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过程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，数值解法被广泛采用。然而传统数值算法计算成本高昂，且在处理多物理场系统时面临挑战。近年来，利用神经网络求解偏微分方程取得了重要进展，这类方法被称为物理信息神经网络（PINNs）。PINNs将物理定律编码进神经网络，学习偏微分方程的连续解。针对PINNs的训练，现有方法因需计算偏微分方程残差的自动微分而存在效率低下和收敛不稳定问题。本文提出基于动态网格的重要性采样方法（DMIS）来解决上述问题。DMIS是一种基于重要性采样的新型采样方案，通过构建动态三角网格高效估计样本权重。该方法具有广泛适用性，可便捷嵌入现有方法。在三个广泛使用基准上的评估表明，DMIS能同时提升收敛速度与精度。尤其在求解高度非线性的薛定谔方程时，相较于当前最优方法，DMIS均方根误差降低达46%，收敛速度提升五倍。相关代码已开源至https://github.com/MatrixBrain/DMIS。