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.

翻译：物理信息神经网络（PINNs）已被证明是解决偏微分方程（PDEs）正问题和反问题的有效方法。同时，神经算子方法（包括深度算子网络（DeepONet）和傅里叶神经算子（FNO）等方法）已被引入并广泛应用于逼近PDEs的解。然而，当处理包含尖锐解的问题时，这两种方法均面临重大挑战。为解决这一问题，本文提出了一种名为算子学习增强物理信息神经网络（OL-PINN）的新框架。首先，利用DeepONet学习与具有尖锐解的PDEs相关的一组光滑问题的解算子。随后，将预训练的DeepONet与PINN相结合，以求解目标尖锐解问题。通过成功解决多种问题（例如非线性扩散-反应方程、Burgers方程以及高雷诺数不可压缩Navier-Stokes方程），我们展示了OL-PINN的有效性。与原始PINN相比，所提方法仅需少量残差点即可实现强泛化能力。此外，该方法显著提高了精度，同时确保了稳健的训练过程。进一步地，OL-PINN继承了PINN在求解反问题方面的优势。为此，我们将OL-PINN方法应用于仅具有部分边界条件的问题（这类问题通常无法用经典数值方法求解），展示了其解决不适定问题以及更复杂反问题的能力。