Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of efficient tools for dealing with the interfaces between two adjacent sub-domains heavily hinders the training effects, even leads to the discontinuity of the learned solutions. In this paper, we propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. Specifically, for the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly and are used to divide the whole training domain into a finite number of non-overlapping sub-domains, then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs. For the inverse problem, we first utilize the symmetry group acting on the data of the initial and boundary conditions to generate labeled data in the interior domain of PDEs and then find the undetermined parameters as well as the solution by only training the neural networks in a sub-domain. Consequently, the proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains. Numerical results of the Korteweg-de Vries equation with a translation symmetry and the nonlinear viscous fluid equation with a scaling symmetry show that the accuracies of the learned solutions are improved largely.

翻译：区域分解为解决物理信息神经网络（PINN）在准确高效求解整个域中偏微分方程（PDEs）时的困境提供了有效途径，但缺乏处理两个相邻子区域间界面的高效工具严重阻碍了训练效果，甚至导致学习解的不连续性。本文提出一种基于对称群的区域分解策略来增强PINN，用于求解具有李对称群的PDEs的正问题和反问题。具体而言，对于正问题，首先利用对称群生成具有已知解信息且可灵活调整的分割线，将整个训练域划分为有限个不重叠的子域，然后采用PINN和对称增强PINN方法学习每个子域中的解，最后将其拼接为PDEs的整体解。对于反问题，首先利用对称群作用于初边值条件数据，在PDEs内部域中生成带标签数据，然后仅通过在子域中训练神经网络来确定待定参数及解。因此，所提方法能够预测出在完整域中普通PINN以及相同子域中扩展物理信息神经网络均无法成功求解的高精度PDEs解。具有平移对称性的Korteweg-de Vries方程和具有缩放对称性的非线性粘性流体方程的数值结果表明，学习解的精度得到了大幅提升。