Bertoluzza, Silvia;
Burman, Erik;
He, Cuiyu;
(2024)
WAN Discretization of PDEs: Best Approximation, Stabilization, and Essential Boundary Conditions.
SIAM Journal on Scientific Computing
, 46
(6)
C688-C715.
10.1137/23m1588196.
Preview |
Text (Published version)
23m1588196.pdf - Published Version Download (3MB) | Preview |
Preview |
Text (Author accepted manuscript)
Burman_SISC_M158819.pdf Download (2MB) | Preview |
Abstract
In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method’s effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. We also devise a pseudotime XNODE neural network for static PDE problems, yielding significantly faster convergence results than the classical deep neural networks.
| Type: | Article |
|---|---|
| Title: | WAN Discretization of PDEs: Best Approximation, Stabilization, and Essential Boundary Conditions |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/23m1588196 |
| Publisher version: | https://doi.org/10.1137/23m1588196 |
| Language: | English |
| Additional information: | The second author was partially funded by EPSRC grants EP/W007460/1, EP/V050400/1, and EP/T033126/1; for the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising. - The published version of record is made available in line with the publisher's green open access policy; for information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | weak adversarial network, pseudotime XNODE, Cea’s lemma |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10203010 |
Archive Staff Only
 |
View Item |
