Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion
This work addresses accuracy limitations in PINNs for solving PDEs, which is important for researchers in scientific computing and machine learning, though it is incremental as it builds on existing PINN methods.
The authors identified limitations in physics-informed neural networks (PINNs) due to their soft penalty formulation and proposed a constrained optimization framework using the augmented Lagrangian method to enforce PDE constraints and incorporate multi-fidelity data. Their approach achieved orders of magnitude improvements in accuracy compared to state-of-the-art PINNs for forward and inverse problems.
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.