The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: a deep learning approach
Song Y, Yuan X, Yue H (2024)
Publication Language: English
Publication Status: In review
Publication Type: Unpublished / Preprint
Future Publication Type: Journal article
Publication year: 2024
URI: https://arxiv.org/abs/2302.08309
Open Access Link: https://arxiv.org/abs/2302.08309
Abstract
We study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) for a general class of nonsmooth partial differential equation (PDE)-constrained optimization problems, where additional regularization can be employed for constraints on the control or design variables. The resulting ADMM-PINNs algorithmic framework substantially enlarges the applicable range of PINNs to nonsmooth cases of PDE-constrained optimization problems. The application of the ADMM makes it possible to untie the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization which can be efficiently solved by PINNs, and the other is a simple nonsmooth optimization problem which usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs algorithmic framework does not require to solve PDEs repeatedly, and it is mesh-free, easy to implement, and scalable to different PDE settings. We validate the efficiency of the ADMM-PINNs algorithmic framework by different prototype applications, including inverse potential problems, source identification in elliptic equations, control constrained optimal control of the Burgers equation, and sparse optimal control of parabolic equations.
Authors with CRIS profile
Yongcun Song
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
Hong Kong (HK)
China (CN)How to cite
APA:
Song, Y., Yuan, X., & Yue, H. (2024). The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: a deep learning approach. (Unpublished, In review).
MLA:
Song, Yongcun, Xiaoming Yuan, and Hangrui Yue. The ADMM-PINNs algorithmic framework for nonsmooth PDE-constrained optimization: a deep learning approach. Unpublished, In review. 2024.
BibTeX: Download