Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schr\"odinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed. Code are available at https://github.com/MatrixBrain/DMIS.

翻译：以偏微分方程形式建模动力学是理解真实世界物理过程的有效途径。对于复杂物理系统，解析解难以获取，数值解被广泛采用。然而，传统数值算法计算成本高昂，且难以处理多物理场系统。近年来，利用神经网络求解偏微分方程取得了显著进展，即物理信息神经网络（PINNs）。PINNs将物理定律编码到神经网络中，学习偏微分方程的连续解。在PINNs训练过程中，由于需要计算偏微分方程残差的自动微分，现有方法存在效率低下和收敛不稳定的问题。本文提出基于动态网格的重要采样方法（DMIS）以解决上述问题。DMIS是一种基于重要采样的新型采样方案，通过构建动态三角网格高效估算样本权重。该方法具有广泛适用性，可轻松集成到现有方法中。在三个广泛使用的基准测试上的评估表明，DMIS同时提升了收敛速度与求解精度。特别是在求解高度非线性的薛定谔方程时，与最先进方法相比，DMIS的均方根误差降低高达46%，收敛速度提升五倍。代码已开源至https://github.com/MatrixBrain/DMIS。