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

TL;DR

This paper introduces a constrained optimization framework using augmented Lagrangian methods for physics-informed neural networks, significantly improving accuracy in solving forward and inverse PDE problems with multi-fidelity data.

Contribution

It proposes a novel equality-constrained neural network approach that overcomes limitations of traditional PINNs, enhancing accuracy and versatility in PDE solutions.

Findings

Achieves orders of magnitude improvement in accuracy over existing PINNs.

Effective for both forward and inverse PDE problems.

Handles multi-fidelity data fusion seamlessly.

Abstract

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.