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We study the combination of the alternating direction method of multipliers(ADMM) with physics-informed neural networks (PINNs) for a general class ofnonsmooth partial differential equation (PDE)-constrained optimizationproblems, where additional regularization can be employed for constraints onthe control or design variables. The resulting ADMM-PINNs algorithmic frameworksubstantially enlarges the applicable range of PINNs to nonsmooth cases ofPDE-constrained optimization problems. The application of the ADMM makes itpossible to untie the PDE constraints and the nonsmooth regularization termsfor iterations. Accordingly, at each iteration, one of the resultingsubproblems is a smooth PDE-constrained optimization which can be efficientlysolved by PINNs, and the other is a simple nonsmooth optimization problem whichusually has a closed-form solution or can be efficiently solved by variousstandard optimization algorithms or pre-trained neural networks. The ADMM-PINNsalgorithmic framework does not require to solve PDEs repeatedly, and it ismesh-free, easy to implement, and scalable to different PDE settings. Wevalidate the efficiency of the ADMM-PINNs algorithmic framework by differentprototype applications, including inverse potential problems, sourceidentification in elliptic equations, control constrained optimal control ofthe Burgers equation, and sparse optimal control of parabolic equations.

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