On the Assessment of Domain Decomposition Approaches in Extended Physics-Informed Neural Networks for the Shock-Dominated Burgers Equation
Abstract
Physics-Informed Neural Networks (PINNs) provide a mesh-free framework for solving partial differential equations but often struggle with sharp gradients and shock-dominated solutions. Extended Physics-Informed Neural Networks (XPINNs) address this challenge through domain decomposition; however, the effects of decomposition design, overlap, and local network capacity on solution accuracy remain insufficiently characterized. This work presents a systematic assessment of domain decomposition configurations for XPINNs applied to the one-dimensional viscous Burgers equation (ν=0.01/π). We compare a standard PINN, a conventional XPINN, a shock-focused XPINN with increased network capacity in the shock-containing subdomain, and overlapping XPINNs with different overlap widths. All configurations are trained using a common protocol based on Adam, L-BFGS refinement, and ten Residual-based Adaptive Refinement (RAR) cycles. Performance is evaluated against a reference solution using global and shock-region relative L2 and L∞ errors, maximum PDE residuals, and interface continuity errors. The results show that decomposition design strongly affects XPINN accuracy and shock resolution. Among the overlapping configurations, the smallest overlap provides the best accuracy, while larger overlaps degrade performance. The shock-focused XPINN achieves the lowest overall solution errors, indicating that increased local network capacity can provide greater benefit than increasing overlap alone. RAR produces the largest improvements during early refinement stages, followed by increasingly non-monotonic changes in accuracy. These findings highlight the importance of jointly considering decomposition, interface coupling, network capacity, and adaptive refinement for shock-dominated problems.
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BibTeX
@misc{bolanosacosta2026assessment,
title = {{On the Assessment of Domain Decomposition Approaches in Extended Physics-Informed Neural Networks for the Shock-Dominated Burgers Equation}},
author = {Andrés F. Bolaños-Acosta and Julián C. Restrepo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20621},
eprint = {2610.20621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20621},
}
RIS
TY - UNPB
TI - On the Assessment of Domain Decomposition Approaches in Extended Physics-Informed Neural Networks for the Shock-Dominated Burgers Equation
AU - Andrés F. Bolaños-Acosta
AU - Julián C. Restrepo
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.20621
UR - https://arxiv.org/abs/2610.20621
AB - Physics-Informed Neural Networks (PINNs) provide a mesh-free framework for solving partial differential equations but often struggle with sharp gradients and shock-dominated solutions. Extended Physics-Informed Neural Networks (XPINNs) address this challenge through domain decomposition; however, the effects of decomposition design, overlap, and local network capacity on solution accuracy remain insufficiently characterized. This work presents a systematic assessment of domain decomposition configurations for XPINNs applied to the one-dimensional viscous Burgers equation (ν=0.01/π). We compare a standard PINN, a conventional XPINN, a shock-focused XPINN with increased network capacity in the shock-containing subdomain, and overlapping XPINNs with different overlap widths. All configurations are trained using a common protocol based on Adam, L-BFGS refinement, and ten Residual-based Adaptive Refinement (RAR) cycles. Performance is evaluated against a reference solution using global and shock-region relative L_2 and L_∞ errors, maximum PDE residuals, and interface continuity errors. The results show that decomposition design strongly affects XPINN accuracy and shock resolution. Among the overlapping configurations, the smallest overlap provides the best accuracy, while larger overlaps degrade performance. The shock-focused XPINN achieves the lowest overall solution errors, indicating that increased local network capacity can provide greater benefit than increasing overlap alone. RAR produces the largest improvements during early refinement stages, followed by increasingly non-monotonic changes in accuracy. These findings highlight the importance of jointly considering decomposition, interface coupling, network capacity, and adaptive refinement for shock-dominated problems.
ER -