  Physics-informed neural networks (PINNs) have been proposed to learn the
solution of partial differential equations (PDE). In PINNs, the residual form
of the PDE of interest and its boundary conditions are lumped into a composite
objective function as soft penalties. Here, we show that this specific way of
formulating the objective function is the source of severe limitations in the
PINN approach when applied to different kinds of PDEs. To address these
limitations, we propose a versatile framework based on a constrained
optimization problem formulation, where we use the augmented Lagrangian method
(ALM) to constrain the solution of a PDE with its boundary conditions and any
high-fidelity data that may be available. Our approach is adept at forward and
inverse problems with multi-fidelity data fusion. We demonstrate the efficacy
and versatility of our physics- and equality-constrained deep-learning
framework by applying it to several forward and inverse problems involving
multi-dimensional PDEs. Our framework achieves orders of magnitude improvements
in accuracy levels in comparison with state-of-the-art physics-informed neural
networks.
