The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach

TL;DR

This paper introduces an ADMM-PINNs framework that combines ADMM and physics-informed neural networks to efficiently solve nonsmooth PDE-constrained optimization problems, expanding PINNs applicability to more complex scenarios.

Contribution

The paper presents a novel algorithmic framework that unites ADMM with PINNs, enabling the handling of nonsmooth PDE-constrained optimization problems with improved efficiency and scalability.

Findings

Enlarges PINNs applicability to nonsmooth PDE problems

Efficiently solves subproblems with closed-form or standard algorithms

Validated on inverse, control, and source identification problems

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.