Operator Learning Enhanced Physics-informed Neural Networks for Solving
  Partial Differential Equations Characterized by Sharp Solutions
Bin Lin Zhiping Mao Zhicheng Wang George Em Karniadakis
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.