In this study, we propose a novel phase field smoothing-physics informed neural network (PFS-PINN) approach to efficiently solve partial differential equations (PDEs) with discontinuous coefficients. This method combines the phase field model and the PINN model to overcome the difficulty of low regularity solutions and eliminate the limitations of interface constraints in existing neural network solvers for PDEs with discontinuous coefficients. The proposed PFS-PINN approach includes two key parts. In the first part, the low regularity solutions to PDEs with discontinuous coefficients create challenges for solving the problem by neural networks. A neural network solver for phase field models is presented for constructing approximate PDEs with smooth coefficients, and the associated convergence results are proven. In the second part, the mixed PINN model is introduced to solve the approximate PDEs. Both the solution and the first-order derivatives of the solution are used as the outputs of the neural network, and the time-consuming second-order derivatives in loss functions are avoided effectively. In addition, in order to improve the performance of the PFS-PINN method, the adaptive sampling strategies are also investigated. We carry out some numerical experiments, including the elliptic equations defined over various complex microstructures. The numerical examples demonstrate that the PFS-PINN approach is accurate and effective in solving the elliptic equation with discontinuous coefficients. Besides, the proposed PFS-PINN approach has the potential to solve other PDEs with discontinuous coefficients in science and engineering.

中文翻译：

---

相场平滑-PINN：具有不连续系数的偏微分方程的神经网络求解器


在这项研究中，我们提出了一种新颖的相场平滑物理通知神经网络（PFS-PINN）方法来有效求解具有不连续系数的偏微分方程（PDE）。该方法将相场模型和PINN模型相结合，克服了低正则性求解的困难，消除了现有神经网络求解器对不连续系数偏微分方程的界面约束的限制。所提出的 PFS-PINN 方法包括两个关键部分。在第一部分中，具有不连续系数的偏微分方程的低正则性解给神经网络解决问题带来了挑战。提出了一种用于相场模型的神经网络求解器，用于构造具有平滑系数的近似偏微分方程，并证明了相关的收敛结果。在第二部分中，引入混合PINN模型来求解近似偏微分方程。将解及其一阶导数都作为神经网络的输出，有效避免了损失函数中耗时的二阶导数。此外，为了提高PFS-PINN方法的性能，还研究了自适应采样策略。我们进行了一些数值实验，包括在各种复杂微观结构上定义的椭圆方程。数值算例表明PFS-PINN方法在求解具有不连续系数的椭圆方程时是准确有效的。此外，所提出的 PFS-PINN 方法有潜力解决科学和工程中具有不连续系数的其他偏微分方程。