PreprintarXiv licencearXiv
Singular parameters and missing limits in neural PDE solvers
Abstract
Neural solvers for partial differential equations (PDEs) can approach an accurate solution while their parameters grow without bound. In such cases, the limiting solution may have no finite representation in the chosen model, leaving the best loss unattained. Our analysis connects missing limits in deep neural tanh- networks to unbounded hidden parameters or increasingly redundant neurons. For a class of models built from translated kernels, we describe the missing functions and recover them by adding kernel derivatives to the model. This completion makes the best approximation attainable under standard assumptions. Numerical studies follow the associated parameter growth and explore how completion affects PDE optimization.
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BibTeX
@misc{fernandez2026singular,
title = {{Singular parameters and missing limits in neural PDE solvers}},
author = {Daniel Fernández},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06770},
eprint = {2610.06770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06770},
}
RIS
TY - UNPB
TI - Singular parameters and missing limits in neural PDE solvers
AU - Daniel Fernández
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06770
UR - https://arxiv.org/abs/2610.06770
AB - Neural solvers for partial differential equations (PDEs) can approach an accurate solution while their parameters grow without bound. In such cases, the limiting solution may have no finite representation in the chosen model, leaving the best loss unattained. Our analysis connects missing limits in deep neural tanh- networks to unbounded hidden parameters or increasingly redundant neurons. For a class of models built from translated kernels, we describe the missing functions and recover them by adding kernel derivatives to the model. This completion makes the best approximation attainable under standard assumptions. Numerical studies follow the associated parameter growth and explore how completion affects PDE optimization.
ER -