Comparison of neural and spline representations for physics-informed learning
Abstract
Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.
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BibTeX
@misc{duvigneau2026comparison,
title = {{Comparison of neural and spline representations for physics-informed learning}},
author = {Régis Duvigneau},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06294},
eprint = {2610.06294},
archivePrefix = {arXiv},
url = {https://inria.hal.science/hal-05774225v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Comparison of neural and spline representations for physics-informed learning
AU - Régis Duvigneau
PY - 2026
DA - 2026/10/01
PB - HAL
DO - 10.48550/arxiv.2610.06294
UR - https://inria.hal.science/hal-05774225v1
AB - Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.
ER -