  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.
