Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions

TL;DR

This paper introduces OL-PINN, a novel framework combining neural operator learning with physics-informed neural networks to effectively solve PDEs with sharp solutions, improving accuracy and robustness over traditional PINNs.

Contribution

The paper proposes OL-PINN, integrating DeepONet with PINN to better handle sharp solutions and inverse problems, demonstrating significant improvements in accuracy and training stability.

Findings

OL-PINN outperforms vanilla PINN in accuracy and efficiency.

Effective in solving high Reynolds number Navier-Stokes equations.

Capable of addressing ill-posed inverse problems with partial boundary data.

Abstract

Physics-informed Neural Networks (PINNs) have been shown as a promising approach for solving both forward and inverse problems of partial differential equations (PDEs). Meanwhile, the neural operator approach, including methods such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has been introduced and extensively employed in approximating solution of PDEs. Nevertheless, to solve problems consisting of sharp solutions poses a significant challenge when employing these two approaches. To address this issue, we propose in this work a novel framework termed Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize DeepONet to learn the solution operator for a set of smooth problems relevant to the PDEs characterized by sharp solutions. Subsequently, we integrate the pre-trained DeepONet with PINN to resolve the target sharp solution problem. We showcase the efficacy of OL-PINN by successfully addressing various problems, such as the nonlinear diffusion-reaction equation, the Burgers equation and the incompressible Navier-Stokes equation at high Reynolds number. Compared with the vanilla PINN, the proposed method requires only a small number of residual points to achieve a strong generalization capability. Moreover, it substantially enhances accuracy, while also ensuring a robust training process. Furthermore, OL-PINN inherits the advantage of PINN for solving inverse problems. To this end, we apply the OL-PINN approach for solving problems with only partial boundary conditions, which usually cannot be solved by the classical numerical methods, showing its capacity in solving ill-posed problems and consequently more complex inverse problems.