TY - JOUR
T1 - Multilevel domain decomposition-based architectures for physics-informed neural networks
AU - Dolean, Victorita
AU - Heinlein, Alexander
AU - Mishra, Siddhartha
AU - Moseley, Ben
N1 - Publisher Copyright:
© 2024 The Author(s)
PY - 2024/9/1
Y1 - 2024/9/1
N2 - Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.
AB - Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.
KW - Differential equations
KW - Forward modeling
KW - Multi-scale modeling
KW - Multilevel methods
KW - Overlapping domain decomposition methods
KW - Physics-informed neural networks
KW - Spectral bias
UR - http://www.scopus.com/inward/record.url?scp=85196360869&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2024.117116
DO - 10.1016/j.cma.2024.117116
M3 - Article
AN - SCOPUS:85196360869
SN - 0045-7825
VL - 429
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
M1 - 117116
ER -