The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
Yongcun Song, Xiaoming Yuan, Hangrui Yue
TL;DR
The paper introduces the ADMM-PINNs framework to solve nonsmooth PDE-constrained optimization problems by decoupling smooth PDE constraints from nonsmooth regularizers via ADMM. It provides two PINN-based strategies for the smooth subproblem: an approximate-then-optimize (AtO) approach and an optimize-then-approximate (OtA) approach, along with proximal updates for the nonsmooth term. The framework is demonstrated on four prototype problems—inverse potential, Burgers equation control, discontinuous source identification, and sparse parabolic control—showing mesh-free, scalable performance with competitive accuracy against traditional FEM-based solvers. These results highlight the method’s flexibility in handling a range of nonsmooth terms (TV, L1, bounds) and PDE types, with practical implications for efficient PDE-constrained optimization in science and engineering.
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.