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Physics and equality constrained artificial neural networks (PECANN) aregrounded in methods of constrained optimization to properly constrain thesolution of partial differential equations (PDEs) with their boundary andinitial conditions and any high-fidelity data that may be available. To thisend, adoption of the augmented Lagrangian method within the PECANN framework isparamount for learning the solution of PDEs without manually balancing theindividual loss terms in the objective function used for determining theparameters of the neural network. Generally speaking, ALM combines the meritsof the penalty and Lagrange multiplier methods while avoiding the illconditioning and convergence issues associated singly with these methods . Inthe present work, we apply our PECANN framework to solve forward and inverseproblems that have an expanded and diverse set of constraints. We show that ALMwith its conventional formulation to update its penalty parameter and Lagrangemultipliers stalls for such challenging problems. To address this issue, wepropose an adaptive ALM in which each constraint is assigned a unique penaltyparameter that evolve adaptively according to a rule inspired by the adaptivesubgradient method. Additionally, we revise our PECANN formulation for improvedcomputational efficiency and savings which allows for mini-batch training. Wedemonstrate the efficacy of our proposed approach by solving several forwardand PDE-constrained inverse problems with noisy data, including simulation ofincompressible fluid flows with a primitive-variables formulation of theNavier-Stokes equations up to a Reynolds number of 1000.

An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks