Challenges in Automatic Differentiation and Numerical Integration in Physics-Informed Neural Networks Modelling

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F25%3A43976348" target="_blank" >RIV/49777513:23520/25:43976348 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1002/nme.70148" target="_blank" >https://doi.org/10.1002/nme.70148</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/nme.70148" target="_blank" >10.1002/nme.70148</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Challenges in Automatic Differentiation and Numerical Integration in Physics-Informed Neural Networks Modelling

Popis výsledku v původním jazyce

In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is often critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary, but using higher precision arithmetic with neural networks does not have to bring an improvement at all. Three problematic use-cases for solving differential equations using PINNs are analyzed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.

Název v anglickém jazyce

Challenges in Automatic Differentiation and Numerical Integration in Physics-Informed Neural Networks Modelling

Popis výsledku anglicky

In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is often critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary, but using higher precision arithmetic with neural networks does not have to bring an improvement at all. Three problematic use-cases for solving differential equations using PINNs are analyzed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

ISSN

0029-5981

e-ISSN

1097-0207

Svazek periodika

126

Číslo periodika v rámci svazku

20

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

28

Strana od-do

"e70148"

Kód UT WoS článku

001607184200002

EID výsledku v databázi Scopus

2-s2.0-105019396045