Operator learning augmented physics-informed neural networks for partial differential equations exhibiting sharp...

Area of Science:

• Computational fluid dynamics
• Scientific machine learning
• Numerical analysis

Background:

• Physics-informed neural networks (PINNs) are effective for solving differential equations but struggle with sharp solution features.
• Deep operator networks (DeepONets) can extrapolate but may lack stability.
• Operator learning offers a path to enhance PINN capabilities.

Purpose of the Study:

• To develop a novel operator learning augmented physics-informed neural network (OL-PINN) framework.
• To address challenges in solving partial differential equations with sharp solution features and singular behaviors.
• To improve stability, efficiency, and accuracy compared to existing methods.

Main Methods:

• Integration of a pretrained DeepONet as an operator prior into the standard PINN architecture.
• Utilizing the DeepONet prior as a regularization mechanism for enhanced stability and extrapolation.
• Systematic validation on benchmark problems: nonlinear diffusion-reaction, Burgers, lid-driven cavity flow, and 2D Navier-Stokes equations.

Main Results:

• OL-PINN demonstrates superior predictive accuracy and stable convergence over vanilla PINNs and DeepONet extrapolations.
• The method requires fewer residual points for training, enhancing efficiency.
• OL-PINN effectively handles ill-posed problems with incomplete boundary conditions, outperforming classical numerical methods.
• Successful prediction of emergent small-scale flow structures, like vortices at higher Reynolds numbers.

Conclusions:

• OL-PINN provides a robust and efficient framework for solving complex partial differential equations, especially those with challenging solution features.
• The operator learning augmentation significantly enhances the performance and applicability of PINNs.
• OL-PINN shows great promise for simulating turbulent and near-turbulent flows, including predicting novel flow structures.