TY - JOUR
T1 - A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods
AU - Makridakis , Charalambos G.
AU - Pim , Aaron
AU - Pryer , Tristan
JO - Journal of Machine Learning
VL - 3
SP - 166
EP - 191
PY - 2025
DA - 2025/09
SN - 4
DO - http://doi.org/10.4208/jml.250107
UR - https://global-sci.org/intro/article_detail/jml/24379.html
KW - Physics-informed neural networks, Deep Ritz method, Uzawa algorithm, Lagrange multipliers, Boundary condition enforcement.
AB - <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>