@Article{JML-4-166,
author = {Makridakis , Charalambos G.Pim , Aaron and Pryer , Tristan},
title = {A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods},
journal = {Journal of Machine Learning},
year = {2025},
volume = {4},
number = {3},
pages = {166--191},
abstract = {<p style="text-align: justify;">We introduce a deep learning-based framework for weakly enforcing boundary conditions in the
numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational
adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary
conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis
demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm
through numerical experiments, including high dimensional, singularly perturbed problems and those posed
over non-convex domains.</p>},
issn = {2790-2048},
doi = {https://doi.org/10.4208/jml.250107},
url = {http://global-sci.org/intro/article_detail/jml/24379.html}
}