Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

Understand

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.