One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A  long list of numerical methods have been proposed and  successfully used for various applications. In recent years, deep learning methods have shown their superiority for high-dimensional  PDEs  where  traditional methods fail.  However, for low dimensional problems,  it  remains unclear whether these methods have a real advantage over traditional algorithms as a direct solver.  In this work, we propose  the  random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms.  RFM is  based on a combination of well-known ideas:  1. representation of the approximate solution using random feature functions;   2. collocation method  to take care of the PDE; 3. penalty method to treat the boundary conditions, which allows us to treat the boundary condition  and  the PDE in the same footing. We find it crucial to add several additional components including multi-scale representation and adaptive  weight rescaling  in the loss  function. We demonstrate that the method exhibits spectral accuracy  and  can compete with  traditional  solvers in terms of both accuracy and efficiency. In addition, we find that RFM is particularly suited  for  problems with complex geometry,  where both traditional and machine learning-based algorithms encounter difficulties.

陈景润，中国科学技术大学教授。主要研究方向为材料性质的多尺度建模、分析、算法与仿真，机器学习与偏微分方程。主要工作发表在J. Comput. Phys.，Math. Comp.，SIAM系列期刊等期刊上。