana Carpio},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.13574},
eprint = {2607.13574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13574},
}
RIS
TY - UNPB
TI - Approximation of solutions of parameter-dependent problems by residual neural networks
AU - Ana Carpio
PY - 2026
DA - 2026/07/15
PB - arXiv
DO - 10.48550/arxiv.2607.13574
UR - https://arxiv.org/abs/2607.13574
AB - We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.
ER -